Document Type
Dissertation
Date of Award
8-31-2024
Degree Name
Doctor of Philosophy in Mathematical Sciences - (Ph.D.)
Department
Mathematical Sciences
First Advisor
Yu Wang
Second Advisor
Cristina Frederick
Third Advisor
David Shirokoff
Fourth Advisor
Travis Askham
Fifth Advisor
Xiaoming He
Abstract
Current numerical methods for Monge-Ampere-type equations and Optimal Transport problems face challenges when handling higher dimensions and large-scale data. This dissertation aims to develop and analyze efficient, highly parallelizable numerical algorithms for solving the Monge-Ampere equation to address these issues. Two approaches are employed:
(1) The discretization method is enhanced by introducing an integral repre-sentation of the Monge-Ampere operator. This integral can be discretized using higher-order quadrature, yielding a more efficient, higher-order monotone scheme that allows for narrower stencils. An additional advantage of this discretization is its natural extension to arbitrary dimensions.
(2) The nonlinear solvers for this scheme are improved by applying Domain Decomposition Methods (DDM) in conjunction with Newton's Method (NM). This results in a highly parallelizable, fast nonlinear solver better suited for large-scale applications. The solver's performance is further enhanced by recasting it as a nonlinear preconditioner.
The combination of these approaches enables solving Monge-Ampere equations with higher precision in less time. These methods can be integrated to provide an efficient approach for three-dimensional (3D) problems.
Recommended Citation
Brusca, Jake S., "Efficient numerical methods for monge-ampere type equations" (2024). Dissertations. 1800.
https://digitalcommons.njit.edu/dissertations/1800
