Publications of the Department of Mathematics

Abstract

We present a novel model for fluid-driven fracture propagation in poro-elastic media. Our approach combines ideas from dimensionally reduced discrete fracture models with diffuse phase-field models. The main advantage of this combined approach is that the fracture geometry is always represented explicitly, while the propagation remains geometrically flexible. We prove that our model is thermodynamically consistent. In order to solve our model numerically, we propose a mixed-dimensional discontinuous Galerkin scheme with a computational grid fully conforming to the fractures. As the fracture propagates, the diffuse phase-field acts as indicator to identify new fracture facets to be added to the discrete fracture network. Numerical experiments demonstrate that our approach reproduces classical scenarios for fracturing porous media

Abstract

We study single-phase flow in a fractured porous medium at a macroscopic scale that allows to model fractures individually. The flow is governed by Darcy's law in both fractures and porous matrix. We derive a new mixed-dimensional model, where fractures are represented by $(n-1)$-dimensional interfaces between $n$-dimensional subdomains for $n2$. In particular, we suggest a generalization of the model in~22 by accounting for asymmetric fractures with spatially varying aperture. Thus, the new model is particularly convenient for the description of surface roughness or for modeling curvilinear or winding fractures. The wellposedness of the new model is proven under appropriate conditions. Further, we formulate a discontinuous Galerkin discretization of the new model and validate the model by performing two- and three-dimensional numerical experiments.

Abstract

We study the gradient-flow structure of a non-Newtonian thin film equation with power-law rheology. The equation is quasilinear, of fourth order and doubly-degenerate parabolic. By adding a singular potential to the natural Dirichlet energy, we introduce a modified version of the thin-film equation. Then, we set up a minimising-movement scheme that converges to global positive weak solutions to the modified problem. These solutions satisfy an energy-dissipation equality and follow a gradient flow. In the limit of a vanishing singularity of the potential, we obtain global non-negative weak solutions to the power-law thin-film equation ∂tu+∂x(m(u)|∂3xu−G′′(u)∂xu|α−1(∂3xu−G′′(u)∂xu))=0 with potential G in the shear-thinning (α>1), Newtonian (α=1) and shear-thickening case (0<α<1). The latter satisfy an energy-dissipation inequality. Finally, we derive dissipation bounds in the case G≡0 which imply that solutions emerging from initial values with low energy lift up uniformly in finite time.

Abstract

Building on the well-known total variation, this paper develops a general regularization technique based on nonlinear isotropic diffusion (NID) for inverse problems with piecewise smooth solutions. The novelty of our approach is to be adaptive (we speak of A-NID), i.e., the regularization varies during the iterates in order to incorporate prior information on the edges, deal with the evolution of the reconstruction and circumvent the limitations due to the nonconvexity of the proposed functionals. After a detailed analysis of the convergence and well-posedness of the method, the latter is validated by simulations performed on synthetic and real data on computerized tomography.

Abstract

Abstract The viscous flow of two immiscible fluids in a porous medium on the Darcy scale is governed by a system of nonlinear parabolic equations. If infinite mobility of one phase can be assumed (e.g., in soil layers in contact with the atmosphere) the system can be substituted by the scalar Richards model. Thus, the porous medium domain may be partitioned into disjoint subdomains where either the full two-phase or the simplified Richards model dynamics are valid. Extending the previously considered one-model situations we suggest coupling conditions for this hybrid model approach. Based on an Euler implicit discretization, a linear iterative (L-type) domain decomposition scheme is proposed, and proved to be convergent. The theoretical findings are verified by a comparative numerical study that in particular confirms the efficiency of the hybrid ansatz as compared to full two-phase model computations.

Abstract

Summary: "We study the flow of two immiscible fluids located on a solid bottom, where the lower fluid is Newtonian and the upper fluid is a non-Newtonian Ellis fluid. Neglecting gravitational effects, we consider the formal asymptotic limit of small film heights in the twop-hase Navier-Stokes system. This leads to a strongly coupled system of two parabolic equations of fourth order with merely Hölder-continuous dependence on the coefficients. For the case of strictly positive initial film heights we prove local existence of strong solutions by abstract semigroup theory. Uniqueness is proved by energy methods. Under additional regularity assumptions, we investigate asymptotic stability of the unique equilibrium solution, which is given by constant film heights.''

Abstract

Python Code to generate the meshes and FEM solutions to Section 5 of the paper Uncertainty visualization: Fundamentals and recent developments.Commentaries are in the code to explain it.Paraview is used for the visualization.

Abstract

In this paper, we consider a non-linear fourth-order evolution equation of Cahn–Hilliard-type on evolving surfaces with prescribed velocity, where the non-linear terms are only assumed to have locally Lipschitz derivatives. High-order evolving surface finite elements are used to discretise the weak equation system in space, and a modified matrix–vector formulation for the semi-discrete problem is derived. The anti-symmetric structure of the equation system is preserved by the spatial discretisation. A new stability proof, based on this structure, combined with consistency bounds proves optimal-order and uniform-in-time error estimates. The paper is concluded by a variety of numerical experiments.

Abstract

Symplectic model order reduction is a structure-preserving reduction technique for Hamiltonian systems. Apart from theoretical results like the preservation of stability, it has been demonstrated to give improved numerical results compared to classical MOR techniques. A key element in this procedure is the choice of a good symplectic reduced order basis (ROB). In our work, we introduce so-called canonizable Hamiltonian systems in energy coordinates. For such systems with the assumption of a periodic solution, we derive a globally optimal symplectic ROB in the sense of the proper symplectic decomposition (PSD). To this end, we show that the proper orthogonal decomposition (POD) of a canonizable Hamiltonian system is automatically symplectic from which we deduce optimality of the PSD. To verify our findings numerically, we consider a reproduction experiment for the linear wave equation.

Abstract

Multiphase flow in fractured porous media can be described by discrete fracture matrix models that represent the fractures as dimensionally reduced manifolds embedded in the bulk porous medium. Generalizing earlier work on this approach we focus on immiscible two-phase flow in time-dependent fracture geometries, i.e., the fracture itself and the aperture of the fractures might evolve in time. For dynamic fracture geometries of that kind, neglecting capillary forces, we deduce by transversal averaging of a full dimensional description a dimensionally reduced model that governs the geometric evolution and the flow dynamics. The core computational contribution is a mixed-dimensional finite-volume discretization based on a conforming moving-mesh ansatz. This finite-volume moving-mesh (FVMM) algorithm is tracking the fractures' motions as a family of unions of facets of the mesh. Notably, the method permits arbitrary movement of facets of the triangulation while keeping the mass conservation constraint. In a series of numerical examples we investigate the modeling error of the reduced model as it compares to the original full dimensional model. Moreover, we show the performance of the finite-volume moving-mesh algorithm for the complex wave pattern that is induced by the interaction of saturation fronts and evolving fractures.

Abstract

The analysis of eigenvalues of Laplace and Schrödinger operators is an important and classical topic in mathematical physics with many applications. This book presents a thorough introduction to the area, suitable for masters and graduate students, and includes an ample amount of background material on the spectral theory of linear operators in Hilbert spaces and on Sobolev space theory. Of particular interest is a family of inequalities by Lieb and Thirring on eigenvalues of Schrödinger operators, which they used in their proof of stability of matter. The final part of this book is devoted to the active research on sharp constants in these inequalities and contains state-of-the-art results, serving as a reference for experts and as a starting point for further research.

Abstract

We present an integrated approach for the use of simulated data from full order discretization as well as projection-based Reduced Basis reduced order models for the training of machine learning approaches, in particular Kernel Methods, in order to achieve fast, reliable predictive models for the chemical conversion rate in reactive flows with varying transport regimes.

Abstract

This paper considers the problem of designing accelerated gradient-based algorithms for optimization and saddle-point problems. The class of objective functions is defined by a generalized sector condition. This class of functions contains strongly convex functions with Lipschitz gradients but also non-convex functions, which allows not only to address optimization problems but also saddle-point problems. The proposed design procedure relies on a suitable class of Lyapunov functions and on convex semi-definite programming. The proposed synthesis allows the design of algorithms that reach the performance of state-of-the-art accelerated gradient methods and beyond.

Abstract

Quantum systems composed of N distinct particles in \$\$\backslashmathbb \R\^2\$\$with two-body contact interactions of TMS type are shown to arise as limits---in the norm resolvent sense---of Schrödinger operators with suitably rescaled pair potentials.

Abstract

We establish a framework for systematically analyzing and designing output-feedback controllers for linear impulsive and related hybrid systems that might even be affected by various types of uncertainties. In particular, the framework encompasses uncertain switched and sampled-data systems as well as networked systems with switching communication topologies. The framework is based on recently developed convex criteria involving a so-called clock for analyzing impulsive systems under dwell-time constraints. We elaborate on the extension of those criteria for dynamic output-feedback controller synthesis by means of convex optimization and generalize the so-called dual iteration to impulsive systems. The latter originally and still constitutes a promising heuristic procedure for the challenging and non-convex design of static output-feedback controllers for standard linear time-invariant systems. Moreover, for uncertain impulsive systems as modeled in terms of linear fractional representations, we generalize the nominal analysis criteria by providing novel robust analysis conditions based on a novel time-domain and clock-dependent formulation of integral quadratic constraints. Finally, by combining the insights on nominal synthesis and robust analysis, we are able to tackle challenging output-feedback designs of practical relevance, such as the design of gain-scheduled, robust or robust gain-scheduled controllers for impulsive systems. Most of the obtained analysis and synthesis conditions involve infinite-dimensional (differential) linear matrix inequalities which can be numerically solved by using relaxation methods based on, e.g., linear splines, B-splines or matrix sum-of-squares that we discuss as well.

Abstract

We show how to compose robust stability tests for uncertain systems modeled as linear fractional representations and affected by various types of dynamic uncertainties. Our results are formulated in terms of linear matrix inequalities and rest on the recently established notion of finite-horizon integral quadratic constraints with a terminal cost. The construction of such constraints is motivated by an unconventional application of the S-procedure in the frequency domain with dynamic Lagrange multipliers. Our technical contribution reveals how this construction can be performed by dissipativity arguments in the time-domain and in a lossless fashion. This opens the way for generalizations to time-varying or hybrid systems.

Abstract

Starting from a linear fractional representation of a linear system affected by constant parametric uncertainties, we demonstrate how to enhance standard robust analysis tests by taking available (noisy) input-output data of the uncertain system into account. Our approach relies on a lifting of the system and on the construction of data-dependent multipliers. It leads to a test in terms of linear matrix inequalities which guarantees stability and performance for all systems compatible with the observed data if it is in the affirmative. In contrast to many other data-based approaches, prior physical knowledge is included at the outset due to the underlying linear fractional representation.

