Christoph Schwab

Josef Dick, Frances Y. Kuo, Quoc T. Le Gia et al.

We construct quasi-Monte Carlo methods to approximate the expected values of linear functionals of Galerkin discretizations of parametric operator equations which depend on a possibly infinite sequence of parameters. Such problems arise in the numerical solution of differential and integral equations with random field inputs. We analyze the regularity of the solutions with respect to the parameters in terms of the rate of decay of the fluctuations of the input field. If $p\in (0,1]$ denotes the "summability exponent" corresponding to the fluctuations in affine-parametric families of operators, then we prove that deterministic "interlaced polynomial lattice rules" of order $α= \lfloor 1/p \rfloor+1$ in $s$ dimensions with $N$ points can be constructed using a fast component-by-component algorithm, in $\mathcal{O}(α\,s\, N\log N + α^2\,s^2 N)$ operations, to achieve a convergence rate of $\mathcal{O}(N^{-1/p})$, with the implied constant independent of $s$. This dimension-independent convergence rate is superior to the rate $\mathcal{O}(N^{-1/p+1/2})$, for $2/3\leq p\leq 1$ recently established for randomly shifted lattice rules under comparable assumptions. In our analysis we use a non-standard Banach space setting and introduce "smoothness-driven product and order dependent (SPOD)" weights for which we show fast CBC construction.

Viet Ha Hoang, Christoph Schwab, Andrew M. Stuart

We study Bayesian inversion for a model elliptic PDE with unknown diffusion coefficient. We provide complexity analyses of several Markov Chain-Monte Carlo (MCMC) methods for the efficient numerical evaluation of expectations under the Bayesian posterior distribution, given data $δ$. Particular attention is given to bounds on the overall work required to achieve a prescribed error level $\varepsilon$. Specifically, we first bound the computational complexity of "plain" MCMC, based on combining MCMC sampling with linear complexity multilevel solvers for elliptic PDE. Our (new) work versus accuracy bounds show that the complexity of this approach can be quite prohibitive. Two strategies for reducing the computational complexity are then proposed and analyzed: first, a sparse, parametric and deterministic generalized polynomial chaos (gpc) "surrogate" representation of the forward response map of the PDE over the entire parameter space, and, second, a novel Multi-Level Markov Chain Monte Carlo (MLMCMC) strategy which utilizes sampling from a multilevel discretization of the posterior and of the forward PDE. For both of these strategies we derive asymptotic bounds on work versus accuracy, and hence asymptotic bounds on the computational complexity of the algorithms. In particular we provide sufficient conditions on the regularity of the unknown coefficients of the PDE, and on the approximation methods used, in order for the accelerations of MCMC resulting from these strategies to lead to complexity reductions over "plain" MCMC algorithms for Bayesian inversion of PDEs.}

Josef Dick, Robert N. Gantner, Quoc T. Le Gia et al.

We propose and analyze deterministic multilevel approximations for Bayesian inversion of operator equations with uncertain distributed parameters, subject to additive Gaussian measurement data. The algorithms use a multilevel (ML) approach based on deterministic, higher order quasi-Monte Carlo (HoQMC) quadrature for approximating the high-dimensional expectations, which arise in the Bayesian estimators, and a Petrov-Galerkin (PG) method for approximating the solution to the underlying partial differential equation (PDE). This extends the previous single-level approach from [J. Dick, R. N. Gantner, Q. T. Le Gia and Ch. Schwab, Higher order Quasi-Monte Carlo integration for Bayesian Estimation. Report 2016-13, Seminar for Applied Mathematics, ETH Zürich (in review)]. We obtain sufficient conditions which allow us to achieve arbitrarily high, algebraic convergence rates in terms of work, which are independent of the dimension of the parameter space. The convergence rates are limited only by the spatial regularity of the forward problem,the discretization order achieved by the Petrov Galerkin discretization, and by the sparsity of the uncertainty parametrization. We provide detailed numerical experiments for linear elliptic problems in two space dimensions, with $s=1024$ parameters characterizing the uncertain input, confirming the theory and showing that the ML HoQMC algorithms outperform, in terms of error vs.~computational work, both multilevel Monte Carlo (MLMC) methods and single-level (SL) HoQMC methods.

