arXiv subjects

Christoph Schwab

Publications and source records attributed to Christoph Schwab.

At least 19 recordsLinked to original sources

arXiv2026

Neural and Spectral Operator Surrogates on Gaussian Spaces

We prove expression rate bounds of finite-parametric, spectral and neural surrogates for holomorphic maps between separable Hilbert spaces. The surrogates have an encoder-approximator-decoder architecture, with Karhunen-Loéve encoders and frame decoders. We prove expression rate bounds for two classes of finite-parametric surrogates: i) spectral surrogates obtained by N-term truncations of Wiener polynomial chaos expansions and ii) neural surrogates obtained by approximation of parametric maps with deep feedforward neural networks, ReLU and RePU activation functions and uniformly bounded weights. We work under an algebraic decay assumption on the eigenvalues of the covariance of the Gaussian measure on the input space. We obtain convergence rates for mean-square errors, and additionally in first-order Gaussian Sobolev spaces, to account for errors in the approximation of gradients.

math.NA↗

arXiv2026

Optimized multilevel Monte Carlo methods in Banach spaces

We present a theoretical and numerical analysis of Monte Carlo methods for the estimation of statistical moments of random variables $X:Ω\rightarrow E$ taking values in a Banach space $E$. For practical computation, we consider finite-dimensional approximation subspaces ${(E_\ell)_{\ell\in\mathbb{N}}\subset E}$ of increasing dimension. We develop a refined error analysis that explicitly accounts for a dependence of the Rademacher type constants on the dimension of $E_\ell$, leading to novel complexity results for single- and multilevel Monte Carlo methods to estimate the mean and injective moments of arbitrary order, which are, in certain cases, sharper than those derived in [Kirchner, Schwab, J. Funct. Anal, 2024]. Moreover, we show that, in favorable cases, the resulting error-vs.-work bounds are independent of the Rademacher type of $E$. We then focus on $L^p(S)$-valued random variables for a $σ$-finite measure space satisfying certain approximation properties, and prove that for a random variable $X\in L^q(Ω;L^p(S))\cap L^p(S;L^q(Ω))$, with $q\in (1,\infty)$ and $p\in [1,\infty)$, the $L^q$-convergence rate of a Monte Carlo estimator is determined exclusively by the integrability parameter $\min\{q,2\}$, with no dependence on the Rademacher type $\min\{p,2\}$ of $L^p(S)$. We further investigate the impact of measuring the (multilevel) Monte Carlo error in the $L^q(Ω;L^p(S))$-norm while $X$ possesses additional regularity, $X\in L^{\tilde{q}}(Ω;L^p(S))\cap L^p(S;L^{\tilde{q}}(Ω))$ with $\tilde{q}\in [q,\infty)$. This analysis reveals an interplay between the sampling error and the strong approximation error, and leads to optimized error-vs.-work bounds for both single- and multilevel Monte Carlo methods. Numerical experiments confirm the sharpness of the analyses presented.

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↗

arXiv2026

Quantum Circuit Encodings of Polynomial Chaos Expansions

This work investigates the expressive power of quantum circuits in approximating high-dimensional, real-valued functions. We focus on countably-parametric holomorphic maps $u:U\to \mathbb{R}$, where the parameter domain is $U=[-1,1]^{\mathbb{N}}$. We establish dimension-independent quantum circuit approximation rates via the best $n$-term truncations of generalized polynomial chaos (gPC) expansions of these parametric maps, demonstrating that these rates depend solely on the summability exponent of the gPC expansion coefficients. The key to our findings is based on the fact that so-called ``$(\boldsymbol{b},ε)$-holomorphic'' functions, where $\boldsymbol{b}\in (0,1]^\mathbb N \cap \ell^p(\mathbb N)$ for some $p\in(0,1)$, permit structured and sparse gPC expansions. Then, $n$-term truncated gPC expansions are known to admit approximation rates of order $n^{-1/p + 1/2}$ in the $L^2$ norm and of order $n^{-1/p + 1}$ in the $L^\infty$ norm. We show the existence of parameterized quantum circuit (PQC) encodings of these $n$-term truncated gPC expansions, and bound PQC depth and width via (i) tensorization of univariate PQCs that encode Chebyšev-polynomials in $[-1,1]$ and (ii) linear combination of unitaries (LCU) to build PQC emulations of $n$-term truncated gPC expansions. The results provide a rigorous mathematical foundation for the use of quantum algorithms in high-dimensional function approximation. As countably-parametric holomorphic maps naturally arise in parametric PDE models and uncertainty quantification (UQ), our results have implications for quantum-enhanced algorithms for a wide range of maps in applications.

