Factorizing the Stochastic Galerkin System

Recent work has explored solver strategies for the linear system of equations arising from a spectral Galerkin approximation of the solution of PDEs with parameterized (or stochastic) inputs. We consider the related problem of a matrix equation whose matrix and right hand side depend on a set of parameters (e.g. a PDE with stochastic inputs semidiscretized in space) and examine the linear system arising from a similar Galerkin approximation of the solution. We derive a useful factorization of this system of equations, which yields bounds on the eigenvalues, clues to preconditioning, and a flexible implementation method for a wide array of problems. We complement this analysis with (i) a numerical study of preconditioners on a standard elliptic PDE test problem and (ii) a fluids application using existing CFD codes; the MATLAB codes used in the numerical studies are available online.Comment: 13 pages, 4 figures, 2 table

Computational aspects of scalar dispersion modeling and simulation in complex flows

We present an overview of nowadays modeling capabilities and numerical challanges in the simulation of scalar dispersion phenomena in complex flows. Results from the simulation of a passive plume emitted from a line source downstream of a square obstacle are summarized to provide an example of a basic test case where the reliability of computational techniques can be carefully established

Simplex stochastic collocation with ENO-type stencil selection for robust uncertainty quantification

Multi-element uncertainty quantification approaches can robustly resolve the high sensitivities caused by discontinuities in parametric space by reducing the polynomial degree locally to a piecewise linear approximation. It is important to extend the higher degree interpolation in the smooth regions up to a thin layer of linear elements that contain the discontinuity to maintain a highly accurate solution. This is achieved here by introducing Essentially Non-Oscillatory (ENO) type stencil selection into the Simplex Stochastic Collocation (SSC) method. For each simplex in the discretization of the parametric space, the stencil with the highest polynomial degree is selected from the set of candidate stencils to construct the local response surface approximation. The application of the resulting SSC–ENO method to a discontinuous test function shows a sharper resolution of the jumps and a higher order approximation of the percentiles near the singularity. SSC–ENO is also applied to a chemical model problem and a shock tube problem to study the impact of uncertainty both on the formation of discontinuities in time and on the location of discontinuities in space