Efficient Numerical Method for Models Driven by Lévy Process via Hierarchical Matrices

Abstract

Modeling via fractional partial differential equations or a Lévy process has been an active area of research and has many applications. However, the lack of efficient numerical computation methods for general nonlocal operators impedes people from adopting such modeling tools. We proposed an efficient solver for the convection-diffusion equation whose operator is the infinitesimal generator of a Lévy process based on H-matrix technique. The proposed Crank Nicolson scheme is unconditionally stable and has a theoretical O(h 2 + ∆t 2 ) convergence rate. The H-matrix technique has theoretical O(N ) space and computational complexity compared to O(N 2 ) and O(N 3 ) respectively for the direct method. Numerical experiments demonstrate the efficiency of the new algorithm.

Data-driven learning of robust nonlocal physics from high-fidelity synthetic data

“…In Section 8, we see that one may extract computationally efficient compactly supported models providing sparse discrete operators that accurately approximate fractional operators with infinite horizons. This is particularly promising as a means of deriving preconditioners and O(n) solvers for fractional systems, as naive discretizations provide dense matrices which require complex hierarchical preconditioners to solve efficiently [62][63][64]…”

Numerical methods for nonlocal and fractional models

“…In this case, discretization results in dense and unstructured matrices, and hence the storage complexity will be O(N 2 ) whereas the computational complexity will be O(N 3 ). One promising strategy for efficiently solving the nonlocal problem (11.1) with unstructured meshes is to apply hierarchical matrices so that only linear storage and computational complexity are required; see (Xu and Darve 2018a).…”