Christoph Schwab

• PhD
• ETH Zurich

We consider the problem of Lagrange polynomial interpolation in high or countably infinite dimension, motivated by the fast computation of solutions to partial differential equations (PDEs) depending on a possibly large number of parameters which result from the application of generalised polynomial chaos discretisations to random and stochastic PD...

L'approximation numérique des équations aux dérivées partielles paramétriques D(u,y)=0D(u,y)=0 constitue un défi de calcul lorsque la dimension d du vecteur de paramètre y est grande, en raison de ce qui est com- munément appelé plaie des grandes dimensions. Dans and , Il a été démontré que, pour une certaine classe d'EDP elliptiques à coefficients...

We review the recent results of D. Schötzau et al. [SIAM J. Numer. Anal. 51, No. 3, 1610–1633 (2013; Zbl 1276.65084); ibid. 51, No. 4, 2005–2035 (2013; Zbl 06219392)], and establish the exponential convergence of hp-version discontinuous Galerkin finite element methods for the numerical approximation of linear second-order elliptic boundary value p...

Isotropic Gaussian random fields on the sphere are characterized by Karhunen-Lo\`eve 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 H\"older continuity and sample differentiability of the r...

We introduce and analyze hp-version discontinuous Galerkin (dG) finite element methods for the numerical approximation of linear second-order elliptic boundary-value problems in three-dimensional polyhedral domains. To resolve possible corner-, edge- and corner-edge singularities, we consider hexahedral meshes that are geometrically and anisotropic...

We analyze the convergence and complexity of multilevel Monte Carlo discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales. We show under low regularity assumptions on the solution that the judicious combination of low order Galerkin discretizations in space and an Euler–Maruyama discretizatio...

The numerical approximation of parametric partial differential equations is a computational challenge, in particular when the number of involved parameter is large. This paper considers a model class of second order, linear, parametric, elliptic PDEs on a bounded domain D with diffusion coefficients depending on the parameters in an affine manner....

The goal of this paper is to establish exponential convergence of hp-version interior penalty (IP) discontinuous Galerkin (dG) finite element methods for the numerical approximation of linear second-order elliptic boundary-value problems with homogeneous Dirichlet boundary conditions and piecewise analytic data in three-dimensional polyhedral domai...

We prove exponential rates of convergence of hp-version discontinuous Galerkin (dG) interior penalty finite element methods for second-order elliptic problems with mixed Dirichlet-Neumann boundary conditions in axiparallel polyhedra. The dG discretizations are based on axiparallel, σ-geometric anisotropic meshes of mapped hexahedra and anisotropic...

In this paper parabolic random partial differential equations and parabolic stochastic partial differential equations driven by a Wiener process are considered. A deterministic, tensorized evolution equation for the second moment and the covariance of the solutions of the parabolic stochastic partial differential equations is derived. Well-posednes...

We extend the multi-level Monte Carlo (MLMC) in order to quantify uncertainty in the solutions of multi-dimensional hyperbolic systems of conservation laws with uncertain initial data. The algorithm is presented and several issues arising in the massively parallel numerical implementation are addressed. In particular, we present a novel load balanc...

For a nonlinear functional f , and a function u from the span of a set of tensor product interpolets, it is shown how to compute the interpolant of f (u) from the span of this set of tensor product interpolets in linear complexity, assuming that the index set has a certain multiple tree structure. Applications are found in the field of (adaptive) t...

Locally periodic, elliptic multiscale problems in a bounded Lipschitz domain D⊂Rn with K⩾2 separated scales are reduced to an elliptic system of K coupled, anisotropic elliptic one-scale problems in a cartesian product domain of total dimension Kn (e.g. [2,3,11,26]). In [23,31], it has been shown how these coupled elliptic problems could be solved...

In Monte Carlo methods quadrupling the sample size halves the error. In simulations of stochastic partial differential equations (SPDEs), the total work is the sample size times the solution cost of an instance of the partial differential equation. A Multi-level Monte Carlo method is introduced which allows, in certain cases, to reduce the overall...