Josef Dick, Robert N. Gantner, Quoc T. Le Gia et al.

We analyze combined Quasi-Monte Carlo quadrature and Finite Element approximations in Bayesian estimation of solutions to countably-parametric operator equations with holomorphic dependence on the parameters as considered in [Cl.~Schillings and Ch.~Schwab: Sparsity in Bayesian Inversion of Parametric Operator Equations. Inverse Problems, {\bf 30}, (2014)]. Such problems arise in numerical uncertainty quantification and in Bayesian inversion of operator equations with distributed uncertain inputs, such as uncertain coefficients, uncertain domains or uncertain source terms and boundary data. We show that the parametric Bayesian posterior densities belong to a class of weighted Bochner spaces of functions of countably many variables, with a particular structure of the QMC quadrature weights: up to a (problem-dependent, and possibly large) finite dimension $S$ product weights can be used, and beyond this dimension, weighted spaces with so-called SPOD weights are used to describe the solution regularity. We establish error bounds for higher order Quasi-Monte Carlo quadrature for the Bayesian estimation based on [J.~Dick, Q.T.~LeGia and Ch.~Schwab, Higher order Quasi-Monte Carlo integration for holomorphic, parametric operator equations, Report 2014-23, SAM, ETH Zürich]. It implies, in particular, regularity of the parametric solution and of the countably-parametric Bayesian posterior density in SPOD weighted spaces. This, in turn, implies that the Quasi-Monte Carlo quadrature methods in [J. Dick, F.Y.~Kuo, Q.T.~Le Gia, D.~Nuyens, Ch.~Schwab, Higher order QMC Galerkin discretization for parametric operator equations, SINUM (2014)] are applicable to these problem classes, with dimension-independent convergence rates $\calO(N^{-1/p})$ of $N$-point HoQMC approximated Bayesian estimates, where $0<p<1$ depends only on the sparsity class of the uncertain input in the Bayesian estimation.

Jean-Luc Bouchot, Holger Rauhut, Christoph Schwab

We analyze a novel multi-level version of a recently introduced compressed sensing (CS) Petrov-Galerkin (PG) method from [H. Rauhut and Ch. Schwab: Compressive Sensing Petrov-Galerkin approximation of high-dimensional parametric operator equations, Math. Comp. 304(2017) 661-700] for the solution of many-parametric partial differential equations. We propose to use multi-level PG discretizations, based on a hierarchy of nested finite dimensional subspaces, and to reconstruct parametric solutions at each level from level-dependent random samples of the high-dimensional parameter space via CS methods such as weighted l1-minimization. For affine parametric, linear operator equations, we prove that our approach allows to approximate the parametric solution with (almost) optimal convergence order as specified by certain summability properties of the coefficient sequence in a general polynomial chaos expansion of the parametric solution and by the convergence order of the PG discretization in the physical variables. The computations of the parameter samples of the PDE solution is "embarrassingly parallel", as in Monte-Carlo Methods. Contrary to other recent approaches, and as already noted in [A. Doostan and H. Owhadi: A non-adapted sparse approximation of PDEs with stochastic inputs. JCP 230(2011) 3015-3034] the optimality of the computed approximations does not require a-priori assumptions on ordering and structure of the index sets of the largest gpc coefficients (such as the "downward closed" property). We prove that under certain assumptions work versus accuracy of the new algorithms is asymptotically equal to that of one PG solve for the corresponding nominal problem on the finest discretization level up to a constant.

Lukas Herrmann, Christoph Schwab, Jakob Zech

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.

