arXiv subjects

Christoph Schwab

Publications and source records attributed to Christoph Schwab.

At least 55 records · Page 3Linked to original sources

arXiv2021

Deep ReLU Network Expression Rates for Option Prices in high-dimensional, exponential Lévy models

We study the expression rates of deep neural networks (DNNs for short) for option prices written on baskets of $d$ risky assets, whose log-returns are modelled by a multivariate Lévy process with general correlation structure of jumps. We establish sufficient conditions on the characteristic triplet of the Lévy process $X$ that ensure $\varepsilon$ error of DNN expressed option prices with DNNs of size that grows polynomially with respect to $\mathcal{O}(\varepsilon^{-1})$, and with constants implied in $\mathcal{O}(\cdot)$ which grow polynomially with respect $d$, thereby overcoming the curse of dimensionality and justifying the use of DNNs in financial modelling of large baskets in markets with jumps. In addition, we exploit parabolic smoothing of Kolmogorov partial integrodifferential equations for certain multivariate Lévy processes to present alternative architectures of ReLU DNNs that provide $\varepsilon$ expression error in DNN size $\mathcal{O}(|\log(\varepsilon)|^a)$ with exponent $a \sim d$, however, with constants implied in $\mathcal{O}(\cdot)$ growing exponentially with respect to $d$. Under stronger, dimension-uniform non-degeneracy conditions on the Lévy symbol, we obtain algebraic expression rates of option prices in exponential Lévy models which are free from the curse of dimensionality. In this case the ReLU DNN expression rates of prices depend on certain sparsity conditions on the characteristic Lévy triplet. We indicate several consequences and possible extensions of the present results.

math.NA↗

arXiv2021

Multilevel approximation of Gaussian random fields: Covariance compression, estimation and spatial prediction

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

math.ST↗

arXiv2020

Exponential Convergence of $hp$ FEM for Spectral Fractional Diffusion in Polygons

For the spectral fractional diffusion operator of order $2s\in (0,2)$ in bounded, curvilinear polygonal domains $Ω$ we prove exponential convergence of two classes of $hp$ discretizations under the assumption of analytic data, without any boundary compatibility, in the natural fractional Sobolev norm $\mathbb{H}^s(Ω)$. The first $hp$ discretization is based on writing the solution as a co-normal derivative of a $2+1$-dimensional local, linear elliptic boundary value problem, to which an $hp$-FE discretization is applied. A diagonalization in the extended variable reduces the numerical approximation of the inverse of the spectral fractional diffusion operator to the numerical approximation of a system of local, decoupled, second order reaction-diffusion equations in $Ω$. Leveraging results on robust exponential convergence of $hp$-FEM for second order, linear reaction diffusion boundary value problems in $Ω$, exponential convergence rates for solutions $u\in \mathbb{H}^s(Ω)$ of $\mathcal{L}^s u = f$ follow. Key ingredient in this $hp$-FEM are boundary fitted meshes with geometric mesh refinement towards $\partialΩ$. The second discretization is based on exponentially convergent sinc quadrature approximations of the Balakrishnan integral representation of $\mathcal{L}^{-s}$, combined with $hp$-FE discretizations of a decoupled system of local, linear, singularly perturbed reaction-diffusion equations in $Ω$. The present analysis for either approach extends to polygonal subsets $\widetilde{\mathcal{M}}$ of analytic, compact $2$-manifolds $\mathcal{M}$. Numerical experiments for model problems in nonconvex polygonal domains and with incompatible data confirm the theoretical results. Exponentially small bounds on Kolmogoroff $n$-widths of solutions sets for spectral fractional diffusion in polygons are deduced.

math.NA↗

arXiv2020

DNN Expression Rate Analysis of High-dimensional PDEs: Application to Option Pricing

We analyze approximation rates by deep ReLU networks of a class of multi-variate solutions of Kolmogorov equations which arise in option pricing. Key technical devices are deep ReLU architectures capable of efficiently approximating tensor products. Combining this with results concerning the approximation of well behaved (i.e. fulfilling some smoothness properties) univariate functions, this provides insights into rates of deep ReLU approximation of multi-variate functions with tensor structures. We apply this in particular to the model problem given by the price of a European maximum option on a basket of $d$ assets within the Black-Scholes model for European maximum option pricing. We prove that the solution to the $d$-variate option pricing problem can be approximated up to an $\varepsilon$-error by a deep ReLU network with depth $\mathcal{O}\big(\ln(d)\ln(\varepsilon^{-1})+\ln(d)^2\big)$ and $\mathcal{O}\big(d^{2+\frac{1}{n}}\varepsilon^{-\frac{1}{n}}\big)$ non-zero weights, where $n\in \mathbb{N}$ is arbitrary (with the constant implied in $\mathcal{O}(\cdot)$ depending on $n$). The techniques developed in the constructive proof are of independent interest in the analysis of the expressive power of deep neural networks for solution manifolds of PDEs in high dimension.

math.FA↗

arXiv2020

Exponential ReLU Neural Network Approximation Rates for Point and Edge Singularities

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.

math.NA↗

arXiv2020

$hp$-FEM for reaction-diffusion equations II. Robust exponential convergence for multiple length scales in corner domains

