arXiv subjects

Jürgen Dölz

Publications and source records attributed to Jürgen Dölz.

At least 19 recordsLinked to original sources

arXiv2026

Data sparse multilevel covariance estimation in optimal complexity

We consider the $\calH^2$-formatted compression and computational estimation of covariance functions on a compact set in $\bbR^d$. The classical sample covariance or Monte Carlo estimator is prohibitively expensive for many practically relevant problems, where often approximation spaces with many degrees of freedom and many samples for the estimator are needed. In this article, we propose and analyze a data sparse multilevel sample covariance estimator, i.e., a multilevel Monte Carlo estimator. For this purpose, we generalize the notion of asymptotically smooth kernel functions to a Gevrey type class of kernels for which we derive new variable-order $\calH^2$-approximation rates. These variable-order $\calH^2$-approximations can be considered as a variant of $hp$-approximations. Our multilevel sample covariance estimator then uses an approximate multilevel hierarchy of variable-order $\calH^2$-approximations to compress the sample covariances on each level. The non-nestedness of the different levels makes the reduction to the final estimator nontrivial and we present a suitable algorithm which can handle this task in linear complexity. This allows for a data sparse multilevel estimator of Gevrey covariance kernel functions in the best possible complexity for Monte Carlo type multilevel estimators, which is quadratic. Numerical examples which estimate covariance matrices with tens of billions of entries are presented.

math.NA↗

arXiv2026

Beyond Lipschitz: Data-Driven Robustness via Discrete Modulus of Continuity

Robustness of neural networks is commonly quantified via local or global Lipschitz constants. However, Lipschitz continuity can be overly coarse or overly restrictive as global robustness measure, failing to capture nuanced, data-dependent behavior. We propose a data-driven, architecture-agnostic framework based on the discrete modulus of continuity (DMOC), a non linear generalization of Lipschitz continuity that provides a finer notion of robustness. Unlike many existing approaches, DMOC does not require access to model internals and instead evaluates regularity relative to the data distribution. This shifts the focus from the model to the data, which provide a data-driven baseline of regularity against which the network's robustness is assessed. We establish convergence results for DMOC-induced seminorms with explicit data-driven rates in terms of the separation distance, and introduce a scalable minibatch algorithm that reduces the quadratic cost of exact computation, enabling application to large-scale data sets such as ImageNet. Empirically, DMOC serves as an architecture independent diagnostic: it distinguishes trained from untrained networks, reveals underfitting and overfitting regimes, and yields, as a special case, tight Lipschitz estimates comparable to state-of-the-art method such as ECLipsE and ECLipsE-fast.

stat.ML↗

arXiv2026

Data-intrinsic approximation in metric spaces

Analysis and processing of data is a vital part of our modern society and requires vast amounts of computational resources. To reduce the computational burden, compressing and approximating data has become a central topic. We consider the approximation of labeled data samples, mathematically described as site-to-value maps between finite metric spaces. Within this setting, we identify the discrete modulus of continuity as an effective data-intrinsic quantity to measure regularity of site-to-value maps without imposing further structural assumptions. We investigate the consistency of the discrete modulus of continuity in the infinite data limit and propose an algorithm for its efficient computation. Building on these results, we present a sample based approximation theory for labeled data. For data subject to statistical uncertainty we consider multilevel approximation spaces and a variant of the multilevel Monte Carlo method to compute statistical quantities of interest. Our considerations connect approximation theory for labeled data in metric spaces to the covering problem for (random) balls on the one hand and the efficient evaluation of the discrete modulus of continuity to combinatorial optimization on the other hand. We provide extensive numerical studies to illustrate the feasibility of the approach and to validate our theoretical results.

math.NA↗

arXiv2025

Local sensitivity analysis for Bayesian inverse problems

We present an extension of local sensitivity analysis, also referred to as the perturbation approach for uncertainty quantification, to Bayesian inverse problems. More precisely, we show how moments of random variables with respect to the posterior distribution can be approximated efficiently by asymptotic expansions. This is under the assumption that the measurement operators and prediction functions are sufficiently smooth and their corresponding stochastic moments with respect to the prior distribution exist. Numerical experiments are presented to the illustrate the theoretical results.

math.NA↗

arXiv2025

Fully discrete analysis of the Galerkin POD neural network approximation with application to 3D acoustic wave scattering