Abstract

We propose novel quadratic performance tests for linear discrete-time impulsive systems based on viewing these systems as feedback interconnections of some non-impulsive linear system with an impulsive operator. In order to systematically analyze such interconnections, we employ the framework of integral quadratic constraints and propose novel constraints of this kind for capturing the behavior of the involved impulsive operator. As a major benefit, the modularity of this framework permits seamless extensions to interconnections affected by heterogeneous uncertainties in a straightforward fashion. This contrasts with alternative approaches which are based on capturing the system's impulsive behavior by means of a clock. Building upon the developed analysis criteria, we characterize the existence of non-impulsive estimators for such impulsive interconnections in a lossless fashion and in terms of linear matrix inequalities. Finally, our approach is illustrated by means of several numerical examples.

Abstract

Keywords: Transient wave scattering · Elastodynamics · Time-dependent boundary integral equations · Convolution quadrature

Abstract

This paper provides a brief overview of uncertainty visualization along with some fundamental considerations on uncertainty propagation and modeling. Starting from the visualization pipeline, we discuss how the different stages along this pipeline can be affected by uncertainty and how they can deal with this and propagate uncertainty information to subsequent processing steps. We illustrate recent advances in the field with a number of examples from a wide range of applications: uncertainty visualization of hierarchical data, multivariate time series, stochastic partial differential equations, and data from linguistic annotation.

Abstract

We study the long-time behaviour of solutions to quasilinear doubly degenerate parabolic problems of fourth order. The equations model for instance the dynamic behaviour of a non-Newtonian thin-film flow on a flat impermeable bottom and with zero contact angle. We consider a shear-rate dependent fluid the rheology of which is described by a constitutive power-law or Ellis-law for the fluid viscosity. In all three cases, positive constants (i.e. positive flat films) are the only positive steady-state solutions. Moreover, we can give a detailed picture of the long-time behaviour of solutions with respect to the H1(Ω)-norm. In the case of shear-thickening power-law fluids, one observes that solutions which are initially close to a steady state, converge to equilibrium in finite time. In the shear-thinning power-law case, we find that steady states are polynomially stable in the sense that, as time tends to infinity, solutions which are initially close to a steady state, converge to equilibrium at rate 1/t1/β for some β>0. Finally, in the case of an Ellis-fluid, steady states are exponentially stable in H1(Ω).

Abstract

Process metadata constitute a relevant part of the documentation of research processes and the creation and use of research data. As an addition to publications on the research question, they capture details needed for reusability and comparability and are thus important for a sustainable handling of research data. In the DH project SDC4Lit we want to capture process metadata when researchers work with literary material, conducting manual and automatic processing steps, which need to be treated with equal emphasis. We present a content-related mapping between two process metadata schemas from the area of Digital Humanities (GRAIN, RePlay-DH) and one from Computational Engineering (EngMeta) and find that there are no basic obstacles preventing the use of any of them for our purposes. Actually a basic difference rather exists between GRAIN on the one hand and RePlay-DH and EngMeta on the other, regarding the treatment of tools and actors in a workflow step.

Abstract

In this note it is shown that the famous multiplier absolute stability test of R. O'Shea, G. Zames and P. Falb is necessary and sufficient if the set of Lur'e interconnections is lifted to a Kronecker structure and an explicit method to construct the destabilizing static nonlinearity is presented.

Abstract

Summary: "We study the dynamic behaviour of two viscous fluid films confined between two concentric cylinders rotating at a small relative velocity. It is assumed that the fluids are immiscible and that the volume of the outer fluid film is large compared to the volume of the inner one. Moreover, while the outer fluid is considered to have constant viscosity, the rheological behaviour of the inner thin film is determined by a strain-dependent power-law. Starting from a Navier-Stokes system, we formally derive evolution equations for the interface separating the two fluids. Two competing effects drive the dynamics of the interface, namely the surface tension and the shear stresses induced by the rotation of the cylinders. When the two effects are comparable, the solutions behave, for large times, as in the Newtonian regime. We also study the regime in which the surface tension effects dominate the stresses induced by the rotation of the cylinders. In this case, we prove local existence of positive weak solutions both for shear-thinning and shear-thickening fluids. In the latter case, we show that interfaces which are initially close to a circle converge to a circle in finite time and keep that shape for later times.''

Abstract

We study the dynamic behaviour of solutions to a fourth-order quasilinear degenerate parabolic equation for large times arising in fluid dynamical applications. The degeneracy occurs both with respect to the unknown and with respect to the sum of its first and third spatial derivative. The modelling equation appears as a thin-film limit for the interface separating two immiscible viscous fluid films confined between two cylinders rotating at small relative angular velocity. More precisely, the fluid occupying the layer next to the outer cylinder is considered to be Newtonian, i.e. it has constant viscosity, while we assume that the layer next to the inner cylinder is filled by a shear-thinning power-law fluid. Using energy methods, Fourier analysis and suitable regularity estimates for higher-order parabolic equations, we prove global existence of positive weak solutions in the case of low initial energy. Moreover, these global solutions are polynomially stable, in the sense that interfaces which are initially close to a circle, converge at rate 1/t1/β for some β>0 to a circle, as time tends to infinity. In addition, we provide regularity estimates for general nonlinear degenerate parabolic equations of fourth order.

Abstract

The dynamics of compressible liquid–vapor flow depends sensitively on the microscale behavior at the phase boundary. We consider a sharp-interface approach, and propose a multiscale model to describe liquid–vapor flow accurately, without imposing ad-hoc closure relations on the continuum scale. The multiscale model combines the Euler equations on the continuum scale with molecular-scale particle simulations that govern the interface motion. We rely on an interface-preserving moving mesh finite volume method to discretize the continuum-scale sharp-interface flow in a conservative manner. Computational efficiency, while preserving physical properties, is achieved by a surrogate solver for the interface dynamics based on constraint-aware neural networks. The multiscale model is presented in its general form, and applied to regimes of temperature-dependent liquid–vapor flow which have not been accessible before.

Abstract

The modelling of liquid--vapour flow with phase transition poses many challenges, both on the theoretical level, as well as on the level of discretisation methods. Therefore, accurate mathematical models and efficient numerical methods are required. In that, we focus on two modelling approaches: the sharp-interface (SI) approach and the diffuse-interface (DI) approach. For the SI-approach, representing the phase boundary as a co-dimension-1 manifold, we develop and validate analytical Riemann solvers for basic isothermal two-phase flow scenarios. This ansatz becomes cumbersome for increasingly complex thermodynamical settings. A more versatile multiscale interface solver, that is based on molecular dynamics simulations, is able to accurately describe the evolution of phase boundaries in the temperature-dependent case. It is shown to be even applicable to two-phase flow of multiple components. Despite the successful developments for the SI approach, these models fail if the interface undergoes topological changes. To understand merging and splitting phenomena for droplet ensembles, we consider DI models of second gradient type. For these Navier--Stokes--Korteweg systems, that can be seen as a third order extension of the Navier--Stokes equations, we propose variants that are more accessible to standard numerical schemes. More precisely, we reformulate the capillarity operator to restore the hyperbolicity of the Euler operator in the full system.

Abstract

Results of our proposed optical flow method on the Sintel and KITTI datasets. We provide the benchmark results before and after applying our refinement approach.Additionally, we provide a supplementary material to our paper with more details on the minimisation, the numerical solution and additional results.The data is structured as follows:The file supplement.pdf contains the supplementary material of the paper.Additionally, there are 10 data folders according to the test results in Table 3 of our paper. They are named using the scheme input/output_datasetName_methodName:input/output: whether the files are the input for our method or the output of our methoddatasetName: one of kitti, sintelClean or sintelFinal relating to either the KITTI dataset Menze, CVPR 2015 or the Sintel dataset Butler, ECCV 2012methodName: one of raft or raftWarm Teed, ECCV 2020 or the refined variants from our method.The data can be used to build on the output of our method or to compare a novel approach against our method by using the same input data.The dataformat is pdf for the supplementary material and png and flo for the optical flow files as used in the respective benchmarks.

Abstract

This work presents a framework to synthesize structured gain-scheduled controllers for structured plants that are affected by time-varying parametric scheduling blocks. Using a so-called lifting approach, we are able to handle several structured gain-scheduling problems arising from a nested inner and outer loop configuration with partial or full dependence on the scheduling block. Our resulting design conditions are formulated in terms of convex linear matrix inequalities and permit to handle multiple performance objectives.

Abstract

We present efficient reduced basis (RB) methods for the simulation of a coupled problem consisting of a rigid robot hand interacting with soft tissue material. The soft tissue is modeled by the linear elasticity equation and discretized with the Finite Element Method. We look at two different scenarios: (i) the forward simulation and (ii) a feedback control formulation of the model. In both cases, large-scale systems of equations appear, which need to be solved in real-time. This is essential in practice for the implementation in a real robot. For the feedback-scenario, we encounter a high-dimensional Algebraic Riccati Equation (ARE) in the context of the linear quadratic regulator. To overcome the real-time constraint by significantly reducing the computational complexity, we use several structure-preserving and non-structure-preserving reduction methods. These include reduced basis techniques based on the Proper Orthogonal Decomposition. For the ARE, we compute a low-rank-factor and hence solve a low-dimensional ARE instead of solving a full dimensional problem. Numerical examples for both cases (i) and (ii) are provided. These illustrate the approximation quality of the reduced solution and speedup factors of the different reduction approaches.

Abstract

<p style="text-align: justify;">This articles first investigates boundary integral operators for the three-dimensional isotropic linear elasticity of a biphasic model with piecewise constant Lamé coefficients in the form of a bounded domain of arbitrary shape surrounded by a background material. In the simple case of a spherical inclusion, the vector spherical harmonics consist of eigenfunctions of the single and double layer boundary operators and we provide their spectra. Further, in the case of many spherical inclusions with isotropic materials, each with its own set of Lamé parameters, we propose an integral equation and a subsequent Galerkin discretization using the vector spherical harmonics and apply the discretization to several numerical test cases.</p>

Abstract

We consider a phase-field model for the incompressible flow of two immiscible fluids. This model extends widespread models for two fluid phases by including a third, solid phase, which can evolve due to e.g. precipitation and dissolution. We consider a simple, two-dimensional geometry of a thin strip, which can still be seen as the representation of a single pore throat in a porous medium. Under moderate assumptions on the Péclet number and the capillary number, we investigate the limit case when the ratio between the width and the length of the strip goes to zero. In this way, and employing transversal averaging, we derive an upscaled model. The result is a multi-scale model consisting of the upscaled equations for the total flux and the ion transport, while the phase-field equation has to be solved in cell problems at the pore scale to determine the position of interfaces. We also investigate the sharp-interface limit of the multi-scale model, in which the phase-field parameter approaches 0. The resulting sharp-interface model consists only of Darcy-scale equations, as the cell problems can be solved explicitly. Notably, we find asymptotic consistency, that is, the upscaling process and the sharp-interface limit commute. We use numerical results to investigate the validity of the upscaling when discontinuities are formed in the upscaled model.