Lukas Herrmann, Annika Lang, Christoph Schwab

Numerical solutions of stationary diffusion equations on the unit sphere with isotropic lognormal diffusion coefficients are considered. Hölder regularity in $L^p$ sense for isotropic Gaussian random fields is obtained and related to the regularity of the driving lognormal coefficients. This yields regularity in $L^p$ sense of the solution to the diffusion problem in Sobolev spaces. Convergence rate estimates of multilevel Monte Carlo Finite and Spectral Element discretizations of these problems are then deduced. Specifically, a convergence analysis is provided with convergence rate estimates in terms of the number of Monte Carlo samples of the solution to the considered diffusion equation and in terms of the total number of degrees of freedom of the spatial discretization, and with bounds for the total work required by the algorithm in the case of Finite Element discretizations. The obtained convergence rates are solely in terms of the decay of the angular power spectrum of the (logarithm) of the diffusion coefficient. Numerical examples confirm the presented theory.

Stig Larsson, Christoph Schwab

We study linear parabolic initial-value problems in a space-time variational formulation based on fractional calculus. This formulation uses "time derivatives of order one half" on the bi-infinite time axis. We show that for linear, parabolic initial-boundary value problems on $(0,\infty)$, the corresponding bilinear form admits an inf-sup condition with sparse tensor product trial and test function spaces. We deduce optimality of compressive, space-time Galerkin discretizations, where stability of Galerkin approximations is implied by the well-posedness of the parabolic operator equation. The variational setting adopted here admits more general Riesz bases than previous work; in particular, no stability in negative order Sobolev spaces on the spatial or temporal domains is required of the Riesz bases accommodated by the present formulation. The trial and test spaces are based on Sobolev spaces of equal order $1/2$ with respect to the temporal variable. Sparse tensor products of multi-level decompositions of the spatial and temporal spaces in Galerkin discretizations lead to large, non-symmetric linear systems of equations. We prove that their condition numbers are uniformly bounded with respect to the discretization level. In terms of the total number of degrees of freedom, the convergence orders equal, up to logarithmic terms, those of best $N$-term approximations of solutions of the corresponding elliptic problems.

Frances Y. Kuo, Christoph Schwab, Ian H. Sloan

Quasi-Monte Carlo (QMC) methods are applied to multi-level Finite Element (FE) discretizations of elliptic partial differential equations (PDEs) with a random coefficient, to estimate expected values of linear functionals of the solution. The expected value is considered as an infinite-dimensional integral in the parameter space corresponding to the randomness induced by the random coefficient. We use a multi-level algorithm, with the number of QMC points depending on the discretization level, and with a level-dependent dimension truncation strategy. In some scenarios, we show that the overall error is $\mathcal{O}(h^2)$, where $h$ is the finest FE mesh width, or $\mathcal{O}(N^{-1+δ})$ for arbitrary $δ>0$, where $N$ is the maximal number of QMC sampling points. For these scenarios, the total work is essentially of the order of one single PDE solve at the finest FE discretization level. The analysis exploits regularity of the parametric solution with respect to both the physical variables (the variables in the physical domain) and the parametric variables (the parameters corresponding to randomness). Families of QMC rules with "POD weights" ("product and order dependent weights") which quantify the relative importance of subsets of the variables are found to be natural for proving convergence rates of QMC errors that are independent of the number of parametric variables.

Josef Dick, Frances Y. Kuo, Quoc T. Le Gia et al.

Quasi-Monte Carlo (QMC) rules $1/N \sum_{n=0}^{N-1} f(\boldsymbol{y}_n A)$ can be used to approximate integrals of the form $\int_{[0,1]^s} f(\boldsymbol{y} A) \,\mathrm{d} \boldsymbol{y}$, where $A$ is a matrix and $\boldsymbol{y}$ is row vector. This type of integral arises for example from the simulation of a normal distribution with a general covariance matrix, from the approximation of the expectation value of solutions of PDEs with random coefficients, or from applications from statistics. In this paper we design QMC quadrature points $\boldsymbol{y}_0, ..., \boldsymbol{y}_{N-1} \in [0,1]^s$ such that for the matrix $Y = (\boldsymbol{y}_{0}^\top, ..., \boldsymbol{y}_{N-1}^\top)^\top$ whose rows are the quadrature points, one can use the fast Fourier transform to compute the matrix-vector product $Y \boldsymbol{a}^\top$, $\boldsymbol{a} \in \mathbb{R}^s$, in $\mathcal{O}(N \log N)$ operations and at most $s-1$ extra additions. The proposed method can be applied to lattice rules, polynomial lattice rules and a certain type of Korobov $p$-set. The approach is illustrated computationally by three numerical experiments. The first test considers the generation of points with normal distribution and general covariance matrix, the second test applies QMC to high-dimensional, affine-parametric, elliptic partial differential equations with uniformly distributed random coefficients, and the third test addresses Finite-Element discretizations of elliptic partial differential equations with high-dimensional, log-normal random input data. All numerical tests show a significant speed-up of the computation times of the fast QMC matrix method compared to a conventional implementation as the dimension becomes large.