math.NA↗

arXiv2026

Exponential Convergence of $hp$-FEM for the Integral Fractional Laplacian on cuboids

For the Dirichlet integral fractional Laplacian, we prove root exponential convergence of tensor-product $hp$-finite element approximations on $(0,1)^3$, for forcing $f$ that is analytic in $[0,1]^3$. Exploiting analytic regularity estimates in weighted Sobolev spaces, we prove for $hp$-GLL interpolation approximations with $N$ degrees of freedom the energy norm error bound $\lesssim \exp(-b\sqrt[6]{N})$. Tensor product mesh families which are geometrically refined towards all sides of $(0,1)^3$ are used. Numerical experiments with $hp$-Galerkin FEM confirm the bound.

math.NA↗

arXiv2026

Approximation of PDE solution manifolds: Sparse-grid interpolation and quadrature

We study fully-discrete approximations and quadratures of infinite-variate functions in abstract Bochner spaces associated with a Hilbert space $X$ and an infinite-tensor-product Jacobi measure. For target infinite-variate functions taking values in $X$ which admit absolutely convergent Jacobi generalized polynomial chaos expansions, with suitable weighted summability conditions for the coefficient sequences, we generalize and improve prior results on construction of sequences of finite sparse-grid tensor-product polynomial interpolation approximations and quadratures, based on the univariate Chebyshev points. For a generic stable discretization of $X$ in terms of a dense sequence $(V_m)_{m \in \mathbb{N}_0}$ of finite-dimensional subspaces, we obtain fully-discrete, linear approximations in terms of so-called sparse-grid tensor-product projectors, with convergence rates of approximations as well as of sparse-grid tensor-product quadratures of the target functions. We verify the abstract assumptions in two fundamental application settings: first, a linear elliptic diffusion equation with affine-parametric coefficients and second, abstract holomorphic maps between separable Hilbert spaces with affine-parametric input data encoding. For these settings, as in [37,20], cancellation of anti-symmetric terms in ultra-spherical Jacobi generalized polynomial chaos expansion coefficients implies crucially improved convergence rates of sparse-grid tensor-product quadrature with respect to the infinite-tensor-product Jacobi weight, free from the ``curse-of-dimension". Largely self-contained proofs of all results are developed. Approximation convergence rate results in the present setting which are based on construction of neural network surrogates, for unbounded parameter ranges with Gaussian measures, will be developed in extensions of the present work.

math.NA↗

arXiv2025

ROM for Viscous, Incompressible Flow in Polygons -- exponential $n$-width bounds and convergence rate

We demonstrate exponential convergence of Reduced Order Model (ROM) approximations for mixed boundary value problems of the stationary, incompressible Navier-Stokes equations in plane, polygonal domains $Ω$. Admissible boundary conditions comprise mixed BCs, no-slip, slip and open boundary conditions, subject to corner-weighted analytic boundary data and volume forcing. The small data hypothesis is assumed to ensure existence of a unique weak solution in the sense of Leray-Hopf. Recent results on corner-weighted, analytic regularity of velocity and pressure fields in $Ω$, imply exponential convergence rates of so-called mixed $hp$-Finite Element Methods in $H^1(Ω)^2\times L^2(Ω)$ on sequences of geometric partitions of $Ω$, with corner-refinement. Based on these exponential convergence rate bounds, we infer exponential bounds for the Kolmogorov $n$-widths of solution sets for analytic forcing and boundary data. This implies corresponding exponential convergence rates of POD Galerkin methods that are based on truth solutions which are obtained offline from low-order, divergence stable mixed Finite Element discretizations. Numerical experiments confirm the exponential rates and the theoretical results.

math.NA↗

arXiv2025

Expression Rates of Neural Operators for Linear Elliptic PDEs in Polytopes

We study the approximation rates of a class of deep neural network approximations of operators which arise as data-to-solution maps $\mathcal{S}$ of linear elliptic partial differential equations (PDEs), and act between pairs $X,Y$ of suitable infinite-dimensional spaces. We prove expression rate bounds for approximate neural operators $\mathcal{G}$ with the structure $\mathcal{G} = \mathcal{R} \circ \mathcal{A} \circ \mathcal{E}$, with linear encoders $\mathcal{E}$ and decoders $\mathcal{R}$. We focus in particular on deepONets emulating the coefficient-to-solution maps for elliptic PDEs set in polygons and in some polyhedra. Exploiting the regularity of the solution sets of elliptic PDEs in polytopes, we show algebraic rates of convergence for problems with data with finite regularity, and exponential rates for analytic data.