Abstract

Standard kernel methods for machine learning usually struggle when dealing with large datasets. We review a recently introduced Structured Deep Kernel Network (SDKN) approach that is capable of dealing with high-dimensional and huge datasets - and enjoys typical standard machine learning approximation properties.

Abstract

Data-dependent greedy algorithms in kernel spaces are known to provide fast converging interpolants, while being extremely easy to implement and efficient to run. Despite this experimental evidence, no detailed theory has yet been presented. This situation is unsatisfactory, especially when compared to the case of the data-independent P-greedy algorithm, for which optimal convergence rates are available, despite its performances being usually inferior to the ones of target data-dependent algorithms. In this work, we fill this gap by first defining a new scale of greedy algorithms for interpolation that comprises all the existing ones in a unique analysis, where the degree of dependency of the selection criterion on the functional data is quantified by a real parameter. We then prove new convergence rates where this degree is taken into account, and we show that, possibly up to a logarithmic factor, target data-dependent selection strategies provide faster convergence. In particular, for the first time we obtain convergence rates for target data adaptive interpolation that are faster than the ones given by uniform points, without the need of any special assumption on the target function. These results are made possible by refining an earlier analysis of greedy algorithms in general Hilbert spaces. The rates are confirmed by a number of numerical examples.

Abstract

The accuracy of the training data limits the accuracy of bulk properties from machine-learned potentials. For example, hybrid functionals or wave-function-based quantum chemical methods are readily available for cluster data but effectively out of scope for periodic structures. We show that local, atom-centered descriptors for machine-learned potentials enable the prediction of bulk properties from cluster model training data, agreeing reasonably well with predictions from bulk training data. We demonstrate such transferability by studying structural and dynamical properties of bulk liquid water with density functional theory and have found an excellent agreement with experimental and theoretical counterparts.

Abstract

In this report we present advances in our research on direct aeroacoustics and uncertainty quantification, based on the high-order Discontinuous Galerkin solver FLEXI. Oscillation phenomena triggered by flow over cavities can lead to an unpleasant tonal (whistling) noise, which provides motivation for industry and academia to better understand the underlying mechanisms. We present a numerical setup capable of capturing these phenomena with high efficiency, as we show by comparison to experimental data and results from industry. Some of these phenomena are highly sensitive towards flow conditions, which makes an integrated approach regarding these conditions necessary. This is the goal of uncertainty quantification. We present software for both intrusive and non-intrusive uncertainty quantification methods. We investigate convergence and computational performance. The development of both codes was in parts carried out in cooperation with HLRS. Apart from validation results, we show a non-intrusive simulation of 3D turbulent cavity noise.

Abstract

This volume is dedicated to Ari Laptev on the occasion of his 70th birthday. It collects contributions by his numerous colleagues sharing with him research interests in analysis and spectral theory. In brief, the topics covered include Friedrichs, Hardy, and Lieb–Thirring inequalities, eigenvalue bounds and asymptotics, Feshbach–Schur maps and perturbation theory, scattering theory and orthogonal polynomials, stability of matter, electron density estimates, Bose–Einstein condensation, Wehrl-type entropy inequalities, Bogoliubov theory, wave packet evolution, heat kernel estimates, homogenization, d-bar problems, Brezis–Nirenberg problems, the nonlinear Schrödinger equation in magnetic fields, classical discriminants, and the two-dimensional Euler–Bardina equations. In addition, Ari’s multifaceted service to the mathematical community is also touched upon. Altogether the volume presents a collection of research articles which will be of interest to any active scientist working in one of the above mentioned fields.

Abstract

In this paper we revisit the classical problem of nonparametric regression, but impose local differential privacy constraints. Under such constraints, the raw data (X-1, Y-1), ..., (X-n, Y-n), taking values in R-d x R, cannot be directly observed, and all estimators are functions of the randomised output from a suitable privacy mechanism. The statistician is free to choose the form of the privacy mechanism, and here we add Laplace distributed noise to a discretisation of the location of a feature vector X-i and to the value of its response variable Y-i. Based on this randomised data, we design a novel estimator of the regression function, which can be viewed as a privatised version of the well-studied partitioning regression estimator. The main result is that the estimator is strongly universally consistent, and we further establish an upper bound on the rate of convergence. Our methods and analysis also give rise to a strongly universally consistent binary classification rule for locally differentially private data.

Abstract

The combination of machine learning with control offers many opportunities, in particular for robust control. However, due to strong safety and reliability requirements in many real-world applications, providing rigorous statistical and control-theoretic guarantees is of utmost importance, yet difficult to achieve for learning-based control schemes. We present a general framework for learning-enhanced robust control that allows for systematic integration of prior engineering knowledge, is fully compatible with modern robust control and still comes with rigorous and practically meaningful guarantees. Building on the established Linear Fractional Representation and Integral Quadratic Constraints framework, we integrate Gaussian Process Regression as a learning component and state-of-the-art robust controller synthesis. In a concrete robust control example, our approach is demonstrated to yield improved performance with more data, while guarantees are maintained throughout.

Abstract

The Lorenz attractor is considered which is a well known attractor of a dynamical system. As the Lorenz attractor shows some fractal structure, its box dimension is investigated. A general algorithmic approach is developed which enables us to calculate its box dimension, and special attention will be paid to numerical aspects of the algorithm. The algorithm gets tested with the help of some well known fractals - self-similar fractals and fractal graphs of deterministic as well as random functions. For these examples, the algorithm yields values which are very close to the theoretical values. However, in the case of the Lorenz attractor the algorithm seems to fail. The reason for this phenomenon is discussed, the notion of local box dimension is introduced, and the algorithm is used in order to reveal a hidden multifractal structure of the Lorenz attractor. (C) 2020 Elsevier B.V. All rights reserved.

Abstract

Let G be a connected reductive algebraic group defined over a finite field with q elements. In the 1980's, Kawanaka introduced generalised Gelfand--Graev representations of the finite group \$\$ G\backslashleft(\\backslashmathbbm\F\\\_q\backslashright) \$\$, assuming that q is a power of a good prime for G. These representations have turned out to be extremely useful in various contexts. Here we investigate to what extent Kawanaka's construction can be carried out when we drop the assumptions on q. As a curious by-product, we obtain a new, conjectural characterisation of Lusztig's concept of special unipotent classes of G in terms of weighted Dynkin diagrams.

Abstract

A novel convex state-space solution to the robust output-feedback synthesis problem based on dynamic integral quadratic constraints for a particular class of systems is presented. Convexification rests upon the assumption that the control-to-uncertainty block in the generalized plant commutes with the off-diagonal block of the multiplier. We demonstrate that this commutation property is valid for several by themselves interesting concrete scenarios, such as in extremum control, a generalization of the convex design of first-order optimization algorithms involving filtered gradient evaluations.

Abstract

We provide a convex solution to the robust gain-scheduled estimation problem based on integral quadratic constraints with dynamic D-scalings for both the uncertain and the scheduled component. This closes an important gap since so far merely static scalings or multipliers could be used for the scheduled component in estimation problems. To this end, we provide novel synthesis criteria in terms of linear matrix inequalities for the design of nominal gain-scheduled estimators which allow for a direct combination with available results on robust estimation. We illustrate the benefit of our design approach by means of a numerical example.

Abstract

For a partially unknown linear systems, we present a systematic control design approach based on generated data from measurements of closed-loop experiments with suitable test controllers. These experiments are used to improve the achieved performance and to reduce the uncertainty about the unknown parts of the system. This is achieved through a parametrization of auspicious controllers with convex relaxation techniques from robust control, which guarantees that their implementation on the unknown plant is safe. This approach permits to systematically incorporate available prior knowledge about the system by employing the framework of linear fractional representations.

Abstract

The dual iteration was introduced in a conference paper in 1997 by Iwasaki as an iterative and heuristic procedure for the challenging and non-convex design of static output-feedback controllers. We recall in detail its essential ingredients and go beyond the work of Iwasaki by demonstrating that the framework of linear fractional representations allows for a seamless extension the dual iteration to output-feedback designs of tremendous practical relevance such as the design of robust or robust gain-scheduled controllers. In the paper of Iwasaki the dual iteration is solely based on, and motivated by algebraic manipulations resulting from the elimination lemma. We provide a novel control theoretic interpretation of the individual steps, which paves the way for further generalizations of the powerful scheme to situations where the elimination lemma is not applicable. Exemplary, we extend the dual iteration to the multi-objective design of static output-feedback H∞-controllers, which guarantee that the closed-loop poles are contained in an a priori specified generalized stability region. We demonstrate the approach with numerous numerical examples inspired from the literature.

Abstract

Sparse grids enable accurate function approximation in higher-dimensional settings with a moderate number of grid points. In this paper, we introduce generalized versions of various efficient sparse grid algorithms in a unified notation: For local basis functions, transformations between function values at grid points and interpolation coefficients can be efficiently realized using the well-known unidirectional principle (Balder and Zenger, SIAM Journal on Scientific Computing, 17(3):631--646, May 1996; Bungartz, Finite Elements of Higher Order on Sparse Grids, Universität München, Aachen, November 1998). For general basis functions without spatial adaptivity, Avila and Carrington show in (The Journal of Chemical Physics, 143(21):214108, 2015) and follow-up work that the unidirectional principle can instead be applied to an equivalent incremental basis, and we show that a second application of the unidirectional principle can be used to compute interpolation coefficients for the original, non-incremental basis. Our Open Source implementation for the latter case is able to compute interpolation coefficients for up to 20 million sparse grid points in one second on a single CPU core. We also discuss the efficient application of linear operators occurring in derivative evaluations of interpolants and in the solution of partial differential equations on sparse grids with finite elements, e.g., using the UpDown scheme (Bungartz, Dünne Gitter und deren Anwendung bei der adaptiven Lösung der dreidimensionalen Poisson-Gleichung, PhD thesis, Technische Universität München, 1992; Pflüger, Spatially Adaptive Sparse Grids for High-Dimensional Problems, Verlag Dr. Hut, München, February 2010; Zeiser, Journal of Scientific Computing, 47(3):328--346, 2011; Bungartz et al., Journal of Computational and Applied Mathematics, 236(15):3741--3750, 2012).

