Showing papers by "Christoph Schwab published in 2018"

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

Abstract: 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 (ie 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

Large deformation shape uncertainty quantification in acoustic scattering

Abstract: We address shape uncertainty quantification for the two-dimensional Helmholtz transmission problem, where the shape of the scatterer is the only source of uncertainty. In the framework of the so-called deterministic approach, we provide a high-dimensional parametrization for the interface. Each domain configuration is mapped to a nominal configuration, obtaining a problem on a fixed domain with stochastic coefficients. To compute surrogate models and statistics of quantities of interest, we apply an adaptive, anisotropic Smolyak algorithm, which allows to attain high convergence rates that are independent of the number of dimensions activated in the parameter space. We also develop a regularity theory with respect to the spatial variable, with norm bounds that are independent of the parametric dimension. The techniques and theory presented in this paper can be easily generalized to any elliptic problem on a stochastic domain.

Shape Holomorphy of the stationary Navier-Stokes Equations *

Abstract: We consider the stationary Stokes and Navier--Stokes equations for viscous, incompressible flow in parameter dependent bounded domains $\mathrm{D}_T$, subject to homogeneous Dirichlet (``no-slip'')...

Quantized tensor-structured finite elements for second-order elliptic PDEs in two dimensions

Abstract: We analyze the approximation of the solutions of second-order elliptic problems, which have point singularities and belong to a countably normed space of analytic functions, with a first-order, h-version finite element (FE) method based on uniform tensor-product meshes. The FE solutions are well known to converge with algebraic rate at most 1 / 2 in terms of the number of degrees of freedom, and even slower in the presence of singularities. We analyze the compression of the FE coefficient vectors represented in the so-called quantized-tensor-train format. We prove, in a reference square, that the resulting FE approximations converge exponentially in terms of the effective number N of degrees of freedom involved in the representation: $$N={\mathcal {O}} ( \log ^{5} \varepsilon ^{-1} ) $$ , where $$\varepsilon \in (0,1)$$ is the accuracy measured in the energy norm. Numerically we show for solutions from the same class that the entire process of solving the tensor-structured Galerkin first-order FE discretization can achieve accuracy $$\varepsilon $$ in the energy norm with $$N={\mathcal {O}} ( \log ^{\kappa } \varepsilon ^{-1} ) $$ parameters, where $$\kappa <3$$ .

Quasi--Monte Carlo Integration for Affine-Parametric, Elliptic PDEs: Local Supports and Product Weights

Abstract: We analyze convergence rates of first-order quasi--Monte Carlo (QMC) integration with randomly shifted lattice rules and for higher-order, interlaced polynomial lattice rules for a class of countably parametric integrands that result from linear functionals of solutions of linear, elliptic diffusion equations with affine-parametric, uncertain coefficient function $a(x,{y}) = \bar{a}(x) + \sum_{j\geq 1} y_j \psi_j(x)$ in a bounded domain $D\subset \mathbb{R}^d$. Extending the result in [F. Y. Kuo, C. Schwab, and I. H. Sloan, SIAM J. Numer. Anal., 50 (2012), pp. 3351--3374], where $\psi_j$ was assumed to have global support in the domain $D$, we assume in the present paper that ${supp}(\psi_j)$ is localized in $D$ and that we have control on the overlaps of these supports. Under these conditions we prove dimension-independent convergence rates in [1/2,1) of randomly shifted lattice rules with product weights and corresponding higher-order convergence rates by higher-order, interlaced polynomial lattice rule...

Space–time hp -approximation of parabolic equations

Abstract: A new space–time finite element method for the solution of parabolic partial differential equations is introduced. In a mesh and degree-dependent norm, it is first shown that the discrete bilinear form for the space–time problem is both coercive and continuous, yielding existence and uniqueness of the associated discrete solution. In a second step, error estimates in this mesh-dependent norm are derived. In particular, we show that combining low-order elements for the space variable together with an hp-approximation of the problem with respect to the temporal variable allows us to decrease the optimal convergence rates for the approximation of elliptic problems only by a logarithmic factor. For simultaneous space–time hp-discretization in both, the spatial as well as the temporal variable, overall exponential convergence in mesh-degree dependent norms on the space–time cylinder is proved, under analytic regularity assumptions on the solution with respect to the spatial variable. Numerical results for linear model problems confirming exponential convergence are presented.

Numerical analysis of lognormal diffusions on the sphere

Abstract: Numerical solutions of stationary diffusion equations on the sphere with isotropic lognormal diffusion coefficients are considered. Holder 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 on the sphere 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.

