1,229 research outputs found
Solving rank structured Sylvester and Lyapunov equations
We consider the problem of efficiently solving Sylvester and Lyapunov
equations of medium and large scale, in case of rank-structured data, i.e.,
when the coefficient matrices and the right-hand side have low-rank
off-diagonal blocks. This comprises problems with banded data, recently studied
by Haber and Verhaegen in "Sparse solution of the Lyapunov equation for
large-scale interconnected systems", Automatica, 2016, and by Palitta and
Simoncini in "Numerical methods for large-scale Lyapunov equations with
symmetric banded data", SISC, 2018, which often arise in the discretization of
elliptic PDEs.
We show that, under suitable assumptions, the quasiseparable structure is
guaranteed to be numerically present in the solution, and explicit novel
estimates of the numerical rank of the off-diagonal blocks are provided.
Efficient solution schemes that rely on the technology of hierarchical
matrices are described, and several numerical experiments confirm the
applicability and efficiency of the approaches. We develop a MATLAB toolbox
that allows easy replication of the experiments and a ready-to-use interface
for the solvers. The performances of the different approaches are compared, and
we show that the new methods described are efficient on several classes of
relevant problems
Numerical solution of large-scale linear matrix equations
We are interested in the numerical solution of large-scale linear matrix equations. In particular, due to their occurrence
in many applications, we study the so-called Sylvester and Lyapunov equations.
A characteristic aspect of the
large-scale setting is that although data are sparse, the solution is in general
dense so that storing it may be unfeasible. Therefore, it is necessary that the solution allows for a memory-saving
approximation that can be cheaply stored.
An extensive literature treats the case of the aforementioned equations with low-rank right-hand
side.
This assumption, together with certain hypotheses on the spectral distribution of the matrix coefficients, is a sufficient
condition for proving a fast decay in the singular values of the solution.
This decay motivates the search for a low-rank approximation so that only low-rank
matrices are actually computed and stored remarkably reducing the storage demand.
This is the task of the so-called low-rank methods and a large amount of work in this direction has been
carried out in the last years.
Projection methods have been shown to be among the most effective low-rank methods and in the first part
of this thesis we propose some computational enhanchements of the classical algorithms.
The case of equations with not necessarily low rank right-hand side has not been
deeply analyzed so far and efficient methods are still lacking in the literature.
In this thesis we aim to
significantly contribute to this open problem by introducing solution methods for this kind of equations.
In particular, we address the case when the coefficient matrices and the right-hand side are
banded and we further generalize this structure considering quasiseparable data.
In the last part of the thesis we study large-scale generalized Sylvester equations
and, under some assumptions on the coefficient matrices, novel approximation spaces for their solution
by projection are proposed
Low-rank updates and a divide-and-conquer method for linear matrix equations
Linear matrix equations, such as the Sylvester and Lyapunov equations, play
an important role in various applications, including the stability analysis and
dimensionality reduction of linear dynamical control systems and the solution
of partial differential equations. In this work, we present and analyze a new
algorithm, based on tensorized Krylov subspaces, for quickly updating the
solution of such a matrix equation when its coefficients undergo low-rank
changes. We demonstrate how our algorithm can be utilized to accelerate the
Newton method for solving continuous-time algebraic Riccati equations. Our
algorithm also forms the basis of a new divide-and-conquer approach for linear
matrix equations with coefficients that feature hierarchical low-rank
structure, such as HODLR, HSS, and banded matrices. Numerical experiments
demonstrate the advantages of divide-and-conquer over existing approaches, in
terms of computational time and memory consumption
Greedy low-rank algorithm for spatial connectome regression
Recovering brain connectivity from tract tracing data is an important
computational problem in the neurosciences. Mesoscopic connectome
reconstruction was previously formulated as a structured matrix regression
problem (Harris et al., 2016), but existing techniques do not scale to the
whole-brain setting. The corresponding matrix equation is challenging to solve
due to large scale, ill-conditioning, and a general form that lacks a
convergent splitting. We propose a greedy low-rank algorithm for connectome
reconstruction problem in very high dimensions. The algorithm approximates the
solution by a sequence of rank-one updates which exploit the sparse and
positive definite problem structure. This algorithm was described previously
(Kressner and Sirkovi\'c, 2015) but never implemented for this connectome
problem, leading to a number of challenges. We have had to design judicious
stopping criteria and employ efficient solvers for the three main sub-problems
of the algorithm, including an efficient GPU implementation that alleviates the
main bottleneck for large datasets. The performance of the method is evaluated
on three examples: an artificial "toy" dataset and two whole-cortex instances
using data from the Allen Mouse Brain Connectivity Atlas. We find that the
method is significantly faster than previous methods and that moderate ranks
offer good approximation. This speedup allows for the estimation of
increasingly large-scale connectomes across taxa as these data become available
from tracing experiments. The data and code are available online
From low-rank approximation to an efficient rational Krylov subspace method for the Lyapunov equation
We propose a new method for the approximate solution of the Lyapunov equation
with rank- right-hand side, which is based on extended rational Krylov
subspace approximation with adaptively computed shifts. The shift selection is
obtained from the connection between the Lyapunov equation, solution of systems
of linear ODEs and alternating least squares method for low-rank approximation.
The numerical experiments confirm the effectiveness of our approach.Comment: 17 pages, 1 figure
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
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
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