Abstract

I Mathematische Grundlagen. Elementare Logik -- Mengen und Abbildungen -- Kombinatorik und Wahrscheinlichkeit -- Komplexe Zahlen -- II Vektorrechnung. Vektoren -- Längen, Winkel und Skalarprodukt -- Vektor- und Spatprodukt -- Geraden und Ebenen -- III Differentialrechnung. Polynome und rationale Funktionen -- Exponentialfunktion, Logarithmus und trigonometrische Funktionen -- Grenzwerte, Reihen und Stetigkeit -- Differentiationsregeln und Anwendungen -- Taylor-Entwicklung -- Extremwerte und Funktionsuntersuchung -- IV Integralrechnung. Integral und Stammfunktion -- Partielle Integration, Substitution und spezielle Integranden -- Uneigentliche Integrale -- V Anwendung mathematischer Software. MATLAB® -- Maple™. VI. Formelsammlung. Mathematische Grundlagen -- Vektorrechnung -- Differentialrechnung -- Integralrechnung. Literaturverzeichnis.

Abstract

Naturally reductive spaces, in general, can be seen as an adequate generalization of Riemannian symmetric spaces. Nevertheless, there are some that are closer to symmetric spaces than others. On the one hand, there is the series of Hopf fibrations over complex space forms, including the Heisenberg groups with their metrics of type H. On the other hand, there exist certain naturally reductive spaces in dimensions 6 and 7 whose torsion forms have a distinguished algebraic property. All these spaces generalize geometric or algebraic properties of three-dimensional naturally reductive spaces and have the following point in common: along every geodesic, the Jacobi operator satisfies an ordinary differential equation with constant coefficients which can be chosen independently of the given geodesic.

Abstract

A proper isometric Lie group action on a Riemannian manifold is called polar if there exists a closed connected submanifold which meets all orbits orthogonally. In this article we study polar actions on Damek-Ricci spaces. We prove criteria for isometric actions on Damek-Ricci spaces to be polar, find examples and give some partial classifications of polar actions on Damek-Ricci spaces. In particular, we show that non-trivial polar actions exist on all Damek-Ricci spaces.

Abstract

In this work we present a novel exact Riemann solver for an artificial Equation of State (EoS) based modification of the incompressible Euler equations. Differently from the well known artificial compressibility method, this new approach overcomes the lack of the pressure-velocity coupling by using a suitably designed artificial EoS. The modified set of equations fits into the framework of first order hyperbolic conservation laws. Accordingly, an exact Riemann solver is derived. This new solver has the advantage to avoid a wave pattern violation issue which can affect the exact Riemann problem solution obtained by the standard artificial compressibility approach. The new artificial EoS based Riemann solver can be used as a tool for the definition of the advective Godunov fluxes in a Finite Volume or a discontinuous Galerkin discretization of the incompressible Navier–Stokes (INS) equations. We assess and analyse the new exact Riemann solver on 1D Riemann problems. Its capability and effectiveness are then shown in the context of a high-order accurate discontinuous Galerkin discretization of the INS equations. In particular, we verify the convergence properties, both in space and time, considering two test cases in 2D, namely, the Kovasznay test case and a damped travelling waves test case. Finally, we perform an implicit large eddy simulation of the incompressible turbulent flow over periodic hills where the classic forcing term formulation is modified to deal with variable time steps. Solution comparisons with respect to numerical and experimental results from the literature are given.

Abstract

Approaches based on order-adaptive regularisation belong to the most accurate variational methods for computing the optical flow. By locally deciding between first- and second-order regularisation, they are applicable to scenes with both fronto-parallel and ego-motion. So far, however, existing order-adaptive methods have a decisive drawback. While the involved first- and second-order smoothness terms already make use of anisotropic concepts, the underlying selection process itself is still isotropic in that sense that it locally chooses the same regularisation order for all directions. In our paper, we address this shortcoming. We propose a generalised order-adaptive approach that allows to select the local regularisation order for each direction individually. To this end, we split the order-adaptive regularisation across and along the locally dominant direction and perform an energy competition for each direction separately. This in turn offers another advantage. Since the parameters can be chosen differently for both directions, the approach allows for a better adaption to the underlying scene. Experiments for MPI Sintel and KITTI 2015 demonstrate the usefulness of our approach. They not only show improvements compared to an isotropic selection scheme. They also make explicit that our approach is able to improve the results from state-of-the-art learning-based approaches, if applied as a final refinement step – thereby achieving top results in both benchmarks.

Abstract

We consider the incompressible flow of two immiscible fluids in the presence of a solid phase that undergoes changes in time due to precipitation and dissolution effects. Based on a seminal sharp interface model a phase-field approach is suggested that couples the Navier–Stokes equations and the solid’s ion concentration transport equation with the Cahn–Hilliard evolution for the phase fields. The model is shown to preserve the fundamental conservation constraints and to obey the second law of thermodynamics for a novel free energy formulation. An extended analysis for vanishing interfacial width reveals that in this limit the sharp interface model is recovered, including all relevant transmission conditions. Notably, the new phase-field model is able to realize Navier-slip conditions for solid–fluid interfaces in the limit.

Abstract

We present a convex solution for the design of generalized accelerated gradient algorithms for strongly convex objective functions with Lipschitz continuous gradients. We utilize integral quadratic constraints and the Youla parameterization from robust control theory to formulate a solution of the algorithm design problem as a convex semidefinite program. We establish explicit formulas for the optimal convergence rates and extend the proposed synthesis solution to extremum control problems.

Abstract

Enzymatically induced calcite precipitation (EICP) is an engineering technology that allows for targeted reduction of porosity in a porous medium by precipitation of calcium carbonates. This might be employed for reducing permeability in order to seal flow paths or for soil stabilization. This study investigates the growth of calcium-carbonate crystals in a micro-fluidic EICP setup and relies on experimental results of precipitation observed over time and under flow-through conditions in a setup of four pore bodies connected by pore throats. A phase-field approach to model the growth of crystal aggregates is presented, and the corresponding simulation results are compared to the available experimental observations. We discuss the model's capability to reproduce the direction and volume of crystal growth. The mechanisms that dominate crystal growth are complex depending on the local flow field as well as on concentrations of solutes. We have good agreement between experimental data and model results. In particular, we observe that crystal aggregates prefer to grow in upstream flow direction and toward the center of the flow channels, where the volume growth rate is also higher due to better supply.

Abstract

We consider dispersive systems of the form \$\$\backslashbegin\aligned\ \backslashpartial \_t U = \backslashLambda \_U U + B\_U(U,V) , \backslashqquad \backslashvarepsilon \backslashpartial \_t V = \backslashLambda \_V V + B\_V(U,U) \backslashend\aligned\\$\$∂tU=$Łambda$UU+BU(U,V),$\epsilon$∂tV=$Łambda$VV+BV(U,U)in the singular limit \$\$ \backslashvarepsilon \backslashrightarrow 0 \$\$, where \$\$ \backslashLambda \_U,\backslashLambda \_V \$\$are linear and \$\$ B\_U,B\_V \$\$bilinear mappings. We are interested in deriving error estimates for the approximation obtained through the regular limit system \$\$\backslashbegin\aligned\ \backslashpartial \_t \backslashpsi \_U = \backslashLambda \_U \backslashpsi \_U - B\_U(\backslashpsi \_U, \backslashLambda \_V^\-1\ B\_V(\backslashpsi \_U,\backslashpsi \_U) ) \backslashend\aligned\\$\$∂t$\psi$U=$Łambda$U$\psi$U-BU($\psi$U,$Łambda$V-1BV($\psi$U,$\psi$U))from a more general point of view. Our abstract approximation theorem applies to a number of semilinear systems, such as the Dirac--Klein--Gordon system, the Klein--Gordon--Zakharov system, and a mean field polaron model. It extracts the common features of scattered results in the literature, but also gains an approximation result for the Dirac--Klein--Gordon system which has not been documented in the literature before. We explain that our abstract approximation theorem is sharp in the sense that there exists a quasilinear system of the same structure where the regular limit system makes wrong predictions.

Abstract

We propose a novel $hp$-Multilevel Monte Carlo method for the quantification of uncertainties in compressible flows using the Discontinuous Galerkin method as deterministic solver. The multilevel approach exploits hierarchies of uniformly refined meshes jointly with an increasing polynomial degree for the ansatz space. It allows for a very large range of resolutions in the physical space and thus an efficient decrease of the statistical error. We prove that the overall complexity of the $hp$-Multilevel Monte Carlo method to compute the mean field with prescribed accuracy is of quadratic order with respect to the accuracy. We also propose a novel and simple approach to estimate a lower confidence bound for the optimal number of samples per level, which helps to prevent overshooting the optimal number of samples. The method is in particular adapted to the needs of high-performance computing. Our theoretical results are verified by numerical experiments for the two dimensional compressible Navier-Stokes equations. In particular we consider a cavity flow problem from computational acoustics, demonstrating that the method is suitable to handle complex engineering problems.

Abstract

The classic regularization theory for solving inverse problems is built on the assumption that the forward operator perfectly represents the underlying physical model of the data acquisition. However, in many applications, for instance in microscopy or magnetic particle imaging, this is not the case. Another important example represent dynamic inverse problems, where changes of the searched-for quantity during data collection can be interpreted as model uncertainties. In this article, we propose a regularization strategy for linear inverse problems with inexact forward operator based on sequential subspace optimization methods (SESOP). In order to account for local modelling errors, we suggest to combine SESOP with the Kaczmarz’ method. We study convergence and regularization properties of the proposed method and discuss several practical realizations. Relevance and performance of our approach are evaluated at simulated data from dynamic computerized tomography with various dynamic scenarios.

Abstract

Hamiltonian systems are central in the formulation of non-dissipative physical systems. They are characterized by a phase-space, a symplectic form and a Hamiltonian function. In numerical simulations of Hamiltonian systems, algorithms show improved accuracy when the symplectic structure is preserved 10. For structure-preserving model order reduction (MOR) of Hamiltonian systems, symplectic MOR 17, 14, 11, 3, 16 can be used. It is based on a reduced-order basis that is symplectic, which requires symplectic basis generation techniques. In our work, we discuss greedy algorithms for symplectic basis generation. We complement the procedure presented in 14 with ideas of the POD-greedy 9, which results in a new greedy symplectic basis generation technique, the PSD-greedy. Inspired by POD-greedy, we use compression techniques in the greedy iterations to enrich the basis iteratively. We prove that this algorithm computes a symplectic basis when symplectic techniques are used for compression. In the numerical experiments, we compare the discussed methods for a linear elasticity problem. The results show that improvements of up to one order of magnitude in the relative reduction error are achievable with the new basis generation technique compared to the existing greedy approach from 14.

