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Higher-order convex approximations of Young measures in optimal control
(2001)FIM's preprintsReport -
Sparse Tensor Multi-Level Monte Carlo Finite Volume Methods for hyperbolic conservation laws with random intitial data
(2010)SAM Research ReportWe consider scalar hyperbolic conservation laws in several $(d \ge 1)$ spatial dimensions with stochastic initial data. We prove existence and uniqueness of a random-entropy solution and show existence of statistical moments of any order k of this random entropy solution. We present a class of numerical schemes of multi-level Monte Carlo Finite Volume (MLMC-FVM) type for the approximation of random entropy solutions as well as of their ...Report -
hp-DGFEM for Kolmogorov-Fokker-Planck equations of multivariate Lévy processes
(2011)SAM Research ReportWe analyze the discretization of non-local degenerate integrodifferential equations arising as so-called forward equations for jump-diffusion processes, in particular in option pricing problems when dealing with Lévy driven stochastic volatility models. Well-posedness of the arising equations is addressed. We develop and analyze stable discretization schemes. The discontinuous Galerkin (DG) Finite Element Method is analyzed. In the DG-FEM, ...Report -
Quasi-Monte Carlo finite element methods for elliptic PDEs with log-normal random coefficient
(2013)SAM Research ReportIn this paper we analyze the numerical approximation of diffusion problems over polyhedral domains in $R^d$ (d=1,2,3), with diffusion coefficient a(x,ω) given as a lognormal random field, i.e., a(x,ω)=exp(Z(x,ω)) where x is the spatial variable and Z(x,⋅) is a Gaussian random field. The analysis presents particular challenges since the corresponding bilinear form is not uniformly bounded away from 0 or ∞ over all possible realizations of ...Report -
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 -
Adaptive Petrov-Galerkin methods for first order transport equations
(2011)SAM Research ReportWe propose a general framework for well posed variational formulations of linear unsymmetric operators, taking first order transport \cs{and evolution} equations in bounded domains as primary orientation. We outline a general variational framework for stable discretizations of boundary value problems for these operators. To adaptively resolve anisotropic solution features such as propagating singularities the variational formulations ...Report -
Fully Discrete hp-Finite Elements: Fast Quadrature
(1999)SAM Research ReportA fully discrete hp finite element method is presented. It combines the features of the standard hp finite element method (conforming Galerkin Formulation, variable order quadrature schemes, geometric meshes, static condensation) and of the spectral element method (special shape functions and spectral quadrature techniques). The speed-up (relative to standard hp elements) is analyzed in detail both theoretically and computationally .Report -
A Note on Sparse, Adaptive Smolyak Quadratures for Bayesian Inverse Problems
(2013)SAM Research ReportWe present a novel, deterministic approach to inverse problems for identification of parameters in differential equations from noisy measurements. Based on the parametric deterministic formulation of Bayesian inverse problems with unknown input parameter from infinite dimensional, separable Banach spaces, we develop a practical computational algorithm for the efficient approximation of the infinite-dimensional integrals with respect to ...Report -
Low-rank tensor structure of linear diffusion operators in the TT and QTT formats
(2012)SAM Research ReportWe consider a class of multilevel matrices, which arise from the discretization of linear diffusion operators in a $d$-dimensional hypercube. Under certain assumptions on the structure of the diffusion tensor (motivated by financial models), we derive an explicit representation of such a matrix in the recently introduced Tensor Train (TT) format with the $TT$ ranks bounded from above by $2 + \lfloor \frac{d}{2}\rfloor$. We also show that ...Report -
Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients
(2012)SAM Research ReportThis paper is a sequel to our previous work $({Kuo, Schwab, Sloan, SIAM J.\ Numer.\ Anal., 2013})$ where quasi-Monte Carlo (QMC) methods (specifically, randomly shifted lattice rules) are applied to Finite Element (FE) discretizations of elliptic partial differential equations (PDEs) with a random coefficient represented in a countably infinite number of terms. We estimate the expected value of some linear functional of the solution, as ...Report