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 $\mathcal{H}$-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.

40 pages, 16 figures

References in corpus (4)

• An introduction to Lévy processes with applications in finance
• Turbulence Modeling via the Fractional Laplacian
• Finite difference methods for fractional Laplacians
• A simple solver for the fractional Laplacian in multiple dimensions

Cited by in corpus (2)

• Numerical methods for nonlocal and fractional models
• Data-driven learning of robust nonlocal physics from high-fidelity synthetic data