Abstract

We show that simply connected Riemannian homogeneous spaces of compact semisimple Lie groups with polar isotropy actions are symmetric, generalizing results of Fabio Podestà and the third named author. Without assuming compactness, we give a classification of Riemannian homogeneous spaces of semisimple Lie groups whose linear isotropy representations are polar. We show for various such spaces that they do not have polar isotropy actions. Moreover, we prove that Heisenberg groups and non-symmetric Damek--Ricci spaces have non-polar isotropy actions.

Abstract

The authors consider Euler's equations for free surface waves traveling on a body of density stratified water in the scenario when gravity and surface tension act as restoring forces. The flow is continuously stratified, and the water layer is bounded from below by an impermeable horizontal bed. Three equivalent classical formulations in a suitable setting of strong solutions which may describe nevertheless waves with singular density gradients are established. The availability of a weak formulation of the water wave problem, the regularity properties of the corresponding weak solutions, and methods from nonlinear functional analysis are used. The paper is organized as follows. In Section 2, the three formulations of the problem are introduced and their equivalence is established. In Section 3, the authors first introduce the notion of a weak solution to Dubreil-Jacotin's formulation and establish, by means of a shooting method, the existence of at least one laminar flow solution to this latter formulation. In Section 4, the equations are reformulated as an abstract bifurcation problem by using methods from nonlinear functional analysis.

Abstract

Entropy numbers are an important tool for quantifying the compactness of operators. Besides establishing new upper bounds on the entropy numbers of diagonal operators $D_\sigma$ from $\ell_p$ to $\ell_q$, where $p\not=q$, we investigate the optimality of these bounds. In case of $p<q$ optimality is proven for fast decaying diagonal sequences, which include exponentially decreasing sequences. In case of $p>q$ we show optimality under weaker assumption than previously used in the literature. In addition, we illustrate the benefit of our results with examples not covered in the literature so far.

Abstract

This article considers one-dimensional random systems of hyperbolic conservation laws. Existence and uniqueness of random entropy admissible solutions for initial value problems of conservation laws, which involve random initial data and random flux functions, are established. Based on these results an a posteriori error analysis for a numerical approximation of the random entropy solution is presented. For the stochastic discretization, a non-intrusive approach, namely the Stochastic Collocation method is used. The spatio-temporal discretization relies on the Runge–Kutta Discontinuous Galerkin method. The a posteriori estimator is derived using smooth reconstructions of the discrete solution. Combined with the relative entropy stability framework this yields computable error bounds for the entire space-stochastic discretization error. The estimator admits a splitting into a stochastic and a deterministic (space-time) part, allowing for a novel residual-based space-stochastic adaptive mesh refinement algorithm. The scaling properties of the residuals are investigated and the efficiency of the proposed adaptive algorithms is illustrated in various numerical examples.

Abstract

Bereitstellung einer Sammlung von Android-Apps zum Einsatz in der universitäten Mathematik-Ausbildung, gleichermaßen als Demonstratoren und zum Selbststudium.

Abstract

We are interested in the description of small modulations in time and space of wave-train solutions to the complex Ginzburg-Landau equation partial derivative(T)Psi = (1 + i alpha)partial derivative(2)(X)Psi + Psi - (1+i beta)Psi vertical bar Psi vertical bar(2) near the Eckhaus boundary, that is, when the wave train is near the threshold of its first instability. Depending on the parameters of, alpha, beta, a number of modulation equations can he derived, such as the KdV equation, the Cahn-Hilliard equation, and a family of Ginzburg-Landau based amplitude equations. Here we establish error estimates showing that the Korteweg-de Vries (KdV) approximation makes correct predictions in a certain parameter regime. Our proof is based on energy estimates and exploits the conservation law structure of the critical mode. In order to improve linear damping, we work in spaces of analytic functions.

Abstract

The N-wave interaction (NWI) system appears as an amplitude system in the description of N nonlinearly interacting and linearly transported wave packets in dispersive wave systems such as the water wave problem or problems in nonlinear optics. The purpose of this paper is twofold. First we give a new simplified proof for the failure of the NWI approximation in case of resonances which are located at integer multiples of a basic wave number k(0) in the original 2 pi/k0-spatially periodic dispersive wave system. Secondly, we give a first rigorous proof that an amplitude system fails in the description of an original system, without imposing periodic boundary conditions on the original system.

Abstract

We derive novel criteria for designing stabilizing dynamic output-feedback controllers for a class of aperiodic impulsive systems subject to a range dwell-time condition. Our synthesis conditions are formulated as clock-dependent linear matrix inequalities (LMIs) which can be solved numerically, e.g., by using matrix sum-of-squares relaxation methods. We show that our results allow us to design dynamic output-feedback controllers for aperiodic sample-data systems and illustrate the proposed approach by means of a numerical example.

Abstract

In every point of a Kahler manifold there exist special holomorphic coordinates well adapted to the underlying geometry. Comparing these Kahler normal coordinates with the Riemannian normal coordinates defined via the exponential map we prove that their difference is a universal power series in the curvature tensor and its iterated covariant derivatives and devise an algorithm to calculate this power series to arbitrary order. As a byproduct we generalize Kahler normal coordinates to the class of complex affine manifolds with (1, 1)-curvature tensor. Moreover we describe the Spencer connection on the infinite order Taylor series of the Kahler normal potential and obtain explicit formulas for the Taylor series of all relevant geometric objects on symmetric spaces.

Abstract

"The authors study eigenvalues of the Laplacian (the negative second derivative operator) of a compact metric graph equipped with complex δ-vertex conditions. More precisely: (a) a continuity condition is imposed at each vertex; (b) on a selected set of vertices R, the vertex conditions ∑e∼vj∂∂νf|e(vj)+αjf(vj)=0 are imposed, where the interaction strengths αj are complex parameters and the derivative is taken in the direction towards the vertex; and (c) on the vertices in ∖R, Kirchhoff vertex conditions are imposed. The main focus in this article is on the asymptotic behavior of the (purely discrete) spectrum as certain coefficients αj tend to ∞ in the complex plane. If m of these coefficients tend to infinity within some sector in the open left half-plane while the remaining coefficients tend to infinity in a way such that Reαj remains bounded from below, then the authors show that exactly m eigenvalues diverge away from the positive real semi-axis. Moreover, they provide the asymptotics of these eigenvalues; the leading term is quadratic in the αj, with a coefficient depending on the vertex degrees. As variational principles are not available for the study of such non-self-adjoint problems, the authors use a Birman-Schwinger type characterization of eigenvalues in terms of a parameter-dependent Dirichlet-to-Neumann matrix (or Titchmarsh-Weyl function). In addition, they also obtain estimates for the numerical range and the eigenvalues."

Abstract

We present version 3 of the open-source simulator for flow and transport processes in porous media DuMux. DuMux is based on the modular C++ framework Dune (Distributed and Unified Numerics Environment) and is developed as a research code with a focus on modularity and reusability. We describe recent efforts in improving the transparency and efficiency of the development process and community-building, as well as efforts towards quality assurance and reproducible research. In addition to a major redesign of many simulation components in order to facilitate setting up complex simulations in DuMux, version 3 introduces a more consistent abstraction of finite volume schemes. Finally, the new framework for multi-domain simulations is described, and three numerical examples demonstrate its flexibility.

Abstract

We obtain well-posedness results in Lp-based weighted Sobolev spaces for a transmission problem for anisotropic Stokes and Navier–Stokes systems with L∞ strongly elliptic coefficient tensor, in complementary Lipschitz domains of Rn, n≥3. The strong ellipticity allows to explore the associated pseudostress setting. First, we use a variational approach that reduces the anisotropic Stokes system in the whole Rn to an equivalent mixed variational formulation with data in Lp-based weighted Sobolev spaces. We show that such mixed variational formulation is well-posed in the space Hp1(Rn)n×Lp(Rn), n≥ 3, for any p in an open interval containing 2. Then similar well-posedness results are obtained for two linear transmission problems. These results are used to define the Newtonian and layer potential operators for the considered anisotropic Stokes system and to obtain mapping properties of these operators. The potentials are employed to show the well-posedness of some linear transmission problems, which then is combined with a fixed point theorem in order to show the well-posedness of a nonlinear transmission problem for the anisotropic Stokes and Navier–Stokes systems in Lp-based weighted Sobolev spaces, whenever the given data are small enough.

Abstract

Using octonions and the triality property of Spin(8), we find explicit formulae for the Lie brackets of the exceptional simple real Lie algebras f4 and f∗4, i.e. the Lie algebras of the isometry groups of the Cayley projective plane and the Cayley hyperbolic plane. As an application, we determine all polar actions on the Cayley hyperbolic plane which leave a totally geodesic subspace invariant.

Abstract

In this paper, the authors study a model describing the evolution of the height of a non-Newtonian incompressible thin liquid film that spreads on a solid surface by the Ellis constitutive law. Existence of strong positive solutions in Sobolev-Slobodetskiĭ spaces is shown by using Amann's abstract existence theory. In addition, the authors prove uniqueness of the solutions by the energy approach.

Abstract

We show the existence of breather solutions in a nonlinear Klein-Gordon system on a discrete graph with periodic junctions. The proof is based on the Theorem of Crandall-Rabinowitz.

Abstract

The purpose of this paper is to construct small-amplitude breather solutions for a nonlinear Klein-Gordon equation posed on a periodic metric graph via spatial dynamics and center manifold reduction. The major difficulty occurs from the irregularity of the solutions. The persistence of the approximately constructed pulse solutions under higher order perturbations is obtained by symmetry and reversibility arguments. (C) 2019 Elsevier Inc. All rights reserved.

Abstract

We study the limiting behavior of the Dirichlet and Neumann eigenvalue counting function of generalized second-order differential operators d/d mu d/dx, where mu is a finite atomless Borel measure on some compact interval a, b. Therefore, we firstly recall the results of the spectral asymptotics for these operators received so far. Afterwards, we make a proposition about the convergence behavior for so-called random V-variable Cantor measures.

Abstract

Gaussian process emulators (GPE) are a machine learning approach that replicates computational demanding models using training runs of that model. Constructing such a surrogate is very challenging and, in the context of Bayesian inference, the training runs should be well invested. The current paper offers a fully Bayesian view on GPEs for Bayesian inference accompanied by Bayesian active learning (BAL). We introduce three BAL strategies that adaptively identify training sets for the GPE using information-theoretic arguments. The first strategy relies on Bayesian model evidence that indicates the GPE's quality of matching the measurement data, the second strategy is based on relative entropy that indicates the relative information gain for the GPE, and the third is founded on information entropy that indicates the missing information in the GPE. We illustrate the performance of our three strategies using analytical- and carbon-dioxide benchmarks. The paper shows evidence of convergence against a reference solution and demonstrates quantification of post-calibration uncertainty by comparing the introduced three strategies. We conclude that Bayesian model evidence-based and relative entropy-based strategies outperform the entropy-based strategy because the latter can be misleading during the BAL. The relative entropy-based strategy demonstrates superior performance to the Bayesian model evidence-based strategy.

