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Accession Number
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ADA565202
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Title
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Solving Differential Equations with Random Ultra-Sparse Numerical Discretizations.
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Publication Date
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Sep 2011
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Media Count
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8p
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Personal Author
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A. J. Christlieb D. M. Bortz
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Abstract
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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.
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Keywords
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Approximation(Mathematics)
Boundary value problems
Computation science
Mesh
Nonuniform mesh
Numerical partial differential equations
Partial differential equations
Random sampling
Sampling
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Source Agency
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Non Paid ADAS
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NTIS Subject Category
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72B - Algebra, Analysis, Geometry, & Mathematical Logic
72F - Statistical Analysis
72E - Operations Research
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Corporate Author
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Colorado Univ. at Boulder. Dept. of Applied Mathematics.
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Document Type
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Technical report
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Title Note
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Final rept. Dec 2008-Jun 2011.
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NTIS Issue Number
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1304
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Contract Number
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FA9550-09-1-0403
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