Helmut Harbrecht’s research while affiliated with University of Basel and other places

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.

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.

The distribution of centered Gaussian random fields (GRFs) indexed by compacta such as smooth, bounded Euclidean domains or smooth, compact and orientable manifolds is determined by their covariance operators. We consider centered GRFs given as variational solutions to coloring operator equations driven by spatial white noise, with an elliptic self-adjoint pseudodifferential coloring operator from the Hörmander class. This includes the Matérn class of GRFs as a special case. Using biorthogonal multiresolution analyses on the manifold, we prove that the precision and covariance operators, respectively, may be identified with bi-infinite matrices and finite sections may be diagonally preconditioned rendering the condition number independent of the dimension p of this section. We prove that a tapering strategy by thresholding applied on finite sections of the bi-infinite precision and covariance matrices results in optimally numerically sparse approximations. That is, asymptotically only linearly many nonzero matrix entries are sufficient to approximate the original section of the bi-infinite covariance or precision matrix using this tapering strategy to arbitrary precision. The locations of these nonzero matrix entries can be determined a priori. The tapered covariance or precision matrices may also be optimally diagonally preconditioned. Analysis of the relative size of the entries of the tapered covariance matrices motivates novel, multilevel Monte Carlo (MLMC) oracles for covariance estimation, in sample complexity that scales log-linearly with respect to the number p of parameters. In addition, we propose and analyze novel compressive algorithms for simulating and kriging of GRFs. The complexity (work and memory vs. accuracy) of these three algorithms scales near-optimally in terms of the number of parameters p of the sample-wise approximation of the GRF in Sobolev scales.

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.

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.

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 embedded fast multipole method, which is based on interpolation of the kernel function under consideration on the reference domain, rather than in space. To overcome the prohibitive cost of the potential evaluation in case of many evaluation points, we also accelerate the potential evaluation by a fast multipole method which interpolates in space. The result is a frequency stable algorithm that scales essentially linear in the number of degrees of freedom and potential points. Numerical experiments are performed which show the feasibility and the performance of the approach.

The present article considers the quantification of uncertainty for the equations of linear elasticity on random domains. To this end, we model the random domains as the images of some given fixed, nominal domain under random domain mappings, which are defined by a Karhunen-Loève expansion. We then prove the analytic regularity of the random solution with respect to the countable random input parameters which enter the problem through the Karhunen-Loève expansion of the random domain mappings. In particular, we provide appropriate bounds on arbitrary derivatives of the random solution with respect to those input parameters, which enable the use of state-of-the-art quadrature methods to compute deterministic statistics such as the mean and variance of quantities of interest such as the random solution itself or the random von Mises stress as integrals over the countable random input parameters in a dimensionally robust way. Numerical examples qualify and quantify the theoretical findings.

The present article is devoted to wavelet matrix compression for boundary integral equations when using anisotropic wavelet bases for the discretization. We provide a compression scheme for boundary integral operators of order 2q2q on patchwise smooth and globally Lipschitz continuous mainfolds which amounts to only O(N)$\mathcal {O}(N)$ relevant matrix coefficients in the system matrix without deteriorating the accuracy offered by the underlying Galerkin scheme. Here, NN denotes the degrees of freedom in the related trial spaces. By numerical results we validate our theoretical findings.

This paper 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 [Formula: see text]-Gevrey assumptions on the residual equation, we establish [Formula: see text]-Gevrey bounds on the Fréchet derivatives of the locally defined 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.

Opisthorchis viverrini is a parasitic liver fluke contracted by consumption of raw fish, which affects over 10 million people in Southeast Asia despite sustained control efforts. Chronic infections are a risk factor for the often fatal bile duct cancer, cholangiocarcinoma. Previous modeling predicted rapid elimination of O. viverrini following yearly mass drug administration (MDA) campaigns. However, field data collected in affected populations shows persistence of infection, including heavy worm burden, after many years of repeated interventions. A plausible explanation for this observation is systematic adherence of individuals in health campaigns, such as MDA and education, with some individuals consistently missing treatment. We developed an agent-based model of O. viverrini which allows us to introduce various heterogeneities including systematic adherence to MDA and education campaigns at the individual level. We validate the agent-based model by comparing it to a previously published population-based model. We estimate the degree of systematic adherence to MDA and education campaigns indirectly, using epidemiological data collected in Lao PDR before and after 5 years of repeated MDA, education and sanitation improvement campaigns. We predict the impact of interventions deployed singly and in combination, with and without the estimated systematic adherence. We show how systematic adherence can substantially increase the time required to achieve reductions in worm burden. However, we predict that yearly MDA campaigns alone can result in a strong reduction of moderate and heavy worm burden, even under systematic adherence. We predict latrines and education campaigns to be particularly important for the reduction in overall prevalence, and therefore, ultimately, elimination. Our findings show how systematic adherence can explain the observed persistence of worm burden; while emphasizing the benefit of interventions for the entire population, even under systematic adherence. At the same time, the results highlight the substantial opportunity to further reduce worm burden if patterns of systematic adherence can be overcome.