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Accession Number:
ADA566203
Title:
Solving Differential Equations with Random Ultra-Sparse Numerical Discretizations
Descriptive Note:
Final rept. Dec 2008-Jun 2011
Corporate Author:
MICHIGAN STATE UNIV EAST LANSING
Report Date:
2011-09-29
Pagination or Media Count:
8.0
Abstract:
We proposed a novel approach which employs random sampling to generate an accurate non-uniform mesh for numerically solving Partial Differential Equation Boundary Value Problems PDE-BVPs. From a uniform probability distribution U over a 1D domain, we considered a M discretization of size N where MN. The statistical moments of the solutions to a given BVP on each of the M ulta-sparse meshes provide insight into identifying highly accurate non-uniform meshes. We used the pointwise mean and variance of the coarse-grid solutions to construct a mapping Qx from uniformly to non-uniformly spaced mesh-points. The error convergence properties of the approximate solution to the PDE-BVP on the non-uniform mesh are superior to a uniform mesh for a certain class of BVPs. In particular, the method works well for BVPs with locally non-smooth solutions. We fully developed a framework for studying the sampled sparse-mesh solutions and provided numerical evidence for the utility of this approach as applied to a set of example BVPs.
Distribution Statement:
APPROVED FOR PUBLIC RELEASE
