arXiv subjects

Helmut Harbrecht

Publications and source records attributed to Helmut Harbrecht.

At least 19 recordsLinked to original sources

arXiv2026

Optimal compression of kernel matrices by interpolets with application to high-dimensional approximation

We consider the compression of kernel matrices on the unit interval $[0,1]$ by interpolets that have sufficiently many vanishing moments. We define a compression rule which discards most matrix coefficients without compromising the accuracy offered by the underlying discretization. Since interpolets can be scaled such that the compressed kernel matrices are well conditioned, we derive a fully discrete scheme that solves a kernel interpolation problem under consideration in linear overall complexity. We finally generalize this approach to the unit $n$-cube $[0,1]^n$ by means of the sparse grid combination technique. Numerical experiments are carried out to validate the theoretical findings.

math.NA↗

arXiv2026

Shape optimization for thermoelasticity with temperature-dependent material parameters

We consider the numerical solution of shape optimization problems for thermoelasticity with temperature-dependent material parameters. We show the existence of the shape derivative and derive an expression for generic functionals of domain integral type. Numerical results are presented for two settings: minimization of the compliance under a volume constraint, and minimization of the volume under a constraint on the $L^2$-norm of the von Mises stress. We use the finite element method for solving the underlying boundary value problems and the level set method for the representation of the actual domain.

math.OC↗

arXiv2026

Adaptive space-time BEM for the heat equation with Neumann boundary conditions

We consider the space-time boundary element method (BEM) for the heat equation with prescribed initial and Neumann data. We propose a weighted-residual a posteriori error estimator that is an upper bound for the unknown BEM error. The possibly locally refined meshes are assumed to be parabolically scaled prismatic, i.e., their elements are tensor-products $J\times K$ of elements in time $J$ and space $K$ with $|J| \eqsim \text{diam}(K)^2$. In the considered numerical experiments on two-dimensional domains in space, an adaptive algorithm steered by the derived estimator yields significantly faster convergence compared to uniform refinement, achieving near-optimal rates even in the presence of strong singularities.

math.NA↗

arXiv2026

Simulation of Gaussian random fields on surfaces using the isogeometric finite element method

We are concerned with the fast simulation of random fields on closed surfaces in $\mathbb{R}^3$ which are generated by the (Whittle-) Mat\'ern class of covariance functions. To this end, we solve the underlying fractional stochastic partial differential equation with additive white noise by using an isogeometric finite element method on the surface in combination with the Balakrishnan integral representation of the solution. The solution of the underlying linear system of equations is performed by means of a geometric multigrid method that naturally underlies the isogeometric approach. Numerical results are presented to demonstrate the approach.

math.NA↗

arXiv2026

Neural Shape Operator Surrogates -- Expression Rate Bounds

We prove error bounds for operator surrogates of solution operators for partial differential and boundary integral equations on families of domains which are diffeomorphic to one common reference (or latent) domain $D_{ref}$. The pullback of the PDE to $D_{ref}$ via affine-parametric shape encoding produces a collection of holomorphic parametric PDEs on $D_{ref}$. Sufficient conditions for (uniformly with respect to the parameter) well-posedness are given, implying existence, uniqueness and stability of parametric solution families on $D_{ref}$. We illustrate the abstract hypotheses by reviewing recent holomorphy results for a suite of elliptic and parabolic PDEs. Quantified parametric holomorphy implies existence of finite-parametric, discrete approximations of the parametric solution families with convergence rates in terms of the number $N$ of parameters. We obtain constructive proofs of existence of Neural and Spectral Operator surrogates for the shape-to-solution maps with error bounds and convergence rate guarantees uniform on the collection of admissible shapes. We admit principal-component shape encoders and frame decoders. Our results support in particular the (empirically reported) ability of neural operators to realize data-to-solution maps for elliptic and parabolic PDEs and BIEs that generalize across parametric families of shapes.

cs.LG↗

arXiv2026

Elliptic PDEs on log-Gaussian Shapes: Sparsity and Finite Element Discretization

In this article, we consider the solution to elliptic diffusion problems on a class of random domains obtained by log-Gaussian random homothety of the unit disk respectively an annulus. We model the problem under consideration and verify the existence and uniqueness of the random solution by path-wise pullback to the nominal unit disk respectively annulus. We prove the analytic regularity of the solution with respect to the random input parameter. We consider the numerical approximation of the random diffusion problem by means of continuous, piecewise linear Lagrangian Galerkin Finite Elements with numerical quadrature in the nominal domain, and by sparse grid interpolation and quadrature of Gauss-Hermite Smolyak and Quasi-Monte Carlo type in the parameter domain. The theoretical findings are complemented by numerical results.

math.NA↗

arXiv2025

An inverse problem for the one-phase Stefan problem with varying melting temperature