A multiscale generalised hp-finite element method (MSFEM) for time harmonic wave propagation in bands of locally periodic media of large, but finite extent, e.g., photonic crystal (PhC) bands, is presented. The method distinguishes itself by its size robustness, i.e., to achieve a prescribed error its computational effort does not depend on the num...

We investigate the convergence rate of approximations by finite sums of rank-1 tensors of solutions of multiparametric elliptic PDEs. Such PDEs arise, for example, in the parametric, deterministic reformulation of elliptic PDEs with random field inputs, based, for example, on the $M$-term truncated Karhunen-Loève expansion. Our approach could be re...

Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D⊂ℝ d are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in L 2(D)-orthogonal bases, and on viewing the coefficients of these expansions as...

This work presents a thorough treatment of boundary element methods (BEM) for solving strongly elliptic boundary integral equations obtained from boundary reduction of elliptic boundary value problems in $\mathbb{R}^3$. The book is self-contained, the prerequisites on elliptic partial differential and integral equations being presented in Chapters...

In practice, the description of the “true” physical surface might be very compli- 3 cated or even not available as an exact analytic function and has to be approximated 4 by using, e.g., pointwise measurements of the surface or some geometric mod- 5 elling software. In this chapter, we will address the question how to approximate 6 quite general su...

In order to implement the Galerkin method for boundary integral equations, the approximation of the coefficients of the system matrix and the right-hand side becomes a primary task.

Partial differential equations can be directly discretized by means of difference methods or finite element methods (domain methods).

Homogeneous, linear elliptic boundary value problems with constant coefficients can be transformed into boundary integral equations by using the integral equation method. In this chapter we will introduce the relevant boundary integral operators and we will derive the most important mapping properties and representations. We will also present the...

We review the design and analysis of multiresolution (wavelet) methods for the numerical solution of the Kolmogorov equations arising, among others, in financial engineering when Lévy and Feller or additive processes are used to model the dynamics of the risky assets. In particular, the Dirichlet and free boundary problems connected to barrier and...

Parametric partial differential equations are commonly used to model physical systems. They also arise when Wiener chaos expansions are used as an alternative to Monte Carlo when solving stochastic elliptic problems. This paper considers a model class of second order, linear, parametric, elliptic PDEs in a bounded domain D with coefficients dependi...

Adaptive tensor product wavelet methods are applied for solving Poisson’s equation, as well as anisotropic generalizations, in high space dimensions. It will be demonstrated that the resulting approximations converge in energy norm with the same rate as the best approximations from the span of the best N tensor product wavelets, where moreover the...

We consider the weakly singular boundary integral equation Vu=g(ω) on a deterministic smooth closed curve Γ⊂R2 with random loading g(ω). Given the kth order statistical moment of g, the aim is the efficient deterministic computation of the kth order statistical moment of u. We derive a deterministic formulation for the kth statistical moment. It is...

We introduce and analyze hp-version discontinuous Galerkin (dG) finite element methods for the numerical approximation of linear second-order elliptic boundary value problems in three dimensional polyhedral domains. In order to resolve possible corner-, edge-and corner-edge singularities, we consider hexahedral meshes that are geometrically and ani...

With respect to space-time tensor-product wavelet bases, parabolic initial boundary value problems are equivalently formulated as bi-infinite matrix problems. Adaptive wavelet methods are shown to yield sequences of approximate solutions which converge at the optimal rate. In case the spatial domain is of product type, the use of spatial tensor pro...

We describe the analysis and the implementation of two finite element (FE) algorithms for the deterministic numerical solution of elliptic boundary value problems with stochastic coefficients.They are based on separation of deterministic and stochastic parts of the input data by a Karhunen–Loève expansion, truncated after M terms. With a change of...

We propose and analyze sparse deterministic-stochastic tensor Galerkin finite element methods (sparse sGFEMs) for the numerical solution of elliptic partial differential equations (PDEs) with random coefficients in a physical domain $D\subset\mathbb{R}^d$. In tensor product sGFEMs, the variational solution to the boundary value problem is approxima...

In this paper, we will consider the modelling of problems in linear elasticity on thin plates by the models of Kirchhoff–Love and Reissner–Mindlin. A fundamental investigation for the Kirchhoff plate goes back to Morgenstern (Arch. Ration. Mech. Anal. 4:145–152, 1959) and is based on the two-energies principle of Prager and Synge. This was half a c...

Partial differential equations with nonnegative characteristic form arise in numerous mathematical models in science. In problems of this kind, the exponential growth of computational complexity as a function of the dimension d of the problem domain, the so-called “curse of dimension”, is exacerbated by the fact that the problem may be transport-do...

For Au=f with an elliptic differential operator A:H ® H¢{A:\mathcal{H} \rightarrow \mathcal{H}'} and stochastic data f, the m-point correlation function Mm u{{\mathcal M}^m u} of the random solution u satisfies a deterministic equation with the m-fold tensor product operator A (m) of A. Sparse tensor products of hierarchic FE-spaces in H{\mathcal{H...

We consider the numerical solution of elliptic boundary value problems in domains with random boundary perturbations. Assuming normal perturbations with small amplitude and known mean field and two-point correlation function, we derive, using a second order shape calculus, deterministic equations for the mean field and the two-point correlation fun...

With standard isotropic approximation by (piecewise) polynomials of fixed order in a domain Dsubset mathbb{R}^d , the convergence rate in terms of the number N of degrees of freedom is inversely proportional to the space dimension d . This so-called curse of dimensionality can be circumvented by applying sparse tensor product approximation, when ce...

The formulation of the fluid flow in an unbounded exterior domain Ω is not always convenient for computations and, therefore, the problem is often truncated to a bounded domain Ω−⊂Ω with an artificial exterior boundary Γ. Then the problem of the choice of suitable “transparent” boundary conditions on Γ appears. Another possibility is to simulate th...

A scalar, elliptic boundary-value problem in divergence form with stochastic diffusion coefficient a ( x , ω ) in a bounded domain D ⊂ ℝ d is reformulated as a deterministic, infinite-dimensional, parametric problem by separation of deterministic ( x ∈ D ) and stochastic ( ω ∈ Ω) variables in a ( x , ω ) via Karhúnen–Loève or Legendre expansio...

KL approximation of a possibly instationary random field a(ω, x) ∈ L2(Ω, dP; L∞(D)) subject to prescribed meanfield and covariance in a polyhedral domain D⊂Rd is analyzed. We show how for stationary covariances Va(x, x′) = ga(|x − x′|) with ga(z) analytic outside of z = 0, an M-term approximate KL-expansion aM(ω, x) of a(ω, x) can be computed in lo...

We investigate the analytic regularity of the Stokes problem in a polygonal domain Ω⊂R2 with straight sides and piecewise analytic data. We establish a shift theorem in weighted Sobolev spaces of arbitrary order with explicit control of the order-dependence of the constants. The shift-theorem in the framework of countably weighted Sobolev spaces im...

Let A: V → V′ be a strongly elliptic operator on a d-dimensional manifold D (polyhedra or boundaries of polyhedra are also allowed). An operator equation Au = f with stochastic data f is considered. The goal of the computation is the mean field and higher moments \(\mathcal{M}^1 u \in V,\mathcal{M}^2 u \in V \otimes V,...,\mathcal{M}^k u \in V \oti...

In this paper we propose and analyze a wavelet-based numerical method for exact controllability of the one-dimensional wave equation, concentrating on the particular case of Dirichlet controls. We use the Hilbert uniqueness method (HUM) and prove this result in the context of the finite element space semidiscretization with spline wavelet basis fun...

We describe a deterministic finite element (FE) solution algorithm for a stochastic elliptic boundary value problem (sbvp), whose coefficients are assumed to be random fields with finite second moments and known, piecewise smooth two-point spatial correlation function. Separation of random and deterministic variables (parametrization of the uncerta...

We numerically solve parabolic problems u(t) + Au = 0 in (0, T) x Omega, T < infinity, where Omega subset of R is a bounded interval and A is a strongly elliptic integrodi. erential operator of order rho is an element of [ 0, 2]. A discontinuous Galerkin (dG) discretization in time and a wavelet discretization in space are used. The densely populat...

