Efficient Numerical Method for Models Driven by L\'evy Process via Hierarchical Matrices

TL;DR

This paper introduces a fast, stable numerical solver for Le9vy process-driven models using hierarchical matrices, significantly reducing computational complexity and enabling practical applications.

Contribution

The paper presents a novel - matrix based solver for Le9vy process models that achieves  complexity, improving efficiency over traditional methods.

Findings

The proposed Crank-Nicolson scheme is unconditionally stable.

The -matrix technique reduces computational complexity to linear scale.

Numerical experiments confirm the efficiency and accuracy of the method.

Abstract

Modeling via fractional partial differential equations or a L\'evy 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 L\'evy process based on $\mathcal{H}$-matrix technique. The proposed Crank Nicolson scheme is unconditionally stable and has a theoretical $\mathcal{O}(h^2+\Delta t^2)$ convergence rate. The $\mathcal{H}$-matrix technique has theoretical $\mathcal{O}(N)$ space and computational complexity compared to $\mathcal{O}(N^2)$ and $\mathcal{O}(N^3)$ respectively for the direct method. Numerical experiments demonstrate the efficiency of the new algorithm.