The present article is dedicated to the forward and backward solution of a transient one-phase Stefan problem. In the forward problem, we compute the evolution of the initial domain for a Stefan problem where the melting temperature varies over time. This occurs in practice, for example, when the pressure in the external space changes in time. In the corresponding backward problem, we then reconstruct the time-dependent melting temperature from the knowledge of the evolving geometry. We develop respective numerical algorithms using a moving mesh finite element method and provide numerical simulations.

math.NA↗

arXiv2025

Kernel interpolation on generalized sparse grids

We consider scattered data approximation on product regions of equal and different dimensionality. On each of these regions, we assume quasi-uniform but unstructured data sites and construct optimal sparse grids for scattered data interpolation on the product region. For this, we derive new improved error estimates for the respective kernel interpolation error by invoking duality arguments. An efficient algorithm to solve the underlying linear system of equations is proposed. The algorithm is based on the sparse grid combination technique, where a sparse direct solver is used for the elementary anisotropic tensor product kernel interpolation problems. The application of the sparse direct solver is facilitated by applying a samplet matrix compression to each univariate kernel matrix, resulting in an essentially sparse representation of the latter. In this way, we obtain a method that is able to deal with large problems up to billions of interpolation points, especially in case of reproducing kernels of nonlocal nature. Numerical results are presented to qualify and quantify the approach.

math.NA↗

arXiv2025

Optimization of the cut configuration for skin grafts

The subject of this work is the problem of optimizing the configuration of cuts for skin grafting in order to improve the efficiency of the procedure. We consider the optimization problem in the framework of a linear elasticity model. We choose three mechanical measures that define optimality via related objective functionals: the compliance, the $L^p$-norm of the von Mises stress, and the area of the stretched skin. We provide a proof of the existence of solutions for each problem, but we cannot claim uniqueness. We compute the gradient of the objectives with respect to the cut configuration using shape calculus concepts. To solve the problem numerically, we use the gradient descent method, which performs well under uniaxial stretching. However, in more complex cases, such as multidirectional stretching, its effectiveness is limited due to low sensitivity of the functionals. To avoid this difficulty, we use a combination of the genetic algorithm and the gradient descent method, which leads to a significant improvement in the results.

math.OC↗

arXiv2025

On the Compressibility of Integral Operators in Anisotropic Wavelet Coordinates

The present article is concerned with the s*-compressibility of classical boundary integral operators in anisotropic wavelet coordinates. Having the s*-compressibility at hand, one can design adaptive wavelet algorithms which are asymptotically optimal, meaning that any target accuracy can be achieved at a computational expense that stays proportional to the number of degrees of freedom (within the setting determined by an underlying wavelet basis) that would ideally be necessary for realising that target accuracy if full knowledge about the unknown solution were given. As we consider anisotropic wavelet bases, we can achieve higher convergence rates compared to the standard, isotropic setting. Especially, edge singularities of anisotropic nature can be resolved.

math.NA↗

arXiv2025

Samplets: Wavelet concepts for scattered data

This chapter is dedicated to recent developments in the field of wavelet analysis for scattered data. We introduce the concept of samplets, which are signed measures of wavelet type and may be defined on sets of arbitrarily distributed data sites in possibly high dimension. By employing samplets, we transfer well-known concepts known from wavelet analysis, namely the fast basis transform, data compression, operator compression and operator arithmetics to scattered data problems. Especially, samplet matrix compression facilitates the rapid solution of scattered data interpolation problems, even for kernel functions with nonlocal support. Finally, we demonstrate that sparsity constraints for scattered data approximation problems become meaningful and can efficiently be solved in samplet coordinates.

math.NA↗

arXiv2024

On Sobolev and Besov Spaces of Hybrid Regularity

The present article is concerned with the nonlinear approximation of functions in the Sobolev space H^q with respect to a tensor-product, or hyperbolic wavelet basis on the unit n-cube. Here, q is a real number, which is not necessarily positive. We derive Jackson and Bernstein inequalities to obtain that the approximation classes contain Besov spaces of hybrid regularity. Especially, we show that all functions that can be approximated by classical wavelets are also approximable by tensor-product wavelets at least at the same rate. In particular, this implies that for nonnegative regularity, the classical Besov spaces of regularity q+sn, integrability and weak index t, with 1/t = s + 1/2, are included in the Besov spaces of hybrid regularity with isotropic regularity q and additional mixed regularity s.