In bounded, polygonal domains $Ω\subset \mathbb{R}^2$ with Lipschitz boundary $\partialΩ$ consisting of a finite number of Jordan curves admitting analytic parametrizations, we analyze $hp$-FEM discretizations of linear, second order, singularly perturbed reaction diffusion equations on so-called geometric boundary layer meshes. We prove, under suitable analyticity assumptions on the data, that these $hp$-FEM afford exponential convergence in the natural "energy" norm of the problem, as long as the geometric boundary layer mesh can resolve the smallest length scale present in the problem. Numerical experiments confirm the robust exponential convergence of the proposed $hp$-FEM.

math.NA↗

arXiv2020

Space-time discontinuous Galerkin approximation of acoustic waves with point singularities

We develop a convergence theory of space-time discretizations for the linear, 2nd-order wave equation in polygonal domains $Ω\subset\mathbb{R}^2$, possibly occupied by piecewise homogeneous media with different propagation speeds. Building on an unconditionally stable space-time DG formulation developed in [Moiola, Perugia 2018], we (a) prove optimal convergence rates for the space-time scheme with local isotropic corner mesh refinement on the spatial domain, and (b) demonstrate numerically optimal convergence rates of a suitable \emph{sparse} space-time version of the DG scheme. The latter scheme is based on the so-called \emph{combination formula}, in conjunction with a family of anisotropic space-time DG-discretizations. It results in optimal-order convergent schemes, also in domains with corners, with a number of degrees of freedom that scales essentially like the DG solution of one stationary elliptic problem in $Ω$ on the finest spatial grid. Numerical experiments for both smooth and singular solutions support convergence rate optimality on spatially refined meshes of the full and sparse space-time DG schemes.

math.NA↗

arXiv2020

Analytic regularity for the incompressible Navier-Stokes equations in polygons

In a plane polygon $P$ with straight sides, we prove analytic regularity of the Leray-Hopf solution of the stationary, viscous, and incompressible Navier-Stokes equations. We assume small data, analytic volume force and no-slip boundary conditions. Analytic regularity is quantified in so-called countably normed, corner-weighted spaces with homogeneous norms. Implications of this analytic regularity include exponential smallness of Kolmogorov $N$-widths of solutions, exponential convergence rates of mixed $hp$-discontinuous Galerkin finite element and spectral element discretizations and of model order reduction techniques.

math.AP↗

arXiv2019

Tensor Rank bounds for Point Singularities in $\mathbb{R}^3$

We analyze rates of approximation by quantized, tensor-structured representations of functions with isolated point singularities in ${\mathbb R}^3$. We consider functions in countably normed Sobolev spaces with radial weights and analytic- or Gevrey-type control of weighted semi-norms. Several classes of boundary value and eigenvalue problems from science and engineering are discussed whose solutions belong to the countably normed spaces. It is shown that quantized, tensor-structured approximations of functions in these classes exhibit tensor ranks bounded polylogarithmically with respect to the accuracy $ε\in(0,1)$ in the Sobolev space $H^1$. We prove exponential convergence rates of three specific types of quantized tensor decompositions: quantized tensor train (QTT), transposed QTT and Tucker-QTT. In addition, the bounds for the patchwise decompositions are uniform with respect to the position of the point singularity. An auxiliary result of independent interest is the proof of exponential convergence of $hp$-finite element approximations for Gevrey-regular functions with point singularities in the unit cube $Q=(0,1)^3$. Numerical examples of function approximations and of Schrödinger-type eigenvalue problems illustrate the theoretical results.

math.NA↗

arXiv2019

Improved Efficiency of a Multi-Index FEM for Computational Uncertainty Quantification

We propose a multi-index algorithm for the Monte Carlo (MC) discretization of a linear, elliptic PDE with affine-parametric input. We prove an error vs. work analysis which allows a multi-level finite-element approximation in the physical domain, and apply the multi-index analysis with isotropic, unstructured mesh refinement in the physical domain for the solution of the forward problem, for the approximation of the random field, and for the Monte-Carlo quadrature error. Our approach allows Lipschitz domains and mesh hierarchies more general than tensor grids. The improvement in complexity over multi-level MC FEM is obtained from combining spacial discretization, dimension truncation and MC sampling in a multi-index fashion. Our analysis improves cost estimates compared to multi-level algorithms for similar problems and mathematically underpins the superior practical performance of multi-index algorithms for partial differential equations with random coefficients.

math.NA↗

arXiv2017

Multi-level Compressed Sensing Petrov-Galerkin discretization of high-dimensional parametric PDEs

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.

math.NA↗

arXiv2017

Tensor FEM for spectral fractional diffusion

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.

math.NA↗

arXiv2017

Fully discrete approximation of parametric and stochastic elliptic PDEs

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.

math.NA↗

arXiv2016

Multilevel higher order Quasi-Monte Carlo Bayesian Estimation

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.

math.NA↗

arXiv2016

Numerical analysis of lognormal diffusions on the sphere

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.

math.PR↗

arXiv2016

Multilevel Quasi-Monte Carlo Methods for Lognormal Diffusion Problems

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.

math.NA↗

arXiv2016

Higher order Quasi-Monte Carlo integration for Bayesian Estimation

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.

math.NA↗

arXiv2015

Isotropic Gaussian random fields on the sphere: Regularity, fast simulation and stochastic partial differential equations

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.

math.PR↗