Wolfgang Dahmen, Chunyan Huang, Gitta Kutyniok et al.

We highlight some recent new delevelopments concerning the sparse representation of possibly high-dimensional functions exhibiting strong anisotropic features and low regularity in isotropic Sobolev or Besov scales. Specifically, we focus on the solution of transport equations which exhibit propagation of singularities where, additionally, high-dimensionality enters when the convection field, and hence the solutions, depend on parameters varying over some compact set. Important constituents of our approach are directionally adaptive discretization concepts motivated by compactly supported shearlet systems, and well-conditioned stable variational formulations that support trial spaces with anisotropic refinements with arbitrary directionalities. We prove that they provide tight error-residual relations which are used to contrive rigorously founded adaptive refinement schemes which converge in $L_2$. Moreover, in the context of parameter dependent problems we discuss two approaches serving different purposes and working under different regularity assumptions. For frequent query problems, making essential use of the novel well-conditioned variational formulations, a new Reduced Basis Method is outlined which exhibits a certain rate-optimal performance for indefinite, unsymmetric or singularly perturbed problems. For the radiative transfer problem with scattering a sparse tensor method is presented which mitigates or even overcomes the curse of dimensionality under suitable (so far still isotropic) regularity assumptions. Numerical examples for both methods illustrate the theoretical findings.

Josef Dick, Quoc Thong Le Gia, Christoph Schwab

We review recent results on dimension-robust higher order convergence rates of Quasi-Monte Carlo Petrov-Galerkin approximations for response functionals of infinite-dimensional, parametric operator equations which arise in computational uncertainty quantification.

Christoph Schwab, Andreas Stein, Jakob Zech

We establish universality and expression rate bounds for a class of neural Deep Operator Networks (DON) emulating Lipschitz (or Hölder) continuous maps $\mathcal G:\mathcal X\to\mathcal Y$ between (subsets of) separable Hilbert spaces $\mathcal X$, $\mathcal Y$. The DON architecture considered uses linear encoders $\mathcal E$ and decoders $\mathcal D$ via (biorthogonal) Riesz bases of $\mathcal X$, $\mathcal Y$, and an approximator network of an infinite-dimensional, parametric coordinate map that is Lipschitz continuous on the sequence space $\ell^2(\mathbb N)$. Unlike previous works ([Herrmann, Schwab and Zech: Neural and Spectral operator surrogates: construction and expression rate bounds, SAM Report, 2022], [Marcati and Schwab: Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations, SAM Report, 2022]), which required for example $\mathcal G$ to be holomorphic, the present expression rate results require mere Lipschitz (or Hölder) continuity of $\mathcal G$. Key in the proof of the present expression rate bounds is the use of either super-expressive activations (e.g. [Yarotski: Elementary superexpressive activations, Int. Conf. on ML, 2021], [Shen, Yang and Zhang: Neural network approximation: Three hidden layers are enough, Neural Networks, 2021], and the references there) which are inspired by the Kolmogorov superposition theorem, or of nonstandard NN architectures with standard (ReLU) activations as recently proposed in [Zhang, Shen and Yang: Neural Network Architecture Beyond Width and Depth, Adv. in Neural Inf. Proc. Sys., 2022]. We illustrate the abstract results by approximation rate bounds for emulation of a) solution operators for parametric elliptic variational inequalities, and b) Lipschitz maps of Hilbert-Schmidt operators.