math.FA↗

arXiv2024

On Quasi-Localized Dual Pairs in Reproducing Kernel Hilbert Spaces

In scattered data approximation, the span of a finite number of translates of a chosen radial basis function is used as approximation space and the basis of translates is used for representing the approximate. However, this natural choice is by no means mandatory and different choices, like, for example, the Lagrange basis, are possible and might offer additional features. In this article, we discuss different alternatives together with their canonical duals. We study a localized version of the Lagrange basis, localized orthogonal bases, such as the Newton basis, and multiresolution versions thereof, constructed by means of samplets. We argue that the choice of orthogonal bases is particularly useful as they lead to symmetric preconditioners. All bases under consideration are compared numerically to illustrate their feasibility for scattered data approximation. We provide benchmark experiments in two spatial dimensions and consider the reconstruction of an implicit surface as a relevant application from computer graphics.

math.NA↗

arXiv2024

Two-norm discrepancy and convergence of the stochastic gradient method with application to shape optimization

The present article is dedicated to proving convergence of the stochastic gradient method in case of random shape optimization problems. To that end, we consider Bernoulli's exterior free boundary problem with a random interior boundary. We recast this problem into a shape optimization problem by means of the minimization of the expected Dirichlet energy. By restricting ourselves to the class of convex, sufficiently smooth domains of bounded curvature, the shape optimization problem becomes strongly convex with respect to an appropriate norm. Since this norm is weaker than the differentiability norm, we are confronted with the so-called two-norm discrepancy, a well-known phenomenon from optimal control. We therefore need to adapt the convergence theory of the stochastic gradient method to this specific setting correspondingly. The theoretical findings are supported and validated by numerical experiments.

math.OC↗

arXiv2024

Wavelet compressed, modified Hilbert transform in the space-time discretization of the heat equation

On a finite time interval $(0,T)$, we consider the multiresolution Galerkin discretization of a modified Hilbert transform $\mathcal H_T$ which arises in the space-time Galerkin discretization of the linear diffusion equation. To this end, we design spline-wavelet systems in $(0,T)$ consisting of piecewise polynomials of degree $\geq 1$ with sufficiently many vanishing moments which constitute Riesz bases in the Sobolev spaces $H^{s}_{0,}(0,T)$ and $H^{s}_{,0}(0,T)$. These bases provide stable multilevel splittings of the temporal discretization spaces into "increment" or "detail" spaces. Furthermore, they allow to optimally compress the nonlocal integrodifferential operators which appear in stable space-time variational formulations of initial-boundary value problems, such as the heat equation and the acoustic wave equation. We then obtain sparse space-time tensor-product spaces via algebraic tensor-products of the temporal multilevel discretizations with standard, hierarchic finite element spaces in the spatial domain (with standard Lagrangian FE bases). Hence, the construction of multiresolutions in the spatial domain is not necessary. An efficient multilevel preconditioner is proposed that solves the linear system of equations resulting from the sparse space-time Galerkin discretization with essentially linear complexity (in work and memory). A substantial reduction in the number of the degrees of freedom and CPU time (compared to time-marching discretizations) is demonstrated in numerical experiments.

math.NA↗

arXiv2024

The dimension weighted fast multipole method for scattered data approximation

The present article is concerned scattered data approximation for higher dimensional data sets which exhibit an anisotropic behavior in the different dimensions. Tailoring sparse polynomial interpolation to this specific situation, we derive very efficient degenerate kernel approximations which we then use in a dimension weighted fast multipole method. This dimension weighted fast multipole method enables to deal with many more dimensions than the standard black-box fast multipole method based on interpolation. A thorough analysis of the method is provided including rigorous error estimates. The accuracy and the cost of the approach are validated by extensive numerical results. As a relevant application, we apply the approach to a shape uncertainty quantification problem.

math.NA↗

arXiv2023

The Gevrey class implicit mapping theorem with application to UQ of semilinear elliptic PDEs

This article is concerned with a regularity analysis of parametric operator equations with a perspective on uncertainty quantification. We study the regularity of mappings between Banach spaces near branches of isolated solutions that are implicitly defined by a residual equation. Under $s$-Gevrey assumptions on on the residual equation, we establish $s$-Gevrey bounds on the Fr\'echet derivatives of the local data-to-solution mapping. This abstract framework is illustrated in a proof of regularity bounds for a semilinear elliptic partial differential equation with parametric and random field input.

math.AP↗

arXiv2023

Solving acoustic scattering problems by the isogeometric boundary element method

We solve acoustic scattering problems by means of the isogeometric boundary integral equation method. In order to avoid spurious modes, we apply the combined field integral equations for either sound-hard scatterers or sound-soft scatterers. These integral equations are discretized by Galerkin's method, which especially enables the mathematically correct regularization of the hypersingular integral operator. In order to circumvent densely populated system matrices, we employ the isogeometric fast multipole method. The result is an algorithm that scales essentially linear in the number of boundary elements. Numerical experiments are performed which show the feasibility and the performance of the approach.

math.NA↗