Dissertation/Thesis Abstract

Applications of the Minimum Sobolev Norm and Associated Fast Algorithms
by Gorman, Christopher Henry, Ph.D., University of California, Santa Barbara, 2019, 220; 13896728
Abstract (Summary)

This dissertation focuses on the development, implementation, and analysis of fast algorithms for the Minimum Sobolev norm (MSN). The MSN method obtains a unique solution from an underdetermined linear system by minimizing a derivative norm in the appropriate Hilbert space. We obtain fast algorithms by exploiting the inherent structure of the underlying system. After performing an Inverse Discrete Cosine Transform, a small number of additional operations are required. Results show the method performs as well as Chebyshev interpolation when approximating smooth functions and better than a wide variety of smooth Chebyshev filters when attempting to approximate rough functions.

One chapter is devoted to analyzing a stochastic norm estimate which is useful when computing low-rank approximations of matrices. This estimate allows us to compute approximations with relative error close to machine precision, which previously was not possible.

Indexing (document details)
Advisor: Chandrasekaran, Shivkumar, Yang, Xu
Commitee: Ceniceros, Hector, Li, Xiaoye
School: University of California, Santa Barbara
Department: Mathematics
School Location: United States -- California
Source: DAI-B 81/3(E), Dissertation Abstracts International
Subjects: Applied Mathematics, Mathematics
Keywords: Minimum Sobolev norm, Fast algorithms, Inverse discrete cosine transform
Publication Number: 13896728
ISBN: 9781088309773
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