In this work, we consider the approximation of parametric maps using the so-called Galerkin POD-NN method. This technique combines the computation of a reduced basis via proper orthogonal decomposition (POD) and artificial neural networks (NNs) for the construction of fast surrogates of said parametric maps. In contrast to the existing literature, which has studied the approximation properties of this kind of architecture on a continuous level, we provide a fully discrete error analysis of this approach. More precisely, our estimates also account for discretization errors during the construction of the NN architecture. We consider the number of reduced basis in the approximation of the solution manifold, truncation in the parameter space, and, most importantly, the number of samples in the computation of the reduced space, together with the effect of the use of NNs in the approximation of the reduced coefficients. Following this error analysis, we provide a-priori bounds on the required POD tolerance, the resulting POD ranks, and NN parameters to maintain the order of convergence of quasi Monte Carlo sampling techniques. We conclude this work by showcasing the applicability of this method through a practical industrial application: the sound-soft acoustic scattering problem by a parametrically defined scatterer in three physical dimensions.

math.NA↗

arXiv2025

Quantifying uncertainty in spectral clusterings: expectations for perturbed and incomplete data

Spectral clustering is a popular unsupervised learning technique which is able to partition unlabelled data into disjoint clusters of distinct shapes. However, the data under consideration are often experimental data, implying that the data is subject to measurement errors and measurements may even be lost or invalid. These uncertainties in the corrupted input data induce corresponding uncertainties in the resulting clusters, and the clusterings thus become unreliable. Modelling the uncertainties as random processes, we discuss a mathematical framework based on random set theory for the computational Monte Carlo approximation of statistically expected clusterings in case of corrupted, i.e., perturbed, incomplete, and possibly even additional, data. We propose several computationally accessible quantities of interest and analyze their consistency in the infinite data point and infinite Monte Carlo sample limit. Numerical experiments are provided to illustrate and compare the proposed quantities.

stat.ML↗

arXiv2024

A Low-Frequency-Stable Higher-Order Isogeometric Discretization of the Augmented Electric Field Integral Equation

This contribution investigates the connection between isogeometric analysis and integral equation methods for full-wave electromagnetic problems up to the low-frequency limit. The proposed spline-based integral equation method allows for an exact representation of the model geometry described in terms of non-uniform rational B-splines without meshing. This is particularly useful when high accuracy is required or when meshing is cumbersome for instance during optimization of electric components. The augmented electric field integral equation is adopted and the deflation method is applied, so the low-frequency breakdown is avoided. The extension to higher-order basis functions is analyzed and the convergence rate is discussed. Numerical experiments on academic and realistic test cases demonstrate the high accuracy of the proposed approach.

cs.CE↗

arXiv2024

Parametric Shape Holomorphy of Boundary Integral Operators with Applications

We consider a family of boundary integral operators supported on a collection of parametrically defined bounded Lipschitz boundaries. Consequently, the boundary integral operators themselves also depend on the parametric variables, thus leading to a parameter-to-operator map. The main result of this article is to establish the analytic or holomorphic dependence of said boundary integral operators upon the parametric variables, i.e., of the parameter-to-operator map. As a direct consequence we also establish holomorphic dependence of solutions to boundary integral equations, i.e.,~holomorphy of the parameter-to-solution map. To this end, we construct a holomorphic extension to complex-valued boundary deformations and investigate the \emph{complex} Fréchet differentiability of boundary integral operators with respect to each parametric variable. The established parametric holomorphy results have been identified as a key property to overcome the so-called curse of dimensionality in the approximation of parametric maps with distributed, high-dimensional inputs. To demonstrate the applicability of the derived results, we consider as a concrete example the sound-soft Helmholtz acoustic scattering problem and its frequency-robust boundary integral formulations. For this particular application, we explore the consequences of our results in reduced order modelling, Bayesian shape inversion, and the construction of efficient surrogates using artificial neural networks.

math.NA↗

arXiv2024

Isogeometric Shape Optimization of Multi-Tapered Coaxial Baluns Simulated by an Integral Equation Method

We discuss the advantages of a spline-based freeform shape optimization approach using the example of a multi-tapered coaxial balun connected to a spiral antenna. The underlying simulation model is given in terms of a recently proposed isogeometric integral equation formulation, which can be interpreted as a high-order generalization of the partial element equivalent circuit method. We demonstrate a significant improvement in the optimized design, i.e., a reduction in the magnitude of the scattering parameter over a wide frequency range.

