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Accession Number ADA565202
Title Solving Differential Equations with Random Ultra-Sparse 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 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 M>>N. 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 Q(x) 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.
Keywords Approximation(Mathematics)
Boundary value problems
Computation science
Nonuniform mesh
Numerical partial differential equations
Partial differential equations
Random 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 2008-Jun 2011.
NTIS Issue Number 1304
Contract Number FA9550-09-1-0403

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