In this dissertation, we develop fast algorithms for large scale numerical computations, including the fast multipole method (FMM) in layered media, and the forward-backward stochastic differential equation (FBSDE) based deep neural network (DNN) algorithms for high-dimensional parabolic partial differential equations (PDEs), addressing the issues of real-world challenging computational problems in various computation scenarios.
We develop the FMM in layered media, by first studying analytical and numerical properties of the Green's functions in layered media for the 2-D and 3-D Helmholtz equation, the linearized Poisson--Boltzmann equation, the Laplace's equation, and the tensor Green's functions for the time-harmonic Maxwell's equations and the elastic wave equation. Then, we propose the far-field expansions in layered media as the natural extension of the Graf's addition theorems for the free-space problems. We define a modified far-field polarization distance in layered structure according to the exponential convergence behavior of these far-field expansions, and design the FMM framework for layered media based on the modified polarization distance. Numerical tests are conducted for a massive number of particles in layered media, verifying both the efficiency and the accuracy of the developed FMM algorithms.
Numerical algorithms using the deep neural network (DNN) for the purpose of finding a remedy to address the curse of dimensionality are proposed for the solution to high-dimensional quasi-linear parabolic PDEs based on the Pardoux--Peng theory of the FBSDEs. The algorithms are shown to generate DNN solutions for a 100-dimensional Black--Scholes--Barenblatt equation that are accurate in a finite region in the solution space, and have a convergence rate similar to that of the Euler--Maruyama discretization scheme used for the FBSDEs.
Mathematics, Applied, Mathematics, General/Other
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Zhang, Wenzhong, "Fast Multipole Methods for Wave and Charge Source Interactions in Layered Media and Deep Neural Network Algorithms for High-Dimensional PDEs" (2021). Mathematics Theses and Dissertations. 15.