Abstract

The problem of dynamic 2D vector tomography is considered. Object motion is a combination of rotation and shifting. Properties of the dynamic ray transform operators are investigated. Singular value decomposition of the operators is constructed with usage of classic orthogonal polynomials.

Abstract

We present a novel acceleration method for the solution of parametric ODEs by single-step implicit solvers by means of greedy kernel-based surrogate models. In an offline phase, a set of trajectories is precomputed with a high-accuracy ODE solver for a selected set of parameter samples, and used to train a kernel model which predicts the next point in the trajectory as a function of the last one. This model is cheap to evaluate, and it is used in an online phase for new parameter samples to provide a good initialization point for the nonlinear solver of the implicit integrator. The accuracy of the surrogate reflects into a reduction of the number of iterations until convergence of the solver, thus providing an overall speedup of the full simulation. Interestingly, in addition to providing an acceleration, the accuracy of the solution is maintained, since the ODE solver is still used to guarantee the required precision. Although the method can be applied to a large variety of solvers and different ODEs, we will present in details its use with the Implicit Euler method for the solution of the Burgers equation, which results to be a meaningful test case to demonstrate the method's features.

Abstract

This article deals with the problem of recovering an image from tomographic data corrupted by the motion of an object. Using generalized Radon transforms as modeling for the dynamic imaging operators, we present a novel approach for the reconstruction process. The proposed method, motivated by results on microlocal analysis, approximates the solution via a filtered backprojection-type algorithm which has the advantage of being quick to implement. The numerical results applied on dynamic photoacoustic tomography (PAT) data corroborate the efficiency and relevance of the proposed method.

Abstract

We study symmetric Killing 2-tensors on Riemannian manifolds. We show that several additional conditions can be realized only for Sasakian manifolds and Euclidean spheres. In particular we recover a result of S. Gallot on the characterization of spheres by means of functions satisfying a certain differential equation of order three. (C) 2018 Published by Elsevier B.V.

Abstract

We develop novel criteria for robust stability of a class of hybrid systems that are affected by piecewise constant uncertain parameters with dwell-time constraints. These criteria result from clock-dependent Lyapunov arguments that are based on several new robust stability conditions for continuous-time linear time-invariant systems which are affected by parametric passive uncertainties and do not admit any discrete dynamics. In contrast to other approaches, our stability analysis results rely on separation techniques from robust control involving dynamic resetting scalings. Moreover, these techniques are expected to pave the way for covering much more general uncertainties within the framework of integral quadratic constraints. Building upon the new insights on robust stability, we are able to propose an output-feedback gain-scheduling design methodology similarly to our recently obtained synthesis result involving dynamic resetting $D$-scalings. Most of the obtained conditions are expressed as infinite-dimensional (differential) linear matrix inequalities which can be numerically solved by using relaxation methods based on, e.g., linear splines or matrix sum-of-squares. We apply our results to obtain novel stability criteria and to design distributed controllers for networked systems over undirected switching topologies. Finally, the approach is illustrated with several numerical examples.

Abstract

We propose a novel approach for output-feedback controller design for linear systems affected by arbitrary fast time-varying uncertainties. Our approach is based on consecutively solving two related robust synthesis problems which by themselves admit convex solutions. The first one is robust full information controller design, while the subsequent one is similar to the robust estimator design problem. The latter includes a homotopy parameter that serves to construct a solution of the output-feedback synthesis problem from a given solution of the full information problem. In contrast to alternative approaches, ours requires to solve just two linear matrix inequalities and generates guaranteed bounds for the achieved performance. We illustrate our results with several numerical examples taken from the literature.

Abstract

Mehr als 500 typische Klausur- und Übungsaufgaben zur Höheren Mathematik für Ingenieure, Natur- und Wirtschaftswissenschaftler mit detaillierten Lösungen ermöglichen eine optimale Vorbereitung auf Prüfungen und erleichtern die Bearbeitung von Übungsblättern. Die wesentlich erweiterte, in drei Bände gegliederte zweite Auflage enthält ebenfalls Aufgaben, die den Einsatz von MATLAB(R)und Maple(TM) bei zentralen mathematischen Problemstellungen illustrieren. Inhalt dieses ersten Bandes: Mathematische Grundlagen Vektorrechnung Differentialrechnung Integralrechnung Anwendungen mathematischer Software Ergänzt werden die Bände der Neuauflage durch Präsentationsfolien zu den Themengebieten der Höheren Mathematik sowie das Lexikon und die Aufgabensammlung von Mathematik-Online. Diese Internet-Angebote bieten insbesondere ausführliche Beschreibungen mathematischer Lehrsätze und Methoden und ermöglichen, erlernte Techniken interaktiv zu überprüfen. Die Autoren Klaus Höllig promovierte 1979 in Bonn, lehrte als Professor of Mathematics and Computer Sciences an der University of Wisconsin-Madisonund leitete anschließend den Lehrstuhl für Numerik und Geometrische Modellierung an der Universität Stuttgart. Er ist Mitbegründer von Mathematik-Online, einem Internet-Portal zur Höheren Mathematik. Jörg Hörner ist seit über 20 Jahren an der Universität Stuttgart in der Mathematik-Ausbildung von Ingenieuren und Naturwissenschaftlern tätig. Er ist technischer Leiter von Mathematik-Online und entwickelt unter anderem Software und Demos zur Illustration mathematischer Verfahren

Abstract

In this presentation we address the dynamic concentration reconstruction problem in magnetic particle imaging. The tradeoff between the signal-to-noise ratio in the processed data used for the reconstruction and the concentration dynamics of the tracer requires sophisticated reconstruction techniques to avoid motion artifacts. We incorporate motion priors in the concentration reconstruction and as an extra we can obtain an estimate for a flow field.

Abstract

A variety of methods is available to quantify uncertainties arising within the modeling of flow and transport in carbon dioxide storage, but there is a lack of thorough comparisons. Usually, raw data from such storage sites can hardly be described by theoretical statistical distributions since only very limited data is available. Hence, exact information on distribution shapes for all uncertain parameters is very rare in realistic applications. We discuss and compare four different methods tested for data-driven uncertainty quantification based on a benchmark scenario of carbon dioxide storage. In the benchmark, for which we provide data and code, carbon dioxide is injected into a saline aquifer modeled by the nonlinear capillarity-free fractional flow formulation for two incompressible fluid phases, namely carbon dioxide and brine. To cover different aspects of uncertainty quantification, we incorporate various sources of uncertainty such as uncertainty of boundary conditions, of parameters in constitutive relations, and of material properties. We consider recent versions of the following non-intrusive and intrusive uncertainty quantification methods: arbitrary polynomial chaos, spatially adaptive sparse grids, kernel-based greedy interpolation, and hybrid stochastic Galerkin. The performance of each approach is demonstrated assessing expectation value and standard deviation of the carbon dioxide saturation against a reference statistic based on Monte Carlo sampling. We compare the convergence of all methods reporting on accuracy with respect to the number of model runs and resolution. Finally, we offer suggestions about the methods' advantages and disadvantages that can guide the modeler for uncertainty quantification in carbon dioxide storage and beyond.

Abstract

This article is a follow up of our submitted paper (D. Seus et al, Comput Methods Appl Mech Eng 333:331--355, 2018) in which a decomposition of the Richards equation along two soil layers was discussed. A decomposed problem was formulated and a decoupling and linearisation technique was presented to solve the problem in each time step in a fixed point type iteration. This article extends these ideas to the case of two-phase in porous media and the convergence of the proposed domain decomposition method is rigorously shown."

Abstract

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Abstract

As a simplified model for subsurface flows, elliptic equations may be utilized. Insufficient measurements or uncertainty in those are commonly modeled by a random coefficient, which then accounts for the uncertain permeability of a given medium. As an extension of this methodology to flows in heterogeneous/fractured/porous media, we incorporate jumps in the diffusion coefficient. These discontinuities then represent transitions in the media. More precisely, we consider a second order elliptic problem where the random coefficient is given by the sum of a (continuous) Gaussian random field and a (discontinuous) jump part. To estimate moments of the solution to the resulting random partial differential equation, we use a pathwise numerical approximation combined with multilevel Monte Carlo sampling. In order to account for the discontinuities and improve the convergence of the pathwise approximation, the spatial domain is decomposed with respect to the jump positions in each sample, leading to path-dependent grids. Hence, it is not possible to create a nested sequence of grids which is suitable for each sample path a priori. We address this issue by an adaptive multilevel algorithm, where the discretization on each level is sample-dependent and fulfills given refinement conditions.

Abstract

In this paper approximation methods for infinite-dimensional Levy processes, also called (time-dependent) Levy fields, are introduced. For square integrable fields beyond the Gaussian case, it is no longer given that the one-dimensional distributions in the spectral representation with respect to the covariance operator are independent. When simulated via a Karhunen-Loeve expansion a set of dependent but uncorrelated one-dimensional Levy processes has to be generated. The dependence structure among the one-dimensional processes ensures that the resulting field exhibits the correct point-wise marginal distributions. To approximate the respective (one-dimensional) Levy-measures, a numerical method, called discrete Fourier inversion, is developed. For this method, L-p-convergence rates can be obtained and, under certain regularity assumptions, mean square and L-p-convergence of the approximated field is proved. Further, a class of (time-dependent) Levy fields is introduced, where the point-wise marginal distributions are dependent but uncorrelated subordinated Wiener processes. For this specific class one may derive point-wise marginal distributions in closed form. Numerical examples, which include hyperbolic and normal-inverse Gaussian fields, demonstrate the efficiency of the approach.

Abstract

We propose a novel kernel-based method for image reconstruction from scattered Radon data. To this end, we employ generalized Hermite--Birkhoff interpolation by positive definite kernel functions. For radial kernels, however, a straightforward application of the generalized Hermite--Birkhoff interpolation method fails to work, as we prove in this paper. To obtain a well-posed reconstruction scheme for scattered Radon data, we introduce a new class of weighted positive definite kernels, which are symmetric but not radially symmetric. By our construction, the resulting weighted kernels are combinations of radial positive definite kernels and positive weight functions. This yields very flexible image reconstruction methods, which work for arbitrary distributions of Radon lines. We develop suitable representations for the weighted basis functions and the symmetric positive definite kernel matrices that are resulting from the proposed reconstruction scheme. For the relevant special case, where Gaussian radial kernels are combined with Gaussian weights, explicit formulae for the weighted Gaussian basis functions and the kernel matrices are given. Supporting numerical examples are finally presented.

