Kailai Xu

Kailai Xu, Eric Darve · stanford

A spectral method is considered for approximating the fractional Laplacian and solving the fractional Poisson problem in 2D and 3D unit balls. The method is based on the explicit formulation of the eigenfunctions and eigenvalues of the fractional Laplacian in the unit balls under the weighted $L^2$ space. The resulting method enjoys spectral accuracy for all fractional index $α\in (0,2)$ and is computationally efficient due to the orthogonality of the basis functions. We also proposed a numerical integration strategy for computing the coefficients. Numerical examples in 2D and 3D are shown to demonstrate the effectiveness of the proposed methods.

Kailai Xu, Eric Darve · stanford

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 $\mathcal{O}(h^2+Δt^2)$ 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.

Kailai Xu, Eric Darve

Calibrating a Lévy process usually requires characterizing its jump distribution. Traditionally this problem can be solved with nonparametric estimation using the empirical characteristic functions (ECF), assuming certain regularity, and results to date are mostly in 1D. For multivariate Lévy processes and less smooth Lévy densities, the problem becomes challenging as ECFs decay slowly and have large uncertainty because of limited observations. We solve this problem by approximating the Lévy density with a parametrized functional form; the characteristic function is then estimated using numerical integration. In our benchmarks, we used deep neural networks and found that they are robust and can capture sharp transitions in the Lévy density. They perform favorably compared to piecewise linear functions and radial basis functions. The methods and techniques developed here apply to many other problems that involve nonparametric estimation of functions embedded in a system model.