Fasi, Massimiliano
(2015)

*Weighted geometric mean of large-scale matrices: numerical analysis and algorithms.*
[Laurea magistrale], Università di Bologna, Corso di Studio in

Informatica [LM-DM270], Documento ad accesso riservato.

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## Abstract

Computing the weighted geometric mean of large sparse matrices is an operation that tends to become rapidly intractable, when the size of the matrices involved grows. However, if we are not interested in the computation of the matrix function itself, but just in that of its product times a vector, the problem turns simpler and there is a chance to solve it even when the matrix mean would actually be impossible to compute. Our interest is motivated by the fact that this calculation has some practical applications, related to the preconditioning of some operators
arising in domain decomposition of elliptic problems.
In this thesis, we explore how such a computation can be efficiently performed. First, we exploit the properties of the weighted geometric mean and find several equivalent ways to express it through real powers of a matrix. Hence, we focus
our attention on matrix powers and examine how well-known techniques can be adapted to the solution of the problem at hand. In particular, we consider two broad families of approaches for the computation of f(A) v, namely quadrature
formulae and Krylov subspace methods, and generalize them to the pencil case f(A\B) v.
Finally, we provide an extensive experimental evaluation of the proposed algorithms and also try to assess how convergence speed and execution time are influenced by some characteristics of the input matrices. Our results suggest that a few elements have some bearing on the performance and that, although there is no best choice in general, knowing the conditioning and the sparsity of the arguments beforehand can considerably help in choosing the best strategy to tackle the problem.

Abstract

Computing the weighted geometric mean of large sparse matrices is an operation that tends to become rapidly intractable, when the size of the matrices involved grows. However, if we are not interested in the computation of the matrix function itself, but just in that of its product times a vector, the problem turns simpler and there is a chance to solve it even when the matrix mean would actually be impossible to compute. Our interest is motivated by the fact that this calculation has some practical applications, related to the preconditioning of some operators
arising in domain decomposition of elliptic problems.
In this thesis, we explore how such a computation can be efficiently performed. First, we exploit the properties of the weighted geometric mean and find several equivalent ways to express it through real powers of a matrix. Hence, we focus
our attention on matrix powers and examine how well-known techniques can be adapted to the solution of the problem at hand. In particular, we consider two broad families of approaches for the computation of f(A) v, namely quadrature
formulae and Krylov subspace methods, and generalize them to the pencil case f(A\B) v.
Finally, we provide an extensive experimental evaluation of the proposed algorithms and also try to assess how convergence speed and execution time are influenced by some characteristics of the input matrices. Our results suggest that a few elements have some bearing on the performance and that, although there is no best choice in general, knowing the conditioning and the sparsity of the arguments beforehand can considerably help in choosing the best strategy to tackle the problem.

Tipologia del documento

Tesi di laurea
(Laurea magistrale)

Autore della tesi

Fasi, Massimiliano

Relatore della tesi

Scuola

Corso di studio

Ordinamento Cds

DM270

Parole chiave

Numerical linear algebra, weighted geometric matrix mean, Krylov subspace methods, numerical quadrature

Data di discussione della Tesi

18 Marzo 2015

URI

## Altri metadati

Tipologia del documento

Tesi di laurea
(NON SPECIFICATO)

Autore della tesi

Fasi, Massimiliano

Relatore della tesi

Scuola

Corso di studio

Ordinamento Cds

DM270

Parole chiave

Numerical linear algebra, weighted geometric matrix mean, Krylov subspace methods, numerical quadrature

Data di discussione della Tesi

18 Marzo 2015

URI

Gestione del documento: