Scandalously Parallelizable Mesh Generation

TL;DR

This paper introduces a highly parallelizable sampling-based method for generating accurate non-uniform meshes to solve PDE boundary value problems, especially effective for locally non-smooth solutions, leveraging statistical moments from multiple sparse meshes.

Contribution

The paper presents a novel sampling approach that constructs non-uniform meshes using statistical moments, enabling scalable parallel computation for PDE solutions.

Findings

Method improves error convergence over uniform meshes for certain BVPs.

Approach is highly parallelizable, suitable for GPGPU architectures.

Numerical experiments demonstrate effectiveness on example BVPs.

Abstract

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.