Junaid Aftab, Christoph Schwab, Haizhao Yang et al.

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.

Kristin Kirchner, Fabio Nobile, Christoph Schwab et al.

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.

Joost A. A. Opschoor, Christoph Schwab

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.

Joost A. A. Opschoor, Philipp C. Petersen, Christoph Schwab

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.

Helmut Harbrecht, Christoph Schwab

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.

Dinh DÅ©ng, Helmut Harbrecht, Van Kien Nguyen et al.

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.

Joost A. A. Opschoor, Christoph Schwab

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.

Joost A. A. Opschoor, Christoph Schwab, Christos Xenophontos

We prove deep neural network (DNN for short) expressivity rate bounds for solution sets of a model class of singularly perturbed, elliptic two-point boundary value problems, in Sobolev norms, on the bounded interval $(-1,1)$. We assume that the given source term and reaction coefficient are analytic in $[-1,1]$. We establish expression rate bounds in Sobolev norms in terms of the NN size which are uniform with respect to the singular perturbation parameter for several classes of DNN architectures. In particular, ReLU NNs, spiking NNs, and $\tanh$- and sigmoid-activated NNs. The latter activations can represent ``exponential boundary layer solution features'' explicitly, in the last hidden layer of the DNN, i.e. in a shallow subnetwork, and afford improved robust expression rate bounds in terms of the NN size. We prove that all DNN architectures allow robust exponential solution expression in so-called `energy' as well as in `balanced' Sobolev norms, for analytic input data.

Björn Bahr, Markus Faustmann, Carlo Marcati et al.

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.

Marcello Longo, Joost A. A. Opschoor, Nico Disch et al.

On general regular simplicial partitions $\mathcal{T}$ of bounded polytopal domains $Ω\subset \mathbb{R}^d$, $d\in\{2,3\}$, we construct \emph{exact neural network (NN) emulations} of all lowest order finite element spaces in the discrete de Rham complex. These include the spaces of piecewise constant functions, continuous piecewise linear (CPwL) functions, the classical ``Raviart-Thomas element'', and the ``Nédélec edge element''. For all but the CPwL case, our network architectures employ both ReLU (rectified linear unit) and BiSU (binary step unit) activations to capture discontinuities. In the important case of CPwL functions, we prove that it suffices to work with pure ReLU nets. Our construction and DNN architecture generalizes previous results in that no geometric restrictions on the regular simplicial partitions $\mathcal{T}$ of $Ω$ are required for DNN emulation. In addition, for CPwL functions our DNN construction is valid in any dimension $d\geq 2$. Our ``FE-Nets'' are required in the variationally correct, structure-preserving approximation of boundary value problems of electromagnetism in nonconvex polyhedra $Ω\subset \mathbb{R}^3$. They are thus an essential ingredient in the application of e.g., the methodology of ``physics-informed NNs'' or ``deep Ritz methods'' to electromagnetic field simulation via deep learning techniques. We indicate generalizations of our constructions to higher-order compatible spaces and other, non-compatible classes of discretizations, in particular the ``Crouzeix-Raviart'' elements and Hybridized, Higher Order (HHO) methods.

Carlo Marcati, Christoph Schwab

We construct and analyze approximation rates of deep operator networks (ONets) between infinite-dimensional spaces that emulate with an exponential rate of convergence the coefficient-to-solution map of elliptic second-order partial differential equations. In particular, we consider problems set in $d$-dimensional periodic domains, $d=1, 2, \dots$, and with analytic right-hand sides and coefficients. Our analysis covers linear, elliptic second order divergence-form PDEs as, e.g., diffusion-reaction problems, parametric diffusion equations, and elliptic systems such as linear isotropic elastostatics in heterogeneous materials. We leverage the exponential convergence of spectral collocation methods for boundary value problems whose solutions are analytic. In the present periodic and analytic setting, this follows from classical elliptic regularity. Within the ONet branch and trunk construction of [Chen and Chen, 1993] and of [Lu et al., 2021], we show the existence of deep ONets which emulate the coefficient-to-solution map to a desired accuracy in the $H^1$ norm, uniformly over the coefficient set. We prove that the neural networks in the ONet have size $\mathcal{O}(\left|\log(\varepsilon)\right|^κ)$, where $\varepsilon>0$ is the approximation accuracy, for some $κ>0$ depending on the physical space dimension.