math.NA↗

arXiv2025

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↗

arXiv2025

Sparsity for Infinite-Parametric Holomorphic Functions on Gaussian Spaces

We investigate the sparsity of Wiener polynomial chaos expansions of holomorphic maps $\mathcal{G}$ on Gaussian Hilbert spaces, as arise in the coefficient-to-solution maps of linear, second order, divergence-form elliptic PDEs with log-Gaussian diffusion coefficient. Representing the Gaussian random field input as an affine-parametric expansion, the nonlinear map becomes a countably-parametric, deterministic holomorphic map of the coordinate sequence $\boldsymbol{y} = (y_j)_{j\in\mathbb{N}} \in \mathbb{R}^\infty$. We establish weighted summability results for the Wiener-Hermite coefficient sequences of images of affine-parametric expansions of the log-Gaussian input under $\mathcal{G}$. These results give rise to $N$-term approximation rate bounds for the full range of input summability exponents $p\in (0,2)$. We show that these approximation rate bounds apply to parameter-to-solution maps for elliptic diffusion PDEs with lognormal coefficients.

math.NA↗

arXiv2025

Frequency-Explicit Shape Holomorphy in Uncertainty Quantification for Acoustic Scattering

We consider frequency-domain acoustic scattering at a homogeneous star-shaped penetrable obstacle, whose shape is uncertain and modelled via a radial spectral parameterization with random coefficients. Using recent results on the stability of Helmholtz transmission problems with piecewise constant coefficients from [A. Moiola and E. A. Spence, Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions, Mathematical Models and Methods in Applied Sciences, 29 (2019), pp. 317-354] we obtain frequency-explicit statements on the holomorphic dependence of the scattered field and the far-field pattern on the stochastic shape parameters. This paves the way for applying general results on the efficient construction of high-dimensional surrogate models. We also take into account the effect of domain truncation by means of perfectly matched layers (PML). In addition, spatial regularity estimates which are explicit in terms of the wavenumber $k$ permit us to quantify the impact of finite-element Galerkin discretization using high-order Lagrangian finite-element spaces.

math.NA↗

arXiv2024

First Order System Least Squares Neural Networks

We introduce a conceptual framework for numerically solving linear elliptic, parabolic, and hyperbolic PDEs on bounded, polytopal domains in euclidean spaces by deep neural networks. The PDEs are recast as minimization of a least-squares (LSQ for short) residual of an equivalent, well-posed first-order system, over parametric families of deep neural networks. The associated LSQ residual is a) equal or proportional to a weak residual of the PDE, b) additive in terms of contributions from localized subnetworks, indicating locally ``out-of-equilibrium'' of neural networks with respect to the PDE residual, c) serves as numerical loss function for neural network training, and d) constitutes, even with incomplete training, a computable, (quasi-)optimal numerical error estimator in the context of adaptive LSQ finite element methods. In addition, an adaptive neural network growth strategy is proposed which, assuming exact numerical minimization of the LSQ loss functional, yields sequences of neural networks with realizations that converge rate-optimally to the exact solution of the first order system LSQ formulation.

math.NA↗

arXiv2024

Exponential Expressivity of ReLU$^k$ Neural Networks on Gevrey Classes with Point Singularities

We analyze deep Neural Network emulation rates of smooth functions with point singularities in bounded, polytopal domains $\mathrm{D} \subset \mathbb{R}^d$, $d=2,3$. We prove exponential emulation rates in Sobolev spaces in terms of the number of neurons and in terms of the number of nonzero coefficients for Gevrey-regular solution classes defined in terms of weighted Sobolev scales in $\mathrm{D}$, comprising the countably-normed spaces of I.M. Babuška and B.Q. Guo. As intermediate result, we prove that continuous, piecewise polynomial high order (``$p$-version'') finite elements with elementwise polynomial degree $p\in\mathbb{N}$ on arbitrary, regular, simplicial partitions of polyhedral domains $\mathrm{D} \subset \mathbb{R}^d$, $d\geq 2$ can be exactly emulated by neural networks combining ReLU and ReLU$^2$ activations. On shape-regular, simplicial partitions of polytopal domains $\mathrm{D}$, both the number of neurons and the number of nonzero parameters are proportional to the number of degrees of freedom of the finite element space, in particular for the $hp$-Finite Element Method of I.M. Babuška and B.Q. Guo.

