Special Functions Commons^™

Fast Multipole Methods For Wave And Charge Source Interactions In Layered Media And Deep Neural Network Algorithms For High-Dimensional Pdes, Wenzhong Zhang

Mathematics Theses and Dissertations

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 …

Statistical Methods For Resolving Intratumor Heterogeneity With Single-Cell Dna Sequencing, Alexander Davis

Dissertations & Theses (Open Access)

Tumor cells have heterogeneous genotypes, which drives progression and treatment resistance. Such genetic intratumor heterogeneity plays a role in the process of clonal evolution that underlies tumor progression and treatment resistance. Single-cell DNA sequencing is a promising experimental method for studying intratumor heterogeneity, but brings unique statistical challenges in interpreting the resulting data. Researchers lack methods to determine whether sufficiently many cells have been sampled from a tumor. In addition, there are no proven computational methods for determining the ploidy of a cell, a necessary step in the determination of copy number. In this work, software for calculating probabilities from …