Christoph Schwab, Jakob Zech

For artificial deep neural networks, we prove expression rates for analytic functions $f:\mathbb{R}^d\to\mathbb{R}$ in the norm of $L^2(\mathbb{R}^d,γ_d)$ where $d\in {\mathbb{N}}\cup\{ \infty \}$. Here $γ_d$ denotes the Gaussian product probability measure on $\mathbb{R}^d$. We consider in particular ReLU and ReLU${}^k$ activations for integer $k\geq 2$. For $d\in\mathbb{N}$, we show exponential convergence rates in $L^2(\mathbb{R}^d,γ_d)$. In case $d=\infty$, under suitable smoothness and sparsity assumptions on $f:\mathbb{R}^{\mathbb{N}}\to\mathbb{R}$, with $γ_\infty$ denoting an infinite (Gaussian) product measure on $\mathbb{R}^{\mathbb{N}}$, we prove dimension-independent expression rate bounds in the norm of $L^2(\mathbb{R}^{\mathbb{N}},γ_\infty)$. The rates only depend on quantified holomorphy of (an analytic continuation of) the map $f$ to a product of strips in $\mathbb{C}^d$. As an application, we prove expression rate bounds of deep ReLU-NNs for response surfaces of elliptic PDEs with log-Gaussian random field inputs.

Carlo Marcati, Joost A. A. Opschoor, Philipp C. Petersen et al.

We prove exponential expressivity with stable ReLU Neural Networks (ReLU NNs) in $H^1(Ω)$ for weighted analytic function classes in certain polytopal domains $Ω$, in space dimension $d=2,3$. Functions in these classes are locally analytic on open subdomains $D\subset Ω$, but may exhibit isolated point singularities in the interior of $Ω$ or corner and edge singularities at the boundary $\partial Ω$. The exponential expression rate bounds proved here imply uniform exponential expressivity by ReLU NNs of solution families for several elliptic boundary and eigenvalue problems with analytic data. The exponential approximation rates are shown to hold in space dimension $d = 2$ on Lipschitz polygons with straight sides, and in space dimension $d=3$ on Fichera-type polyhedral domains with plane faces. The constructive proofs indicate in particular that NN depth and size increase poly-logarithmically with respect to the target NN approximation accuracy $\varepsilon>0$ in $H^1(Ω)$. The results cover in particular solution sets of linear, second order elliptic PDEs with analytic data and certain nonlinear elliptic eigenvalue problems with analytic nonlinearities and singular, weighted analytic potentials as arise in electron structure models. In the latter case, the functions correspond to electron densities that exhibit isolated point singularities at the positions of the nuclei. Our findings provide in particular mathematical foundation of recently reported, successful uses of deep neural networks in variational electron structure algorithms.

Lehel Banjai, Jens M. Melenk, Ricardo H. Nochetto et al.