Abstract

We study the problem of estimating the smallest achievable mean-squared error in regression function estimation. The problem is equivalent to estimating the second moment of the regression function of Y on X is an element of R-d . We introduce a nearest-neighbor-based estimate and obtain a normal limit law for the estimate when X has an absolutely continuous distribution, without any condition on the density. We also compute the asymptotic variance explicitly and derive a non-asymptotic bound on the variance that does not depend on the dimension d. The asymptotic variance does not depend on the smoothness of the density of X or of the regression function. A non-asymptotic exponential concentration inequality is also proved. We illustrate the use of the new estimate through testing whether a component of the vector X carries information for predicting Y.

Abstract

Abstract Currently, various hardware and software companies are developing augmented reality devices, most prominently Microsoft with its Hololens. Besides gaming, such devices can be used for serious pervasive applications, like interactive mobile simulations to support engineers in the field. Interactive simulations have high demands on resources, which the mobile device alone is unable to satisfy. Therefore, we propose a framework to support mobile simulations by distributing the computation between the mobile device and a remote server based on the reduced basis method. Evaluations show that we can speed-up the numerical computation by over 131 times while using 73 times less energy.

Abstract

A binary classification problem is considered. The excess error probability of the k-nearest-neighbor classification rule according to the error probability of the Bayes decision is revisited by a decomposition of the excess error probability into approximation and estimation errors. Under a weak margin condition and under a modified Lipschitz condition or a local Lipschitz condition, tight upper bounds are presented such that one avoids the condition that the feature vector is bounded. The concept of modified Lipschitz condition is applied for discrete distributions, too. As a consequence of both concepts, we present the rate of convergence of L-2 error for the corresponding nearest neighbor regression estimate.

Abstract

In this article, the authors prove the existence of a traveling wave solution for a gravity-driven thin film of a viscous incompressible dilatant fluid coated with an insoluble surfactant. The governing system of second-order partial differential equations for the film's height h and the surfactant's concentration γ is derived by means of lubrication theory applied to the non-Newtonian Navier-Stokes equations. (Note that there exists a well-known ansatz for modeling non-Newtonian fluids see, for example, D. Klimov and A. Petrov, Arch. Appl. Mech. 70 (2000), no. 1-3, 3–16, doi:10.1007/s004199900027.) The general ansatz and the estimation of the energy of the wave solution presented here will be of interest for people working in the specific field. The present article is an excellent development in this area of research.

Abstract

The solution of the wave equation in a polyhedral domain admits an asymptotic expansion in a neighborhood of the corners and edges. In this article we formulate boundary and screen problems for the wave equation as an equivalent boundary integral equations in time domain and study the regularity properties and numerical approximation of the solution. Guided by the theory for elliptic equations, graded meshes are shown to recover the optimal approximation rates expected for smooth solutions. Numerical experiments illustrate the theory for screen problems. In particular, we discuss the Dirichlet problem, the Dirichlet-to-Neumann operator and applications to the sound emitted by a tire.

Abstract

Abstract This article considers a unilateral contact problem for the wave equation. The problem is reduced to a variational inequality for the Dirichlet-to-Neumann operator for the wave equation on the boundary, which is solved in a saddle point formulation using boundary elements in the time domain. As a model problem, also a variational inequality for the single layer operator is considered. A priori estimates are obtained for Galerkin approximations both to the variational inequality and the mixed formulation in the case of a flat contact area, where the existence of solutions to the continuous problem is known. Numerical experiments demonstrate the performance of the proposed mixed method. They indicate the stability and convergence beyond flat geometries.

Abstract

Modern simulation scenarios frequently require multi-query or real-time responses of simulation models for statistical analysis, optimization, or process control. However, the underlying simulation models may be very time-consuming rendering the simulation task difficult or infeasible. This motivates the need for rapidly computable surrogate models. We address the case of surrogate modeling of functions from vectorial input to vectorial output spaces. These appear, for instance, in simulation of coupled models or in the case of approximating general input--output maps. We review some recent methods and theoretical results in the field of greedy kernel approximation schemes. In particular, we recall the vectorial kernel orthogonal greedy algorithm (VKOGA) for approximating vector-valued functions. We collect some recent convergence statements that provide sound foundation for these algorithms, in particular quasi-optimal convergence rates in case of kernels inducing Sobolev spaces. We provide some initial experiments that can be obtained with non-symmetric greedy kernel approximation schemes. The results indicate better stability and overall more accurate models in situations where the input data locations are not equally distributed.

Abstract

We derive novel criteria for robust stability and output-feedback gain-scheduling controller synthesis for a class of switched systems that are affected by piecewise constant parameters with dwell-time constraints. Our findings are based on clock-dependent Lyapunov arguments and rely, in contrast to other approaches, on separation techniques from robust control involving dynamic multipliers. The obtained conditions are expressed as infinite-dimensional linear matrix inequalities which can be solved by, e.g., using sum-of-squares relaxation methods. We illustrate our results with a numerical example.

Abstract

Nonparametric estimation of a quantile of a random variable m(X) is considered, where is a function which is costly to compute and X is a -valued random variable with a given density. An importance sampling quantile estimate of m(X), which is based on a suitable estimate of m, is defined, and it is shown that this estimate achieves a rate of convergence of order . The finite sample size behavior of the estimate is illustrated by simulated data.

Abstract

This paper is concerned with the study of Riesz means of the eigenvalues of the sub-Laplacian A(Ω)=−X^21−X^22, in the Heisenberg group H1 with Dirichlet boundary conditions on bounded domains Ω of ℝ3 (X1,X2 form a basis for the associated Heisenberg Lie algebra). The spectrum of the sub-Laplacian is purely discrete, and Hansson and Laptev proved an estimate for Tr(A(Ω)−λ)− in terms of the measure of Ω and λ3. The authors of this article improve this estimate and obtain an inequality with a sharp leading term and an additional lower-order term. The method that they employ does not rely on a Hardy inequality involving the distance to the boundary; instead they exploit the properties of the Carnot-Carathéodory metric associated to the Heisenberg setting.

Abstract

We investigate the influence of uncertain input parameters on the aeroacoustic feedback of cavity flows. The so-called Rossiter feedback requires a direct numerical computation of the acoustic noise, which solves hydrodynamics and acoustics simultaneously, in order to capture the interaction of acoustic waves and the hydrodynamics of the flow. Due to the large bandwidth of spatial and temporal scales, a high order numerical scheme with low dissipation and dispersion error is necessary to preserve important small scale information. Therefore, the open-source CFD solver FLEXI, which is based on a high-order discontinuous Galerkin spectral element method, is used to perform the aforementioned direct simulations of an open cavity configuration with a laminar upstream boundary layer. To analyse the precision of the deterministic cavity simulation with respect to random input parameters we establish a framework for uncertainty quantification. In particular, a non-intrusive spectral projection method with Legendre and Hermite polynomial basis functions is employed in order to treat uniform and normal probability distributions of the uncertain input. The results indicate a strong, non-linear dependency of the acoustic feedback mechanism on the investigated uncertain input parameters. An analysis of the stochastic results offers new insights into the noise generation process of open cavity flows and reveals the strength of the implemented uncertainty quantification framework.

Abstract

Summary In this work, we consider two kinds of model reduction techniques to simulate blood flow through the largest systemic arteries, where a stenosis is located in a peripheral artery i.e. in an artery that is located far away from the heart. For our simulations we place the stenosis in one of the tibial arteries belonging to the right lower leg (right post tibial artery). The model reduction techniques that are used are on the one hand dimensionally reduced models (1‐D and 0‐D models, the so‐called mixed‐dimension model) and on the other hand surrogate models produced by kernel methods. Both methods are combined in such a way that the mixed‐dimension models yield training data for the surrogate model, where the surrogate model is parametrised by the degree of narrowing of the peripheral stenosis. By means of a well‐trained surrogate model, we show that simulation data can be reproduced with a satisfactory accuracy and that parameter optimisation or state estimation problems can be solved in a very efficient way. Furthermore it is demonstrated that a surrogate model enables us to present after a very short simulation time the impact of a varying degree of stenosis on blood flow, obtaining a speedup of several orders over the full model.

Abstract

Intrusive Uncertainty Quantification methods such as stochastic Galerkin are gaining popularity, whereas the classical stochastic Galerkin approach is not ensured to preserve hyperbolicity of the underlying hyperbolic system. We present a modification of this method that uses a slope limiter to retain admissible solutions of the system, while providing high-order approximations in the physical and stochastic space. This is done using spatial discontinuous Galerkin and a Multi-Element stochastic Galerkin ansatz in the random space. We analyze the convergence of the resulting scheme and apply it to the compressible Euler equations with different uncertain initial states. The numerical results underline the strength of our method if discontinuities are present in the uncertainty of the solution.

Abstract

The motion of a viscous drop is investigated when the interface is fully covered with a stagnant layer of surfactant in an arbitrary unsteady Stokes flow for the low surface Péclet number limit. The effect of the interfacial slip coefficient on the behavior of the flow field is also considered. The hydrodynamic problem is solved by the solenoidal decomposition method and the drag force is computed in terms of Faxen's laws using a perturbation ansatz in powers of the surface Péclet number. The analytical expressions for the migration velocity of the drop are also obtained in powers of the surface Péclet number. Further instances corresponding to a given ambient flow as uniform flow, Couette flow, Poiseuille flow are analyzed. Moreover, it is observed that, a surfactant-induced cross-stream migration of the drop occur towards the centre-line in both Couette flow and Poiseuille flow cases. The variation of the drag force and migration velocity is computed for different parameters such as Péclet number, Marangoni number etc.

Abstract

Compton scattering imaging is a nascent concept arising from the current development of high-sensitive energy detectors and is devoted to exploit the scattering radiation to image the electron density of the studied medium. Such detectors are able to collect incoming photons in terms of energy. This paper introduces potential 3D modalities in Compton scattering imaging (CSI). The associated measured data are modeled using a class of generalized Radon transforms. The study of this class of operators leads to build a filtered backprojection kind algorithm preserving the contours of the sought-for function and offering a fast approach to partially solve the associated inverse problems. Simulation results including Poisson noise demonstrate the potential of this new imaging concept as well as the proposed image reconstruction approach.

Abstract

Mathematical models for compressible two-phase flow of homogeneous fluids that occur in a liquid and a vapour phase can be classified as either belonging to the class of sharp interface models or to the class of diffuse interface models. Sharp interface models display the phase boundary as a sharp front separating two bulk model domains while diffuse interface models consist of a single model on the complete domain of interest such that phase boundaries are represented as transition zones. This contribution is devoted to a self-consistent introduction to both model classes.