cs.CE↗

arXiv2024

Shape uncertainty quantification of Maxwell eigenvalues and -modes with application to TESLA cavities

We consider Maxwell eigenvalue problems on uncertain shapes with perfectly conducting TESLA cavities being the driving example. Due to the shape uncertainty the resulting eigenvalues and eigenmodes are also uncertain and it is well known that the eigenvalues may exhibit crossings or bifurcations under perturbation. We discuss how the shape uncertainties can be modelled using the domain mapping approach and how the deformation mapping can be expressed as coefficients in Maxwell's equations. Using derivatives of these coefficients and derivatives of the eigenpairs, we follow a perturbation approach to compute approximations of mean and covariance of the eigenpairs. For small perturbations these approximations are faster and more accurate than sampling or surrogate model strategies. For the implementation we use an approach based on isogeometric analysis, which allows for straightforward modelling of the domain deformations and computation of the required derivatives. Numerical experiments for a three-dimensional 9-cell TESLA cavity are presented.

math.NA↗

arXiv2023

$p$-multilevel Monte Carlo for acoustic scattering from large deviation rough random surfaces

We study time harmonic acoustic scattering on large deviation rough random scatterers. Therein, the roughness of the scatterers is caused by a low Sobolev regularity in the covariance function of their deformation field. The motivation for this study arises from physical phenomena where small-scale material defects can potentially introduce non-smooth deviations from a reference domain. The primary challenge in this scenario is that the scattered wave is also random, which makes computational predictions unreliable. Therefore, it is essential to quantify these uncertainties to ensure robust and well-informed design processes. While existing methods for uncertainty quantification typically rely on domain mapping or perturbation approaches, it turns out that large and rough random deviations are not satisfactory covered. To close this gap, and although counter intuitive at first, we show that the $p$-multilevel Monte Carlo method can provide an efficient tool for uncertainty quantification in this setting. To this end, we discuss the stable implementation of higher-order polynomial approximation of the deformation field by means of barycentric interpolation and provide a cost-to-accuracy analysis. Our considerations are complemented by numerical experiments in three dimensions on a complex scattering geometry.

math.NA↗

arXiv2023

On uncertainty quantification of eigenvalues and eigenspaces with higher multiplicity

We consider generalized operator eigenvalue problems in variational form with random perturbations in the bilinear forms. This setting is motivated by variational forms of partial differential equations with random input data. The considered eigenpairs can be of higher but finite multiplicity. We investigate stochastic quantities of interest of the eigenpairs and discuss why, for multiplicity greater than 1, only the stochastic properties of the eigenspaces are meaningful, but not the ones of individual eigenpairs. To that end, we characterize the Fréchet derivatives of the eigenpairs with respect to the perturbation and provide a new linear characterization for eigenpairs of higher multiplicity. As a side result, we prove local analyticity of the eigenspaces. Based on the Fréchet derivatives of the eigenpairs we discuss a meaningful Monte Carlo sampling strategy for multiple eigenvalues and develop an uncertainty quantification perturbation approach. Numerical examples are presented to illustrate the theoretical results.

math.NA↗

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↗

arXiv2021

A fast and oblivious matrix compression algorithm for Volterra integral operators

The numerical solution of dynamical systems with memory requires the efficient evaluation of Volterra integral operators in an evolutionary manner. After appropriate discretisation, the basic problem can be represented as a matrix-vector product with a lower diagonal but densely populated matrix. For typical applications, like fractional diffusion or large scale dynamical systems with delay, the memory cost for storing the matrix approximations and complete history of the data then would become prohibitive for an accurate numerical approximation. For Volterra-integral operators of convolution type, the \emph{fast and oblivious convolution quadrature} method of Schädle, Lopez-Fernandez, and Lubich allows to compute the discretized valuation with $N$ time steps in $O(N \log N)$ complexity and only requiring $O(\log N)$ active memory to store a compressed version of the complete history of the data. We will show that this algorithm can be interpreted as an $\mathcal{H}$-matrix approximation of the underlying integral operator and, consequently, a further improvement can be achieved, in principle, by resorting to $\mathcal{H}^2$-matrix compression techniques. We formulate a variant of the $\mathcal{H}^2$-matrix vector product for discretized Volterra integral operators that can be performed in an evolutionary and oblivious manner and requires only $O(N)$ operations and $O(\log N)$ active memory. In addition to the acceleration, more general asymptotically smooth kernels can be treated and the algorithm does not require a-priori knowledge of the number of time steps. The efficiency of the proposed method is demonstrated by application to some typical test problems.

