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Multilevel MonteCarlo front tracking for random scalar conservation laws
(2012)SAM Research ReportWe consider random scalar hyperbolic conservation laws (RSCLs) in spatial dimension $d\ge 1$ with bounded random flux functions which are $\mathbb{P}$a.s. Lipschitz continuous with respect to the state variable, for which there exists a unique random entropy solution (i.e., a measurable mapping from the probability space into $C(0,T;L^1(\mathbb{R}^d))$ with finite second moments). We present a convergence analysis of a MultiLevel ...Report 
hpDGQTT solution of highdimensional degenerate diffusion equations
(2012)SAM Research ReportWe consider the discretization of degenerate, timeinhomogeneous FokkerPlanck equations for diffusion problems in highdimensional domains. Wellposedness of the problem in timeweighted Bochner spaces is established. Analytic regularity of the timedependence of the solution in countably normed, weighted Sobolev spaces is established. Time discretization by the hpdiscontinuous Galerkin method is shown to converge exponentially. The ...Report 
Numerical solution of scalar conservation laws with random flux functions
(2012)Research ReportWe consider scalar hyperbolic conservation laws in several space dimensions, with a class of random (and parametric) flux functions. We propose a Karhunen–Loève expansion on the state space of the random flux. For random flux functions which are Lipschitz continuous with respect to the state variable, we prove the existence of a unique random entropy solution. Using a Karhunen–Loève spectral decomposition of the random flux into principal ...Report 
Covariance structure of parabolic stochastic partial differential equations
(2012)SAM Research ReportIn 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. Wellposedness of a spacetime weak variational formulation of this tensorized equation is established.Report 
hpDGQTT solution of highdimensional degenerate diffusion equations
(2012)SAM Research ReportWe consider the discretization of degenerate, timeinhomogeneous FokkerPlanck equations for diffusion problems in highdimensional domains. Wellposedness of the problem in timeweighted Bochner spaces is established. Analytic regularity of the timedependence of the solution in countably normed, weighted Sobolev spaces is established. Time discretization by the hpdiscontinuous We consider the discretization of degenerate, timeinhomogeneous ...Report 
Multilevel quasiMonte 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 quasiMonte 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 
Highdimensional adaptive sparse polynomial interpolation and applications to parametric PDEs
(2012)SAM Research ReportWe consider the problem of Lagrange polynomial interpolation in high or countably infinite dimension, motivated by the fast computation of solution to parametric/stochastic PDE’s. In such applications there is a substantial advantage in considering polynomial spaces that are sparse and anisotropic with respect to the different parametric variables. In an adaptive context, the polynomial space is enriched at different stages of the ...Report 
Numerical solution of scalar conservation laws with random flux functions
(2012)SAM Research ReportWe consider scalar hyperbolic conservation laws in several space dimensions, with a class of random (and parametric) flux functions. We propose a KarhunenLoève expansion on the state space of the random flux. For random flux functions which are Lipschitz continuous with respect to the state variable, we prove the existence of a unique random entropy solution. Using a KarhunenLoève spectral decomposition of the random flux into principal ...Report 
Lowrank 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 
Multilevel Monte Carlo methods for stochastic elliptic multiscale PDEs
(2012)SAM Research ReportIn this paper Monte Carlo Finite Element (MC FE) approximations for elliptic homogenization problems with random coefficients which oscillate on $n \in \mathbb{N}$ apriori known, separated length scales are considered. The convergence of multilevel MC FE (MLMC FE) discretizations is analyzed. In particular, it is considered that the multilevel FE discretization resolves the nest physical length scale, but the coarsest FE mesh does not, ...Report