Defense: A. Dektor - Tensor methods for high-dimensional PDEs

Description: The numerical simulation of high-dimensional partial differential equations (PDEs) is a challenging and important problem in science and engineering. Classical methods based on tensor product representations are not viable in high-dimensions, as the number of degrees of freedom grows exponentially fast with the problem dimension. In this dissertation we present low-rank tensor methods for high-dimensional PDEs, which, have a number of degrees of freedom and computational cost that grow linearly with the problem dimension. These methods are based on projecting a given PDE onto a low-rank tensor manifold and then constructing an approximate PDE solution as a path on the manifold. In order to control the accuracy of the low-rank tensor approximation we present a rank-adaptive algorithm that can add or remove tensor modes adaptively from the PDE solution during time integration. We also present a tensor rank reduction method based on coordinate transformations that can greatly increase the efficiency of high-dimensional tensor approximation algorithms. The idea is to determine a coordinate transformation of a given functions domain so that the function in the new coordinate system has smaller tensor rank. We demonstrate each of the presented low-rank tensor methods by providing several numerical applications to multivariate functions and PDEs. 
 
Event Host: Alec Dektor, PhD Candidate, Applied Mathematics
Advisor: Daniele Venturi
 
Join us in person or on Zoom: https://ucsc.zoom.us/j/97898708681?pwd=Q20wc212UjUrQjA0bEJkZExaZ1FDUT09
Passcode: 870356

Friday, June 2, 2023 at 1:30pm

Engineering 2, Rm 506

Event Type

Ph.D. Presentations

Invited Audience

Alumni, Faculty & Staff, Students, Prospective Students, General Public, Graduate Students

Topics

Academic

Group
Applied Math Department, Baskin School of Engineering
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