We present a finite element method for the numerical solution of diffusion problems on rough surfaces. The problem is transformed to an elliptic homogenization problem in a two dimensional parameter domain with a rapidly oscillating diffusion tensor and source term. The finite element method is based on the heterogeneous multiscale methods of Weina...

We consider the Stokes problem of incompressible fluid flow in three-dimensional polyhedral domains discretized oil hexahedral meshes with hp-discontinuous Galerkin finite elements of type Q(k) for the velocity and Q(k-1) for the pressure. We prove that these elements are inf-sup stable on geometric edge meshes that are refined anisotropically and...

We establish multiresolution norm equivalences in weighted spaces L 2 w ((0,1)) with possibly singular weight functions w(x)≥0 in (0,1). Our analysis exploits the locality of the biorthogonal wavelet basis and its dual basis functions. The discrete norms are sums of wavelet coefficients which are weighted with respect to the collocated weight funct...

Prices of European plain vanilla as well as barrier and compound options on one risky asset in a Black-Scholes market with stochastic volatility are expressed as solutions of degenerate parabolic partial differential equations in two spatial variables: the spot price S and the volatility process variable y. We present and analyze a pricing algorith...

We consider the numerical solution of diffusion problems in $(0,T) \times \Omega$ for $\Omega\subset \mathbb{R}^d$ and for $T > 0$ in dimension $d \ge 1$. We use a wavelet based sparse grid space discretization with mesh-width $h$ and order $p \ge 1$, and $hp$ discontinuous Galerkin time-discretization of order $r = O(\left|\log h\right|)$ on a geo...

Arbitrage-free prices $u$ of European contracts on risky assets whose log-returns are modelled by Lévy processes satisfy a parabolic partial integro-differential equation (PIDE) $\partial_t u + {\mathcal{A}}[u] = 0$. This PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by t...

We formulate elliptic boundary value problems with stochastic loading in a bounded domain D d . We show well-posedness of the problem in stochastic Sobolev spaces and we derive a deterministic elliptic PDE in DD for the spatial correlation of the random solution. We show well-posedness and regularity results for this PDE in a scale of weighted Sobo...

We consider stabilized mixed hp-discontinuous Galerkin methods for the discretization of the Stokes problem in three-dimensional polyhedral domains. The methods are stabilized with a term penalizing the pressure jumps. For this approach it is shown that Q(k) - Q(k) and Q(k) - Q(k-1) elements satisfy a generalized inf-sup condition on geometric edge...

The general theory of approximation of (possibly generalized) Young measures is presented, and concrete cases are investigated. An adjoint-operator approach, combined with quasi-interpolation of test integrands, is systematically used. Applicability is demonstrated on an optimal control problem for an elliptic system, together with one-dimensional...

Summary. In this paper, we consider the Stokes problem in a three-dimensional polyhedral domain discretized with hp finite elements of type ℚk for the velocity and ℚk−2 for the pressure, defined on hexahedral meshes anisotropically and non quasi-uniformly refined towards faces, edges, and corners. The inf-sup constant of the discretized problem is...

Multiple scale homogenization problems are reduced to single scale problems in higher dimension. It is shown that sparse tensor product Finite Element Methods (FEM) allow the numerical solution in complexity independent of the dimension and of the length scale. Problems with stochastic input data are reformulated as high dimensional deterministic p...

Die Aufgabe der vorliegenden Arbeit war die Beantwortung der Frage, ob die Verständlichkeit von Sprache, unter experimentellen Bedingungen, mit digitalen Hörhilfen, mit und ohne Mikrofonrichtcharakteristik, differiert. Um möglichst realistische Testbedingungen zu schaffen wurden bei der Untersuchung Umweltgeräusche durch einen Störlärm-Rauschpegel...

The paper is concerned with the numerical solution of nonlinear conservation laws and nonlinear convection–diffusion problems. We discuss two versions of this method: (a) Finite volume discontinuous Galerkin (FVDG) method, which is a generalization of the combined finite volume–finite element (FV–FE) method. Its advantage is the use of only one mes...