Abstract

The stretched Sierpinski gasket, SSG for short, is the space obtained by replacing every branching point of the Sierpinski gasket by an interval. It has also been called the ``deformed Sierpinski gasket'' or ``Hanoi attractor''. As a result, it is the closure of a countable union of intervals and one might expect that a diffusion on SSG is essentially a kind of gluing of the Brownian motions on the intervals. In fact, there have been several works in this direction. There still remains, however, ``reminiscence'' of the Sierpinski gasket in the geometric structure of SSG and the same should therefore be expected for diffusions. This paper shows that this is the case. In this work, we identify all the completely symmetric resistance forms on SSG. A completely symmetric resistance form is a resistance form whose restriction to every contractive copy of SSG in itself is invariant under all geometrical symmetries of the copy, which constitute the symmetry group of the triangle. We prove that completely symmetric resistance forms on SSG can be sums of the Dirichlet integrals on the intervals with some particular weights, or a linear combination of a resistance form of the former kind and the standard resistance form on the Sierpinski gasket.

Abstract

Abstract The Richards equation is a nonlinear parabolic equation that is commonly used for modelling saturated/unsaturated flow in porous media. We assume that the medium occupies a bounded Lipschitz domain partitioned into two disjoint subdomains separated by a fixed interface G . This leads to two problems defined on the subdomains which are coupled through conditions expressing flux and pressure continuity at G . After an Euler implicit discretisation of the resulting nonlinear subproblems, a linear iterative ( L -type) domain decomposition scheme is proposed. The convergence of the scheme is proved rigorously. In the last part we present numerical results that are in line with the theoretical finding, in particular the convergence of the scheme under mild restrictions on the time step size. We further compare the scheme to other approaches not making use of a domain decomposition. Namely, we compare to a Newton and a Picard scheme. We show that the proposed scheme is more stable than the Newton scheme while remaining comparable in computational time, even if no parallelisation is being adopted. After presenting a parametric study that can be used to optimise the proposed scheme, we briefly discuss the effect of parallelisation and give an example of a four-domain implementation.

Abstract

We consider the motion of a viscous drop in an arbitrary unsteady Stokes flow such that the surface of the drop is fully covered with a stagnant surfactant layer. In particular the regime of low surface Péclet number is analyzed and we account for the effect interfacial slip on the overall behavior of the flow field. The hydrodynamic problem is solved by the solenoidal decomposition method and the drag force is computed in terms of Faxen’s laws, using a perturbation ansatz in powers of the surface Péclet number. The surface equation of the deformed sphere has been determined by an iterative method up to the first order approximation. Analytical expressions for the migration velocity of the drop are likewise given. Based on this analysis we can completely characterize various flow situations including a given ambient flow as uniform flow, Couette flow and Poiseuille flow. Moreover, we find out that a surfactant-induced cross-stream migration of the drop occurs towards the center-line in both, Couette and Poiseuille flow cases. The variation of the drag force and the migration velocity is computed for different parameters such as the Péclet number and the Marangoni number. Finally, the theoretical findings are validated on with available experimental data.

Abstract

We are interested in approximating vector-valued functions on a compact set $ØmegaR^d$. We consider reproducing kernel Hilbert spaces of $R^m$-valued functions which each admit a unique matrix-valued reproducing kernel $k$. These spaces seem promising, when modelling correlations between the target function components. The approximation of a function is a linear combination of matrix-valued kernel evaluations multiplied with coefficient vectors. To guarantee a fast evaluation of the approximant the expansion size, i.e. the number of centers $n$ is desired to be small. We thus present three different greedy algorithms by which a suitable set of centers is chosen in an incremental fashion: First, the $P$-Greedy which requires no function evaluations, second and third, the $f$-Greedy and $f/P$-Greedy which require function evaluations but produce centers tailored to the target function. The efficiency of the approaches is investigated on some data from an artificial model.

Abstract

Distributed adaptive conforming refinement requires multiple iterations of the serial refinement algorithm and global communication as the refinement can be propagated over several processor boundaries. We show bounds on the maximum number of iterations. The algorithm is implemented within the open-source software package Dune-ALUGrid.

Abstract

We investigate feedback control for infinite horizon optimal control problems for partial differential equations. The method is based on the coupling between Hamilton-Jacobi-Bellman (HJB) equations and model reduction techniques. It is well-known that HJB equations suffer the so called curse of dimensionality and, therefore, a reduction of the dimension of the system is mandatory. In this report we focus on the infinite horizon optimal control problem with quadratic cost functionals. We compare several model reduction methods such as Proper Orthogonal Decomposition, Balanced Truncation and a new algebraic Riccati equation based approach. Finally, we present numerical examples and discuss several features of the different methods analyzing advantages and disadvantages of the reduction methods.

Abstract

We present a novel acceleration method for the solution of parametric ODEs by single-step implicit solvers by means of greedy kernel-based surrogate models. In an offline phase, a set of trajectories is precomputed with a high-accuracy ODE solver for a selected set of parameter samples, and used to train a kernel model which predicts the next point in the trajectory as a function of the last one. This model is cheap to evaluate, and it is used in an online phase for new parameter samples to provide a good initialization point for the nonlinear solver of the implicit integrator. The accuracy of the surrogate reflects into a reduction of the number of iterations until convergence of the solver, thus providing an overall speedup of the full simulation. Interestingly, in addition to providing an acceleration, the accuracy of the solution is maintained, since the ODE solver is still used to guarantee the required precision. Although the method can be applied to a large variety of solvers and different ODEs, we will present in details its use with the Implicit Euler method for the solution of the Burgers equation, which results to be a meaningful test case to demonstrate the method's features.

Abstract

In the recent paper 1, a new method to compute stable kernel-based interpolants has been presented. This rescaled interpolation method combines the standard kernel interpolation with a properly defined rescaling operation, which smooths the oscillations of the interpolant. Although promising, this procedure lacks a systematic theoretical investigation. Through our analysis, this novel method can be understood as standard kernel interpolation by means of a properly rescaled kernel. This point of view allows us to consider its error and stability properties.

Abstract

We classify polar actions on complex hyperbolic spaces up to orbit equivalence.

Abstract

"Numerical evidence is provided that there are non-constant permittivity profiles which force solutions to a two-dimensional coupled moving boundary problem modelling microelectromechanical systems to be positive, while the corresponding small-aspect ratio model produces solutions which are always non-positive.''

Abstract

"In this survey we review some recent results on microelectromechanical systems with general permittivity profile. Different systems of differential equations are derived by taking various physical modelling aspects into account, according to the particular application. In any case an either semi- or quasilinear hyperbolic or parabolic evolution problem for the displacement of an elastic membrane is coupled with an elliptic moving boundary problem that determines the electrostatic potential in the region occupied by the elastic membrane and a rigid ground plate. Of particular interest in all models is the influence of different classes of permittivity profiles. "The subsequent analytical investigations are restricted to a dissipation dominated regime for the membrane's displacement. For the resulting parabolic evolution problems local well-posedness, global existence, the occurrence of finite-time singularities, and convergence of solutions to those of the so-called small-aspect ratio model, respectively, are investigated. Furthermore, a topic is addressed that is of note not till non-constant permittivity profiles are taken into account—the direction of the membrane's deflection or, in mathematical parlance, the sign of the solution to the evolution problem. The survey is completed by a presentation of some numerical results that in particular justify the consideration of the coupled problem by revealing substantial qualitative differences of the solutions to the widely-used small-aspect ratio model and the coupled problem.''

Abstract

In this article we present an a posteriori error estimator for the spatial-stochastic error of a Galerkin-type discretisation of an initial value problem for a random hyperbolic conservation law. For the stochastic discretisation we use the Stochastic Galerkin method and for the spatial-temporal discretisation of the Stochastic Galerkin system a Runge-Kutta Discontinuous Galerkin method. The estimator is obtained using smooth reconstructions of the discrete solution. Combined with the relative entropy stability framework of Dafermos dafermos2005hyperbolic, this leads to computable error bounds for the space-stochastic discretisation error. \\ Moreover, it turns out that the error estimator admits a splitting into one part representing the spatial error, and a remaining term, which can be interpreted as the stochastic error. This decomposition allows us to balance the errors arising from spatial and stochastic discretisation. We conclude with some numerical examples confirming the theoretical findings.

Abstract

Movements of the object during the data collection in computerized tomography can introduce motion artefacts in the reconstructed image. They can be reduced by employing information about the dynamic behaviour within the reconstruction step. However, inaccuracies concerning the movement are inevitable in practice. In this article, we give an explicit characterization of what is visible in an image obtained by a reconstruction algorithm with incorrect motion information. Then, we use this result to study in detail the situation of locally deforming objects, i.e. individual parts of the object have a different dynamic behaviour. In this context, we prove that additional artefacts arise due to the global nature of the Radon transform, even if the motion is exactly known. Based on our analysis, we propose a numerical scheme to reduce these artefacts in the reconstructed image. All our results are illustrated by numerical examples.

Abstract

A main challenge in computerized tomography consists in imaging moving objects. Temporal changes during the measuring process lead to inconsistent data sets, and applying standard reconstruction techniques causes motion artefacts which can severely impose a reliable diagnostics. Therefore, novel reconstruction techniques are required which compensate for the dynamic behavior. This article builds on recent results from a microlocal analysis of the dynamic setting, which enable us to formulate efficient analytic motion compensation algorithms for contour extraction. Since these methods require information about the dynamic behavior, we further introduce a motion estimation approach which determines parameters of affine and certain non-affine deformations directly from measured motion-corrupted Radon-data. Our methods are illustrated with numerical examples for both types of motion.

Abstract

"In the present article we will give a new proof of the ground state asymptotics of the Dirichlet Laplacian with a Neumann window acting on functions which are defined on a two-dimensional infinite strip or a three-dimensional infinite layer. The proof is based on the analysis of the corresponding Dirichlet-to-Neumann operator as a first order classical pseudo-differential operator. Using the explicit representation of its symbol we prove an asymptotic expansion as the window length decreases."

Abstract

Verl.Beschr.: Dieses Buch bietet Ihnen eine Übersicht aller fundamentalen Problemstellungen in Übungs- und Klausuraufgaben der Mathematikgrundvorlesungen für Ingenieure, Natur- und Wirtschaftswissenschaftler. Es enthält eine Auswahl von mehr als 400 typischen Aufgaben, gegliedert in 50 Themengebiete, mit detaillierten Lösungsskizzen und Beschreibungen der relevanten mathematischen Methoden. Zusätzliche Beispiele und Erläuterungen finden Sie in einer auf das Buch abstimmten internetbasierten Sammlung von Lehrmaterialien, die auch von Dozenten in ihren Vorlesungen verwendet werden kann