We design and analyze several Finite Element Methods (FEMs) applied to the Caffarelli-Silvestre extension that localizes the fractional powers of symmetric, coercive, linear elliptic operators in bounded domains with Dirichlet boundary conditions. We consider open, bounded, polytopal but not necessarily convex domains $Ω\subset \mathbb{R}^d$ with $d=1,2$. For the solution to the extension problem, we establish analytic regularity with respect to the extended variable $y\in (0,\infty)$. We prove that the solution belongs to countably normed, power-exponentially weighted Bochner spaces of analytic functions with respect to $y$, taking values in corner-weighted Kondat'ev type Sobolev spaces in $Ω$. In $Ω\subset \mathbb{R}^d$, we discretize with continuous, piecewise linear, Lagrangian FEM ($P_1$-FEM) with mesh refinement near corners, and prove that first order convergence rate is attained for compatible data $f\in \mathbb{H}^{1-s}(Ω)$. We also prove that tensorization of a $P_1$-FEM in $Ω$ with a suitable $hp$-FEM in the extended variable achieves log-linear complexity with respect to $\mathcal{N}_Ω$, the number of degrees of freedom in the domain $Ω$. In addition, we propose a novel, sparse tensor product FEM based on a multilevel $P_1$-FEM in $Ω$ and on a $P_1$-FEM on radical-geometric meshes in the extended variable. We prove that this approach also achieves log-linear complexity with respect to $\mathcal{N}_Ω$. Finally, under the stronger assumption that the data is analytic in $\overlineΩ$, and without compatibility at $\partial Ω$, we establish exponential rates of convergence of $hp$-FEM for spectral, fractional diffusion operators. We also report numerical experiments for model problems which confirm the theoretical results. We indicate several extensions and generalizations of the proposed methods.

Markus Bachmayr, Albert Cohen, Dinh Dũng et al.

It has recently been demonstrated that locality of spatial supports in the parametrization of coefficients in elliptic PDEs can lead to improved convergence rates of sparse polynomial expansions of the corresponding parameter-dependent solutions. These results by themselves do not yield practically realizable approximations, since they do not cover the approximation of the arising expansion coefficients, which are functions of the spatial variable. In this work, we study the combined spatial and parametric approximability for elliptic PDEs with affine or lognormal parametrizations of the diffusion coefficients and corresponding Taylor, Jacobi, and Hermite expansions, to obtain fully discrete approximations. Our analysis yields convergence rates of the fully discrete approximation in terms of the total number of degrees of freedom. The main vehicle consists in $\ell^p$ summability results for the coefficient sequences measured in higher-order Hilbertian Sobolev norms. We also discuss similar results for non-Hilbertian Sobolev norms which arise naturally when using adaptive spatial discretizations.

Frances Y. Kuo, Robert Scheichl, Christoph Schwab et al.

In this paper we present a rigorous cost and error analysis of a multilevel estimator based on randomly shifted Quasi-Monte Carlo (QMC) lattice rules for lognormal diffusion problems. These problems are motivated by uncertainty quantification problems in subsurface flow. We extend the convergence analysis in [Graham et al., Numer. Math. 2014] to multilevel Quasi-Monte Carlo finite element discretizations and give a constructive proof of the dimension-independent convergence of the QMC rules. More precisely, we provide suitable parameters for the construction of such rules that yield the required variance reduction for the multilevel scheme to achieve an $\varepsilon$-error with a cost of $\mathcal{O}(\varepsilon^{-θ})$ with $θ< 2$, and in practice even $θ\approx 1$, for sufficiently fast decaying covariance kernels of the underlying Gaussian random field inputs. This confirms that the computational gains due to the application of multilevel sampling methods and the gains due to the application of QMC methods, both demonstrated in earlier works for the same model problem, are complementary. A series of numerical experiments confirms these gains. The results show that in practice the multilevel QMC method consistently outperforms both the multilevel MC method and the single-level variants even for non-smooth problems.

Annika Lang, Christoph Schwab

Isotropic Gaussian random fields on the sphere are characterized by Karhunen-Loève expansions with respect to the spherical harmonic functions and the angular power spectrum. The smoothness of the covariance is connected to the decay of the angular power spectrum and the relation to sample Hölder continuity and sample differentiability of the random fields is discussed. Rates of convergence of their finitely truncated Karhunen-Loève expansions in terms of the covariance spectrum are established, and algorithmic aspects of fast sample generation via fast Fourier transforms on the sphere are indicated. The relevance of the results on sample regularity for isotropic Gaussian random fields and the corresponding lognormal random fields on the sphere for several models from environmental sciences is indicated. Finally, the stochastic heat equation on the sphere driven by additive, isotropic Wiener noise is considered, and strong convergence rates for spectral discretizations based on the spherical harmonic functions are proven.

