Page 445 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 445
MATRIX COMPUTATIONS AND NUMERICAL (MS-47)
Infinite Tensor Rings
Lana Periša, lana.perisa@visagetechnologies.com
Visage Technologies, Croatia
Coauthors: Roel Van Beeumen, Chao Yang, Daniel Kressner
While creating a flexible power method for computing the leftmost, i.e., algebraically smallest,
eigenvalue of an infinite dimensional tensor eigenvalue problem, Hx = λx, where the infinite
dimensional symmetric matrix H exhibits a translational invariant structure, we study the theory
of infinite Tensor Rings (iTRs). Under the assumption that the smallest eigenvalue of H is
simple, representing the eigenvector as a translational invariant iTR allows the use of power
iteration of e−H. In order to implement this power iteration, we use a small parameter t so that
the infinite matrix-vector operation e−Htx can efficiently be approximated by the Lie product
formula, also known as Suzuki–Trotter splitting. In this talk we further explain the motivation
for defining iTRs and present their derived and used mathematical properties.
Core reduction: Necessary and sufficient information in linear
approximation problems
Martin Plešinger, martin.plesinger@tul.cz
Technical University of Liberec, Czech Republic
Coauthors: Iveta Hneˇtynková, Jana Žáková
We focus on linear approximation problems Ax ≈ b, where A is a given matrix, x an unknown
vector, and the given right-hand side vector b is not in the range of A, i.e., b ∈ R(A). By solving
of such problem we usually mean replacing it by some minimization. Typically the least squares
(LS) techniques can be used. We focus on the so-called total least squares (TLS) minimization
min [g, E] F s.t. (b + g) ∈ R(A + E).
TLS has been studied since the early eighties. The trouble there is, contrary to the standard LS,
that the minimization may not have a solution for the given (A, b).
The theory of core problem introduced in 2006 by Paige and Strakoš brings a concept of
necessary and sufficient information for solving the TLS minimization. This concept allows us
to distinguish cases having and not having the TLS solution. Moreover, core problem theory
clearly explains why it happens. In recent years the core problems thoery has been applied on
several other linear problems A(X) = B where the linear mapping A as well as the right-hand
side B can have some particular structure.
On the numerical solution of certain linear multiterm matrix equations
and applications
Valeria Simoncini, valeria.simoncini@unibo.it
Alma Mater Studiorum - Università di Bologna, Italy
Linear matrix equations have arisen as the natural algebraic form associated with the discretiza-
tion of a growing number of application problems. The case where the unknown matrix appears
443
Infinite Tensor Rings
Lana Periša, lana.perisa@visagetechnologies.com
Visage Technologies, Croatia
Coauthors: Roel Van Beeumen, Chao Yang, Daniel Kressner
While creating a flexible power method for computing the leftmost, i.e., algebraically smallest,
eigenvalue of an infinite dimensional tensor eigenvalue problem, Hx = λx, where the infinite
dimensional symmetric matrix H exhibits a translational invariant structure, we study the theory
of infinite Tensor Rings (iTRs). Under the assumption that the smallest eigenvalue of H is
simple, representing the eigenvector as a translational invariant iTR allows the use of power
iteration of e−H. In order to implement this power iteration, we use a small parameter t so that
the infinite matrix-vector operation e−Htx can efficiently be approximated by the Lie product
formula, also known as Suzuki–Trotter splitting. In this talk we further explain the motivation
for defining iTRs and present their derived and used mathematical properties.
Core reduction: Necessary and sufficient information in linear
approximation problems
Martin Plešinger, martin.plesinger@tul.cz
Technical University of Liberec, Czech Republic
Coauthors: Iveta Hneˇtynková, Jana Žáková
We focus on linear approximation problems Ax ≈ b, where A is a given matrix, x an unknown
vector, and the given right-hand side vector b is not in the range of A, i.e., b ∈ R(A). By solving
of such problem we usually mean replacing it by some minimization. Typically the least squares
(LS) techniques can be used. We focus on the so-called total least squares (TLS) minimization
min [g, E] F s.t. (b + g) ∈ R(A + E).
TLS has been studied since the early eighties. The trouble there is, contrary to the standard LS,
that the minimization may not have a solution for the given (A, b).
The theory of core problem introduced in 2006 by Paige and Strakoš brings a concept of
necessary and sufficient information for solving the TLS minimization. This concept allows us
to distinguish cases having and not having the TLS solution. Moreover, core problem theory
clearly explains why it happens. In recent years the core problems thoery has been applied on
several other linear problems A(X) = B where the linear mapping A as well as the right-hand
side B can have some particular structure.
On the numerical solution of certain linear multiterm matrix equations
and applications
Valeria Simoncini, valeria.simoncini@unibo.it
Alma Mater Studiorum - Università di Bologna, Italy
Linear matrix equations have arisen as the natural algebraic form associated with the discretiza-
tion of a growing number of application problems. The case where the unknown matrix appears
443