In this paper, we introduce and analyze local discontinuous Galerkin methods for the Stokes system. For arbitrary meshes with hanging nodes and elements of various shapes we derive a priori estimates for the L 2 -norm of the errors in the velocities and the pressure. We show that optimal order estimates are obtained when polynomials of degree k are...

. We study the convergence properties of the hp-version of the local discontinuous Galerkin finite element method for convection-diffusion problems; we consider a model problem in a one-dimensional space domain. We allow arbitrary meshes and polynomial degree distributions and obtain upper bounds for the energy norm of the error which are explicit...

We introduce the concept of generalized Finite Element Method (gFEM) for the numerical treatment of homogenization problems. These problems are characterized by highly oscillatory periodic (or patchwise periodic) pattern in the coefficients of the differential equation and their solutions exhibit a multiple scale behavior: a macroscopic behavior su...

The paper is concerned with the discontinuous Galerkin finite element method for the numerical solution of nonlinear conservation laws and nonlinear convection-diffusion problems with emphasis on applications to the simulation of compressible flows. We discuss two versions of this method: (a) Finite volume discontinuous Galerkin method, which is a...

In this paper, we consider the Stokes problem in a three-dimensional polyhedral domain discretized with hp finite elements of type Qk for the velocity and Qk−2 for the pressure, defined on meshes anisotropically and non-quasi-uniformly refined towards faces, edges, and corners. The inf-sup constant of the discretized problem is independent of arbit...

The Discontinuous Galerkin (DG) time-stepping method for the numerical solution of initial value ODEs is analyzed in the context of the hp-version of the Galerkin method. New a priori error bounds explicit in the time steps and in the approximation orders are derived and it is proved that the DG method gives spectral and exponential accuracy for pr...

We analyze the hp-version of the streamline-diffusion finite element method (SDFEM) and of the discontinuous Galerkin finite element method (DGFEM) for first-order linear hyperbolic problems. For both methods, we derive new error estimates on general finite element meshes which are sharp in the mesh-width h and in the spectral order p of the method...

Two hp–finite element methods for the Stokes problem in polygonal domains are presented: We discuss the Sk×Sk−2 elements which are stable on anisotropic and irregular meshes and introduce a stabilized Galerkin Least Squares approach featuring equal-order interpolation in the velocity and the pressure. Both methods lead to exponential rates of conve...

Inequalities of Jackson and Bernstein type are derived for polynomial approximation on simplices with respect to Sobolev norms. Although we cannot use orthogonal polynomials, sharp estimates are obtained from a decomposition into orthogonal subspaces. The formulas reflect the symmetries of simplices, but analogous estimates on rectangles show that...

The discontinuous Galerkin finite element method (DGFEM) for the time discretization of parabolic problems is analyzed in the context of the hp-version of the Galerkin method. Error bounds which are explicit in the time steps as well as in the approximation orders are derived and it is shown that the hp-DGFEM gives spectral convergence in problems...

The divergence stability of mixed hp Finite Element Methods for incompressible fluid flow is analyzed. A discrete inf-sup condition is proved for a general class of meshes. The meshes may be refined anisotropically, geometrically and may contain hanging nodes on geometric patches. The inf-sup constant is shown to be independent of the aspect ratio...

this paper is to extend the error analysis of the hp-DGFEM, developed in our earlier work [8] for first-order hyperbolic equations, to general second-order partial differential equations with nonnegative characteristic form. In [8] an error bound, optimal both in terms of the local mesh size h and the local polynomial degree p, was derived for the...

The definition, essential properties and formulation of hierarchic models for laminated plates and shells are presented. The hierarchic models satisfy three essential requirements: approximability; asymptotic consistency, and optimality of convergence rate. Aspects of implementation are discussed and the performance characteristics are illustrated...

A stabilized mixed hp-Finite Element Method (FEM) of Galerkin Least Squares type for the Stokes problem in polygonal domains is presented and analyzed. It is proved that for equal order velocity and pressure spaces this method leads to exponential rates of convergence provided that the data is piecewise analytic.

A stabilized mixed hp Finite Element Method (FEM) of Galerkin Least Squares type for the Stokes problem in polygonal domains is presented and analyzed. It is proved that for equal order velocity and pressure spaces this method leads to exponential rates of convergence provided that the data is piecewise analytic.

We analyze multiscale Galerkin methods for strongly elliptic boundary integral equations of order zero on closed surfaces in ℝ3. Piecewise polynomial, discontinuous multiwavelet bases of any polynomial degree are constructed explicitly. We show that optimal convergence rates in the boundary energy norm and in certain negative norms can be achieved...

The Galerkin discretization of a Fredholm integral equation of the second kind on a closed, piecewise analytic surface $\Gamma \subset \hbox{\sf l\kern-.13em R}^3$ is analyzed. High order, $hp$ -boundary elements on grids which are geometrically graded toward the edges and vertices of the surface give exponential convergence, similar to what is...

We study the uniform approximation of boundary layer functions exp(Gammax=d) for x 2 (0; 1), d 2 (0; 1], by the p and hp versions of the finite element method. For the p version (with fixed mesh), we prove super-exponential convergence in the range p+1=2 ? e=(2d). We also establish, for this version, an overall convergence rate of O(p Gamma1 p ln p...

We study the uniform approximation of boundary layer functions exp(Gammax=d) for x 2 (0; 1), d 2 (0; 1], by the p and hp versions of the finite element method. For the p version (with fixed mesh), we prove super-exponential convergence in the range p+1=2 ? e=(2d). We also establish, for this version, an overall convergence rate of O(p Gamma1 p ln p...

The goals of this project were to (1) analyze numerical phenomena such as locking and boundary layers occurring in the modeling of elastic bodies, and obtain methods with robust performance, (2)extend this analysis to hierarchies of models, and (3) continue investigation into the p and h-p FEM. Specifically, the locking of hierarchy of plate models...

We analyze the robustness of various standard finite element schemes for a hierarchy of plate models and obtain asymptotic convergence estimates that are uniform in terms of the thickness d. We identify h version schemes that show locking, i.e., for which the asymptotic convergence rate deteriorates as d→0, and also show that the p version is free...

An elliptic boundary value problem in the interior or exterior of a polygon is transformed into an equivalent first kind boundary integral equation. Its Galerkin discretization with N degrees of freedom on the boundary with spline wavelets as basis functions is analyzed. A truncation strategy is presented which allows to reduce the number of nonzer...

We consider a Galerkin method for an elliptic pseudodifferential operator of order zero on a two-dimensional manifold. We use piecewise linear discontinuous trial functions on a triangular mesh and describe an orthonormal wavelet basis. Using this basis we can compress the stiffness matrix from N 2 to O(N log N) nonzero entries and still obtain (up...

We analyze the robustness of various standard finite element schemes for the Reissner-Mindlin plate and obtain asymptotic convergence estimates that are uniform in terms of the thickness d. We identify h version schemes that show locking, i.e. for which the asymptotic convergence rate deteriorates as d (right arrow) 0 and also show that the p versi...

The problem of locking , which arises in the approximation of parameters dependent problems has been extensively investigated. A general theoretical framework to analyze this phenomenon has been developed, and the locking and robustness of different finite element schemes for various problems has been characterized. Work on the p and h-p versions o...

Vita. Thesis (Ph. D.)--University of Maryland at College Park, 1989. Includes bibliographical references (leaf 246).

We propose a novel class of sparse tensor algorithms for the numerical solution of stochastic elliptic PDEs. The methods are based on a hierarchic discretization in both, physical and probability space. The discretization spaces are then intertwined in a sparse tensor product fashion, leading to algorithms of log-linear complexity. We will present...

We compute the expectation and the two-point correlation of the solution to el- liptic boundary value problems with stochastic input data. Besides stochastic loadings, via perturbation theory, our approach covers also elliptic problems on stochastic domains or with stochastic coecients (1, 2). The solution's two-point correlation satises a determin...