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 bounded physical domain D ⊂ R d. The sparse sGFEMs are based on a separation of stochastic and deterministic input variables by Karhunen-Lò… (More)
We formulate collocation Runge–Kutta time-stepping schemes applied to linear parabolic evolution equations as space-time Petrov–Galerkin discretizations, and investigate their a priori stability for the parabolic space-time norms, that is the continuity constant of the discrete solution mapping. We focus on collocation based on A-stable Gauss–Legendre and… (More)
SUMMARY This paper addresses the solution of parabolic evolution equations simultaneously in space and time as may be of interest in e.g. optimal control problems constrained by such equations. As a model problem we consider the heat equation posed on the unit cube in Euclidean space of moderately high dimension. An a priori stable minimal residual… (More)
An algorithm for a stable parallelizable space-time Petrov-Galerkin discretization for linear parabolic evolution equations is given. Emphasis is on the reusability of spatial finite element codes.
We consider the simultaneous estimation of an optical flow field and an illumination source term in a movie sequence. The particular optical flow equation is obtained by assuming that the image intensity is a conserved quantity up to possible sources and sinks which represent varying illumination. We formulate this problem as an energy minimization problem… (More)
We design and analyze algorithms for the efficient sensitivity computation of eigenpairs of parametric elliptic self-adjoint eigenvalue problems (EVPs) on high-dimensional parameter spaces [AS10]. We quantify the analytic dependence of eigen-pairs on the parameters. For the efficient evaluation of parameter sensitivities of isolated eigenpairs on the entire… (More)
Galerkin discretizations of a class of parametric and random para-bolic partial differential equations (PDEs) are considered. The parabolic PDEs are assumed to depend on a vector y = (y 1 , y 2 , ...) of possibly countably many parameters y j which are assumed to take values in [−1, 1]. Well-posedness of weak formulations of these parametric equation in… (More)
Wavelet-in-time multigrid-in-space preconditioning of parabolic evolution equations Powered by TCPDF (www.tcpdf.org) Abstract. Two different space-time variational formulations of linear parabolic evolution equations are considered, one is symmetric and elliptic on the trial space while the other is not. In each case, space-time Petrov–Galerkin… (More)
Numerical methods for stochastic ordinary differential equations typically estimate moments of the solution from sampled paths. Instead, in this paper we directly target the determin-istic equation satisfied by the first and second moments. For the canonical examples with additive noise (Ornstein–Uhlenbeck process) or multiplicative noise (geometric… (More)
We construct space-time Petrov–Galerkin discretizations of the heat equation on an unbounded temporal interval, either right-unbounded or left-unbounded. The discrete trial and test spaces are defined using Laguerre polynomials in time and are shown to satisfy the discrete inf-sup condition. Numerical examples are provided.