Date of Award
2019
Document Type
Open Access Dissertation
Degree Name
Doctor of Philosophy in Mathematical Sciences (PhD)
Administrative Home Department
Department of Mathematical Sciences
Advisor 1
Mark S. Gockenbach
Committee Member 1
Hassan Masoud
Committee Member 2
Benjamin W. Ong
Committee Member 3
Jiguang Sun
Abstract
Let $X$, $Y$, and $Z$ be real separable Hilbert spaces, let $T:X \to Y$ be a compact operator, and let $L:D(L) \to Z$ be a closed and densely defined linear operator. Then the generalized singular value expansion (GSVE) is an expansion that expresses $T$ and $L$ in terms of a common orthonormal basis. Under certain hypotheses on discretization, the GSVE of an approximate operator pair $(T_j,L_j)$, where $T_j:X_j \to Y_j$ and $L_j:X_j \to Z_j$, converges to the GSVE of $(T,L)$. Error estimates establish a rate of convergence that is consistent with numerical experiments in the case of discretization using piecewise linear finite elements. Further numerical testing suggests that a higher rate of convergence is attained by using higher order elements. However, the theory does not cover this case.
Recommended Citation
Roberts, Matthew Jacob, "Approximation of the Generalized Singular Value Expansion", Open Access Dissertation, Michigan Technological University, 2019.
