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.