Multilevel QMC with Product Weights for Affine-Parametric, Elliptic PDEs

Abstract: We present an error analysis of higher order Quasi-Monte Carlo (QMC) integration and of randomly shifted QMC lattice rules for parametric operator equations with uncertain input data taking values in Banach spaces. Parametric expansions of these input data in locally supported bases such as splines or wavelets was shown in Gantner et al. (SIAM J Numer Anal 56(1):111–135, 2018) to allow for dimension independent convergence rates of combined QMC-Galerkin approximations. In the present work, we review and refine the results in that reference to the multilevel setting, along the lines of Kuo et al. (Found Comput Math 15(2):441–449, 2015) where randomly shifted lattice rules and globally supported representations were considered, and also the results of Dick et al. (SIAM J Numer Anal 54(4):2541–2568, 2016) in the particular situation of locally supported bases in the parametrization of uncertain input data. In particular, we show that locally supported basis functions allow for multilevel QMC quadrature with product weights, and prove new error vs. work estimates superior to those in these references (albeit at stronger, mixed regularity assumptions on the parametric integrand functions than what was required in the single-level QMC error analysis in the first reference above). Numerical experiments on a model affine-parametric elliptic problem confirm the analysis.

Discontinuous Galerkin Methods for Acoustic Wave Propagation in Polygons

Abstract: We analyze space semi-discretizations of linear, second-order wave equations by discontinuous Galerkin methods in polygonal domains where solutions exhibit singular behavior near corners. To resolve these singularities, we consider two families of locally refined meshes: graded meshes and bisection refinement meshes. We prove that for appropriately chosen refinement parameters, optimal asymptotic rates of convergence with respect to the total number of degrees of freedom are obtained, both in the energy norm errors and the $$\mathcal {L}^2$$ -norm errors. The theoretical convergence orders are confirmed in a series of numerical experiments which also indicate that analogous results hold for incompatible data which is not covered by the currently available regularity theory.

Exponential Convergence of hp-FEM for Elliptic Problems in Polyhedra: Mixed Boundary Conditions and Anisotropic Polynomial Degrees

Abstract: We prove exponential rates of convergence of hp-version finite element methods on geometric meshes consisting of hexahedral elements for linear, second-order elliptic boundary value problems in axiparallel polyhedral domains. We extend and generalize our earlier work for homogeneous Dirichlet boundary conditions and uniform isotropic polynomial degrees to mixed Dirichlet–Neumann boundary conditions and to anisotropic, which increase linearly over mesh layers away from edges and vertices. In particular, we construct $$H^1$$ -conforming quasi-interpolation operators with N degrees of freedom and prove exponential consistency bounds $$\exp (-b\root 5 \of {N})$$ for piecewise analytic functions with singularities at edges, vertices and interfaces of boundary conditions, based on countably normed classes of weighted Sobolev spaces with non-homogeneous weights in the vicinity of Neumann edges.

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

Abstract: We analyze approximation rates by deep ReLU networks of a class of multivariate 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 multivariate 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 $${\mathcal {O}}\big (d^{2+\frac{1}{n}}\varepsilon ^{-\frac{1}{n}}\big )$$ nonzero 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.

Multilevel Quasi-Monte Carlo Uncertainty Quantification for Advection-Diffusion-Reaction

Abstract: We survey the numerical analysis of a class of deterministic, higher-order QMC integration methods in forward and inverse uncertainty quantification algorithms for advection-diffusion-reaction (ADR) equations in polygonal domains \(D\subset {\mathbb {R}}^2\) with distributed uncertain inputs. We admit spatially heterogeneous material properties. For the parametrization of the uncertainty, we assume at hand systems of functions which are locally supported in D. Distributed uncertain inputs are written in countably parametric, deterministic form with locally supported representation systems. Parametric regularity and sparsity of solution families and of response functions in scales of weighted Kontrat’ev spaces in D are quantified using analytic continuation.

Correction to: Multilevel Monte Carlo front-tracking for random scalar conservation laws

Abstract: An error in [4, Theorem 4.1, 4.5, Corollary 4.5] is corrected. There, in the Monte Carlo error bounds for front tracking for scalar conservation laws with random input data, 2-integrability in a Banach space of type 1 was assumed. In providing the corrected convergence rate bounds and error versus work analysis of multilevel Monte Carlo front-tracking methods, we also generalize [4] to q-integrability of the random entropy solution for some $$1

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

Abstract: 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.

Uncertainty Quantification for Spectral Fractional Diffusion: Sparsity Analysis of Parametric Solutions

Abstract: In bounded, polygonal domains $D \subset \Bbb{R}^d$, we analyze solution regularity and sparsity for computational uncertainty quantification for spectral fractional diffusion. Several types of unc...