math.NA↗

arXiv2024

Deep ReLU networks and high-order finite element methods II: Chebyshev emulation

We show expression rates and stability in Sobolev norms of deep feedforward ReLU neural networks (NNs) in terms of the number of parameters defining the NN for continuous, piecewise polynomial functions, on arbitrary, finite partitions $\mathcal{T}$ of a bounded interval $(a,b)$. Novel constructions of ReLU NN surrogates encoding function approximations in terms of Chebyshev polynomial expansion coefficients are developed which require fewer neurons than previous constructions. Chebyshev coefficients can be computed easily from the values of the function in the Clenshaw--Curtis points using the inverse fast Fourier transform. Bounds on expression rates and stability are obtained that are superior to those of constructions based on ReLU NN emulations of monomials as considered in [Opschoor, Petersen and Schwab, 2020] and [Montanelli, Yang and Du, 2021]. All emulation bounds are explicit in terms of the (arbitrary) partition of the interval, the target emulation accuracy and the polynomial degree in each element of the partition. ReLU NN emulation error estimates are provided for various classes of functions and norms, commonly encountered in numerical analysis. In particular, we show exponential ReLU emulation rate bounds for analytic functions with point singularities and develop an interface between Chebfun approximations and constructive ReLU NN emulations.

math.NA↗

arXiv2024

Deep ReLU Neural Network Emulation in High-Frequency Acoustic Scattering

We obtain wavenumber-robust error bounds for the deep neural network (DNN) emulation of the solution to the time-harmonic, sound-soft acoustic scattering problem in the exterior of a smooth, convex obstacle in two physical dimensions. The error bounds are based on a boundary reduction of the scattering problem in the unbounded exterior region to its smooth, curved boundary $Γ$ using the so-called combined field integral equation (CFIE), a well-posed, second-kind boundary integral equation (BIE) for the field's Neumann datum on $Γ$. In this setting, the continuity and stability constants of this formulation are explicit in terms of the (non-dimensional) wavenumber $κ$. Using wavenumber-explicit asymptotics of the problem's Neumann datum, we analyze the DNN approximation rate for this problem. We use fully connected NNs of the feed-forward type with Rectified Linear Unit (ReLU) activation. Through a constructive argument we prove the existence of DNNs with an $ε$-error bound in the $L^\infty(Γ)$-norm having a small, fixed width and a depth that increases $\textit{spectrally}$ with the target accuracy $ε>0$. We show that for fixed $ε>0$, the depth of these NNs should increase $\textit{poly-logarithmically}$ with respect to the wavenumber $κ$ whereas the width of the NN remains fixed. Unlike current computational approaches, such as wavenumber-adapted versions of the Galerkin Boundary Element Method (BEM) with shape- and wavenumber-tailored solution $\textit{ansatz}$ spaces, our DNN approximations do not require any prior analytic information about the scatterer's shape.

math.NA↗

arXiv2024

Exponential Convergence of $hp$-ILGFEM for semilinear elliptic boundary value problems with monomial reaction

We study the fully explicit numerical approximation of a semilinear elliptic boundary value model problem, which features a monomial reaction and analytic forcing, in a bounded polygon $Ω\subset\mathbb{R}^2$ with a finite number of straight edges. In particular, we analyze the convergence of $hp$-type iterative linearized Galerkin ($hp$-ILG) solvers. Our convergence analysis is carried out for conforming $hp$-finite element (FE) Galerkin discretizations on sequences of regular, simplicial partitions of $Ω$, with geometric corner refinement, with polynomial degrees increasing in sync with the geometric mesh refinement towards the corners of $Ω$. For a sequence of discrete solutions generated by the ILG solver, with a stopping criterion that is consistent with the exponential convergence of the exact $hp$-FE Galerkin solution, we prove exponential convergence in $\mathrm{H}^1(Ω)$ to the unique weak solution of the boundary value problem. Numerical experiments illustrate the exponential convergence of the numerical approximations obtained from the proposed scheme in terms of the number of degrees of freedom as well as of the computational complexity involved.

math.NA↗

arXiv2024

Neural and spectral operator surrogates: unified construction and expression rate bounds

Approximation rates are analyzed for deep surrogates of maps between infinite-dimensional function spaces, arising e.g. as data-to-solution maps of linear and nonlinear partial differential equations. Specifically, we study approximation rates for Deep Neural Operator and Generalized Polynomial Chaos (gpc) Operator surrogates for nonlinear, holomorphic maps between infinite-dimensional, separable Hilbert spaces. Operator in- and outputs from function spaces are assumed to be parametrized by stable, affine representation systems. Admissible representation systems comprise orthonormal bases, Riesz bases or suitable tight frames of the spaces under consideration. Algebraic expression rate bounds are established for both, deep neural and spectral operator surrogates acting in scales of separable Hilbert spaces containing domain and range of the map to be expressed, with finite Sobolev or Besov regularity. We illustrate the abstract concepts by expression rate bounds for the coefficient-to-solution map for a linear elliptic PDE on the torus.

math.NA↗