Holger Rauhut, Christoph Schwab

We analyze the convergence of compressive sensing based sampling techniques for the efficient evaluation of functionals of solutions for a class of high-dimensional, affine-parametric, linear operator equations which depend on possibly infinitely many parameters. The proposed algorithms are based on so-called "non-intrusive" sampling of the high-dimensional parameter space, reminiscent of Monte-Carlo sampling. In contrast to Monte-Carlo, however, a functional of the parametric solution is then computed via compressive sensing methods from samples of functionals of the solution. A key ingredient in our analysis of independent interest consists in a generalization of recent results on the approximate sparsity of generalized polynomial chaos representations (gpc) of the parametric solution families, in terms of the gpc series with respect to tensorized Chebyshev polynomials. In particular, we establish sufficient conditions on the parametric inputs to the parametric operator equation such that the Chebyshev coefficients of the gpc expansion are contained in certain weighted $\ell_p$-spaces for $0<p\leq 1$. Based on this we show that reconstructions of the parametric solutions computed from the sampled problems converge, with high probability, at the $L_2$, resp. $L_\infty$ convergence rates afforded by best $s$-term approximations of the parametric solution up to logarithmic factors.

Josef Dick, Frances Kuo, Quoc Thong Le Gia et al.

We develop a convergence analysis of a multi-level algorithm combining higher order quasi-Monte Carlo (QMC) quadratures with general Petrov-Galerkin discretizations of countably affine parametric operator equations of elliptic and parabolic type, extending both the multi-level first order analysis in [\emph{F.Y.~Kuo, Ch.~Schwab, and I.H.~Sloan, Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficient} (in review)] and the single level higher order analysis in [\emph{J.~Dick, F.Y.~Kuo, Q.T.~Le~Gia, D.~Nuyens, and Ch.~Schwab, Higher order QMC Galerkin discretization for parametric operator equations} (in review)]. We cover, in particular, both definite as well as indefinite, strongly elliptic systems of partial differential equations (PDEs) in non-smooth domains, and discuss in detail the impact of higher order derivatives of {\KL} eigenfunctions in the parametrization of random PDE inputs on the convergence results. Based on our \emph{a-priori} error bounds, concrete choices of algorithm parameters are proposed in order to achieve a prescribed accuracy under minimal computational work. Problem classes and sufficient conditions on data are identified where multi-level higher order QMC Petrov-Galerkin algorithms outperform the corresponding single level versions of these algorithms. Numerical experiments confirm the theoretical results.

Josef Dick, Quoc T. Le Gia, Christoph Schwab

We analyze the convergence of higher order Quasi-Monte Carlo (QMC) quadratures of solution-functionals to countably-parametric, nonlinear operator equations with distributed uncertain parameters taking values in a separable Banach space $X$ admitting an unconditional Schauder basis. Such equations arise in numerical uncertainty quantification with random field inputs. Unconditional bases of $X$ render the random inputs and the solutions of the forward problem countably parametric, deterministic. We show that these parametric solutions belong to a class of weighted Bochner spaces of functions of countably many variables, with a particular structure of the QMC quadrature weights: up to a (problem-dependent, and possibly large) finite dimension, product weights can be used, and beyond this dimension, weighted spaces with so-called SPOD weights recently introduced in [F.Y.~Kuo, Ch.~Schwab, I.H.~Sloan, Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal., 50, 3351--3374, 2012.] can be used to describe the solution regularity. The regularity results in the present paper extend those in [J. Dick, F.Y.~Kuo, Q.T.~Le Gia, D.~Nuyens, Ch.~Schwab, Higher order QMC (Petrov-)Galerkin discretization for parametric operator equations. SIAM J. Numer. Anal., 52, 2676 -- 2702, 2014.] established for affine parametric, linear operator families; they imply, in particular, efficient constructions of (sequences of) QMC quadrature methods there, which are applicable to these problem classes. We present a hybridized version of the fast component-by-component (CBC for short) construction of a certain type of higher order digital net.