## Dissertations, Master's Theses and Master's Reports

2020

#### Document Type

Open Access Dissertation

#### Degree Name

Doctor of Philosophy in Mathematical Sciences (PhD)

Department of Mathematical Sciences

Mark Gockenbach

Allan Struthers

Cecile Piret

Brian Fick

#### Abstract

Given any bounded linear operator T : X → Y between separable Hilbert spaces X and Y , there exists a measure space (M, Α, µ) and isometries V : L2(M) X, U : L2(M) Y and a nonnegative, bounded, measurable function σ : M [0, ∞) such that

T = UmσV ,

with mσ : L2(M ) L2(M ) defined by mσ(f ) = σf for all f ∈ L2(M ). The expansion T = UmσV is called the singular value expansion (SVE) of T .

The SVE is a useful tool for analyzing a number of problems such as the computation of the generalized inverse T of T , understanding the inverse problem Tx = y and, regularizing Tx = y using methods such as Tikhonov regularization. In fact, many standard Tikhonov regularization results can be derived by making use of the SVE.

The expansion T = UmσV can also be compared to the SVE of a compact operator T : X Y which has the form

T = Σ σnun vn

where the above sum may be finite or infinite depending on the rank of T . The set {σn} is a sequence of positive real numbers that converge to zero if T has infinite rank. Such σn are the singular values of T . The sets {vn} ⊂ X and {un} ⊂ Y are orthonormal sets of vectors that satisfy Tvn = σnun for all n. The vectors vn and un are the right and left singular vectors of T, respectively. If the essential range, denoted Ress(σ), forms a sequence of positive real numbers converging to zero (or is merely a finite set of nonnegative real numbers) and for each nonzero s Ress(σ), the essential preimage of the singleton set s , denoted σess1( {s} ), is finite, then the bounded operator T = UmσV is in fact compact. The converse of this statement is also true.

If the operator T is compact, the singular values and vectors of T may be approximated by discretizing the operator and finding the singular value decomposition of a scaled Galerkin matrix. In general, the approximated singular values and vectors converge at the same rate, which is governed by the error (in the operator norm) in approximating T by the discretized operator. However, when the discretization is accomplished by projection (variational approximation), the computed singular values converge at an increased rate; the typical case is that the errors in the singular values are asymptotically equal to the square of the errors in the singular vectors (this statement must be modified if the approximations to the left and right singular vectors converge at different rates). Moreover, in the case of variational approximation, the error in the singular vectors can be compared with the optimal approximation error, with the two being asymptotically equal in the typical case.

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