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Sparse tensor spherical harmonics approximation in radiative transfer
(2010)SAM Research ReportThe stationary monochromatic radiative transfer equation is a partial differential transport equation stated on a five-dimensional phase space. To obtain a well-posed problem, inflow boundary conditions have to be prescribed. The sparse tensor product discretization has been successfully applied to finite element methods in radiative transfer with wavelet discretization of the angular domain (Widmer2009a). In this report we show that the ...Report -
hp-FEM for second moments of elliptic PDEs with stochastic data Part 1: Analytic regularity
(2010)SAM Research ReportFor a linear second order elliptic partial differential operator $A: V → V'$, we consider the boundary value problems $Au=f$ with stationary Gaussian random data $f$ over the dual $V'$ of the separable Hilbert space $V$ in which the solution u is sought. The operator $A$ is assumed to be deterministic and bijective. The unique solution $u= A^-$$^1f $ is a Gaussian random field over $V$. It is characterized by its mean field $E_u$ and ...Report -
hp-FEM for second moments of elliptic PDEs with stochastic data Part 2: Exponential convergence
(2010)SAM Research ReportWe prove exponential rates of convergence of a class of $hp$ Galerkin Finite Element approximations of solutions to a model tensor non-hypoelliptic equation in the unit square □ = (0,1)$^2$ which exhibit singularities on ∂□ and on the diagonal ∆ = {($x,y$) ∈ □ : $x$ = $y$}, but are otherwise analytic in □. As we explained in the first part [6] of this work, such problems arise as deterministic second moment equations of linear, second ...Report -
Sparse Tensor Approximation of Parametric Eigenvalue Problems
(2010)SAM Research ReportWe design and analyze algorithms for the efficient sensitivity computation of eigenpairs of parametric elliptic self-adjoint eigenvalue problems on high-dimensional parameter spaces. We quantify the analytic dependence of eigenpairs on the parameters. For the efficient approximate evaluation of parameter sensitivities of isolated eigenpairs on the entire parameter space we propose and analyze a sparse tensor spectral collocation method ...Report -
Sparse tensor Galerkin discretizations for parametric and random parabolic PDEs. I: Analytic regularity and gpc-approximation
(2010)SAM Research ReportFor initial boundary value problems of linear parabolic partial differential equations with random coefficients, we show analyticity of the solution with respect to the parameters and give an apriori error analysis for sparse tensor, space-time discretizations. The problem is reduced to a parametric family of deterministic initial boundary value problems on an infinite dimensional parameterspace by Galerkin projection onto finitely supported ...Report -
Numerical analysis of additive, Lévy and Feller processes with applications to option pricing
(2010)SAM Research ReportWe review the design and analysis of multiresolution (wavelet) methods for the numerical solution of the Kolmogoroff 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 American style contracts are specified and solution algorithms based on wavelet representations ...Report -
Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs
(2010)SAM Research ReportParametric 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 depending on possibly countably many parameters. It shows that the dependence of the solution on ...Report -
Analytic regularity and gpc approximation for parametric and random 2nd order hyperbolic PDEs
(2010)SAM Research ReportInitial boundary value problems of linear second order hyperbolic partial differential equations whose coefficients depend on countably many random parameters are reduced to a parametric family of deterministic initial boundary value problems on an infinite dimensional parameter space. This parametric family is approximated by Galerkin projection onto finitely supported polynomial systems in the parameter space. We establish uniform ...Report -
Multi-Level Monte Carlo Finite Element Method for elliptic PDEs with stochastic coefficients
(2010)SAM Research ReportIt is a well-known property of Monte Carlo methods that quadrupling the sample size halves the error. In the case of simulations of a stochastic partial differential equations, this implies that the total work is the sample size times the discretization costs of the equation. This leads to a convergence rate which is impractical for many simulations, namely in finance, physics and geosciences. With the Multi--level Monte Carlo method ...Report -
Fast evaluation of nonlinear functionals of tensor product wavelet expansions
(2010)SAM Research ReportWe investigate the convergence rate of approximations by finite sums of rank-1 tensors of solutions of multi-parametric 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 regarded as either a class of compressed approximations of these solution or as a new class of ...Report