Accession Number

ADA565202

Title

Solving Differential Equations with Random UltraSparse Numerical Discretizations.

Publication Date

Sep 2011

Media Count

8p

Personal Author

A. J. Christlieb D. M. Bortz

Abstract

We proposed a novel approach which employs random sampling to generate an accurate nonuniform mesh for numerically solving Partial Differential Equation Boundary Value Problems (PDEBVPs). From a uniform probability distribution U over a 1D domain, we considered a M discretization of size N where M>>N. The statistical moments of the solutions to a given BVP on each of the M ultasparse meshes provide insight into identifying highly accurate nonuniform meshes. We used the pointwise mean and variance of the coarsegrid solutions to construct a mapping Q(x) from uniformly to non uniformly spaced meshpoints. The error convergence properties of the approximate solution to the PDEBVP on the nonuniform mesh are superior to a uniform mesh for a certain class of BVPs. In particular, the method works well for BVPs with locally nonsmooth solutions. We fully developed a framework for studying the sampled sparsemesh solutions and provided numerical evidence for the utility of this approach as applied to a set of example BVPs.

Keywords

Approximation(Mathematics) Boundary value problems Computation science Mesh Nonuniform mesh Numerical partial differential equations Partial differential equations Random sampling Sampling


Source Agency

Non Paid ADAS

NTIS Subject Category

72B  Algebra, Analysis, Geometry, & Mathematical Logic 72F  Statistical Analysis 72E  Operations Research

Corporate Author

Colorado Univ. at Boulder. Dept. of Applied Mathematics.

Document Type

Technical report

Title Note

Final rept. Dec 2008Jun 2011.

NTIS Issue Number

1304

Contract Number

FA95500910403
