116 research outputs found
The Lyapunov matrix equation. Matrix analysis from a computational perspective
Decay properties of the solution to the Lyapunov matrix equation are investigated. Their exploitation in the understanding of equation
matrix properties, and in the development of new numerical solution strategies
when is not low rank but possibly sparse is also briefly discussed.Comment: This work is a contribution to the Seminar series "Topics in
Mathematics", of the PhD Program of the Mathematics Department, Universit\`a
di Bologna, Ital
Approximation of functions of large matrices with Kronecker structure
We consider the numerical approximation of where and is the sum of Kronecker products, that is . Here is a regular
function such that is well defined. We derive a computational
strategy that significantly lowers the memory requirements and computational
efforts of the standard approximations, with special emphasis on the
exponential function, for which the new procedure becomes particularly
advantageous. Our findings are illustrated by numerical experiments with
typical functions used in applications
Matrix-equation-based strategies for convection-diffusion equations
We are interested in the numerical solution of nonsymmetric linear systems
arising from the discretization of convection-diffusion partial differential
equations with separable coefficients and dominant convection. Preconditioners
based on the matrix equation formulation of the problem are proposed, which
naturally approximate the original discretized problem. For certain types of
convection coefficients, we show that the explicit solution of the matrix
equation can effectively replace the linear system solution. Numerical
experiments with data stemming from two and three dimensional problems are
reported, illustrating the potential of the proposed methodology
Inexact Arnoldi residual estimates and decay properties for functions of non-Hermitian matrices
We derive a priori residual-type bounds for the Arnoldi approximation of a
matrix function and a strategy for setting the iteration accuracies in the
inexact Arnoldi approximation of matrix functions. Such results are based on
the decay behavior of the entries of functions of banded matrices.
Specifically, we will use a priori decay bounds for the entries of functions of
banded non-Hermitian matrices by using Faber polynomial series. Numerical
experiments illustrate the quality of the results
Order reduction methods for solving large-scale differential matrix Riccati equations
We consider the numerical solution of large-scale symmetric differential
matrix Riccati equations. Under certain hypotheses on the data, reduced order
methods have recently arisen as a promising class of solution strategies, by
forming low-rank approximations to the sought after solution at selected
timesteps. We show that great computational and memory savings are obtained by
a reduction process onto rational Krylov subspaces, as opposed to current
approaches. By specifically addressing the solution of the reduced differential
equation and reliable stopping criteria, we are able to obtain accurate final
approximations at low memory and computational requirements. This is obtained
by employing a two-phase strategy that separately enhances the accuracy of the
algebraic approximation and the time integration. The new method allows us to
numerically solve much larger problems than in the current literature.
Numerical experiments on benchmark problems illustrate the effectiveness of the
procedure with respect to existing solvers
Numerical methods for large-scale Lyapunov equations with symmetric banded data
The numerical solution of large-scale Lyapunov matrix equations with
symmetric banded data has so far received little attention in the rich
literature on Lyapunov equations. We aim to contribute to this open problem by
introducing two efficient solution methods, which respectively address the
cases of well conditioned and ill conditioned coefficient matrices. The
proposed approaches conveniently exploit the possibly hidden structure of the
solution matrix so as to deliver memory and computation saving approximate
solutions. Numerical experiments are reported to illustrate the potential of
the described methods
Stability Estimates and Structural Spectral Properties of Saddle Point Problems
For a general class of saddle point problems sharp estimates for
Babu\v{s}ka's inf-sup stability constants are derived in terms of the constants
in Brezzi's theory. In the finite-dimensional Hermitian case more detailed
spectral properties of preconditioned saddle point matrices are presented,
which are helpful for the convergence analysis of common Krylov subspace
methods. The theoretical results are applied to two model problems from optimal
control with time-periodic state equations. Numerical experiments with the
preconditioned minimal residual method are reported
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