4 research outputs found
An asymptotically compatible probabilistic collocation method for randomly heterogeneous nonlocal problems
In this paper we present an asymptotically compatible meshfree method for
solving nonlocal equations with random coefficients, describing diffusion in
heterogeneous media. In particular, the random diffusivity coefficient is
described by a finite-dimensional random variable or a truncated combination of
random variables with the Karhunen-Lo\`{e}ve decomposition, then a
probabilistic collocation method (PCM) with sparse grids is employed to sample
the stochastic process. On each sample, the deterministic nonlocal diffusion
problem is discretized with an optimization-based meshfree quadrature rule. We
present rigorous analysis for the proposed scheme and demonstrate convergence
for a number of benchmark problems, showing that it sustains the asymptotic
compatibility spatially and achieves an algebraic or sub-exponential
convergence rate in the random coefficients space as the number of collocation
points grows. Finally, to validate the applicability of this approach we
consider a randomly heterogeneous nonlocal problem with a given spatial
correlation structure, demonstrating that the proposed PCM approach achieves
substantial speed-up compared to conventional Monte Carlo simulations