2011/03/28 by David M. Bortz, Bortz, David, Andrew Christlieb +1
Computer Science · Decision Sciences · Mathematics · #65N50 #Applications (stat.AP) #Computational Geometry and Mesh Generation #FOS: Computer and information sciences #FOS: Mathematics #Mathematical Approximation and Integration #Numerical Analysis (math.NA) #Simulation Techniques and Applications
paper · pdf · doi:10.48550/arxiv.1103.5268
openalex publication_date 2011/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose a novel approach which employs random sampling to generate an accurate non-uniform mesh for numerically solving Partial Differential Equation Boundary Value Problems (PDE-BVP's). From a uniform probability distribution U over a 1D domain, we sample M discretizations of size N where M>>N. The statistical moments of the solutions to a given BVP on each of the M ultra-sparse meshes provide insight into identifying highly accurate non-uniform meshes. Essentially, we use 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 BVP's. In particular, the method works well for BVP's with locally non-smooth solutions. We present a framework for studying the sampled sparse-mesh solutions and provide numerical evidence for the utility of this approach as applied to a set of example BVP's. We conclude with a discussion of how the near-perfect paralellizability of our approach suggests that these strategies have the potential for highly efficient utilization of massively parallel multi-core technologies such as General Purpose Graphics Processing Units (GPGPU's). We believe that the proposed algorithm is beyond embarrassingly parallel; implementing it on anything but a massively multi-core architecture would be scandalous.