math.NA↗

arXiv2021

A model reduction approach for inverse problems with operator valued data

We study the efficient numerical solution of linear inverse problems with operator valued data which arise, e.g., in seismic exploration, inverse scattering, or tomographic imaging. The high-dimensionality of the data space implies extremely high computational cost already for the evaluation of the forward operator which makes a numerical solution of the inverse problem, e.g., by iterative regularization methods, practically infeasible. To overcome this obstacle, we take advantage of the underlying tensor product structure of the problem and propose a strategy for constructing low-dimensional certified reduced order models of quasi-optimal rank for the forward operator which can be computed much more efficiently than the truncated singular value decomposition. A complete analysis of the proposed model reduction approach is given in a functional analytic setting and the efficient numerical construction of the reduced order models as well as of their application for the numerical solution of the inverse problem is discussed. In summary, the setup of a low-rank approximation can be achieved in an offline stage at essentially the same cost as a single evaluation of the forward operator, while the actual solution of the inverse problem in the online phase can be done with extremely high efficiency. The theoretical results are illustrated by application to a typical model problem in fluorescence optical tomography.

math.NA↗

arXiv2021

On robustly convergent and efficient iterative methods for anisotropic radiative transfer

This paper considers the iterative solution of linear systems arising from discretization of the anisotropic radiative transfer equation with discontinuous elements on the sphere. In order to achieve robust convergence behavior in the discretization parameters and the physical parameters we develop preconditioned Richardson iterations in Hilbert spaces. We prove convergence of the resulting scheme. The preconditioner is constructed in two steps. The first step borrows ideas from matrix splittings and ensures mesh independence. The second step uses a subspace correction technique to reduce the influence of the optical parameters. The correction spaces are build from low-order spherical harmonics approximations generalizing well-known diffusion approximations. We discuss in detail the efficient implementation and application of the discrete operators. In particular, for the considered discontinuous spherical elements, the scattering operator becomes dense and we show that $\mathcal{H}$- or $\mathcal{H}^2$-matrix compression can be applied in a black-box fashion to obtain almost linear or linear complexity when applying the corresponding approximations. The effectiveness of the proposed method is shown in numerical examples.

math.NA↗

arXiv2020

Isogeometric multilevel quadrature for forward and inverse random acoustic scattering

We study the numerical solution of forward and inverse acoustic scattering problems by randomly shaped obstacles in three-dimensional space using a fast isogeometric boundary element method. Within the isogeometric framework, realizations of the random scatterer can efficiently be computed by simply updating the NURBS mappings which represent the scatterer. This way, we end up with a random deformation field. In particular, we show that the knowledge of the deformation field's expectation and covariance at the surface of the scatterer are already sufficient to compute the surface Karhunen-Loève expansion. Leveraging on the isogeometric framework, we utilize multilevel quadrature methods for the efficient approximation of quantities of interest, such as the scattered wave's expectation and variance. Computing the wave's Cauchy data at an artificial, fixed interface enclosing the random obstacle, we can also directly infer quantities of interest in free space. Adopting the Bayesian paradigm, we finally compute the expected shape and the variance of the scatterer from noisy measurements of the scattered wave at the artificial interface. Numerical results for the forward and inverse problem are given to demonstrate the feasibility of the proposed approach.

math.NA↗

arXiv2020

A convolution quadrature method for Maxwell's equations in dispersive media

We study the systematic numerical approximation of Maxwell's equations in dispersive media. Two discretization strategies are considered, one based on a traditional leapfrog time integration method and the other based on convolution quadrature. The two schemes are proven to be equivalent and to preserve the underlying energy-dissipation structure of the problem. The second approach, however, is independent of the number of internal states and allows to handle rather general dispersive materials. Using ideas of fast-and-oblivious convolution quadrature, the method can be implemented efficiently.

math.NA↗