59 research outputs found
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
Rational Krylov for Stieltjes matrix functions: convergence and pole selection
Evaluating the action of a matrix function on a vector, that is , is an ubiquitous task in applications. When is large, one
usually relies on Krylov projection methods. In this paper, we provide
effective choices for the poles of the rational Krylov method for approximating
when is either Cauchy-Stieltjes or Laplace-Stieltjes (or, which is
equivalent, completely monotonic) and is a positive definite
matrix. Relying on the same tools used to analyze the generic situation, we
then focus on the case , and
obtained vectorizing a low-rank matrix; this finds application, for instance,
in solving fractional diffusion equation on two-dimensional tensor grids. We
see how to leverage tensorized Krylov subspaces to exploit the Kronecker
structure and we introduce an error analysis for the numerical approximation of
. Pole selection strategies with explicit convergence bounds are given also
in this case
Inexact Arnoldi residual estimates and decay properties for functions of non-Hermitian matrices
This paper derives a priori residual-type bounds for the Arnoldi approximation of a matrix function together with 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, a priori decay bounds for the entries of functions of banded non-Hermitian matrices will be exploited, using Faber polynomial approximation. Numerical experiments illustrate the quality of the results
Computation of the von Neumann entropy of large matrices via trace estimators and rational Krylov methods
We consider the problem of approximating the von Neumann entropy of a large,
sparse, symmetric positive semidefinite matrix , defined as
where . After establishing some useful
properties of this matrix function, we consider the use of both polynomial and
rational Krylov subspace algorithms within two types of approximations methods,
namely, randomized trace estimators and probing techniques based on graph
colorings. We develop error bounds and heuristics which are employed in the
implementation of the algorithms. Numerical experiments on density matrices of
different types of networks illustrate the performance of the methods.Comment: 32 pages, 10 figure
Rational Krylov for Stieltjes matrix functions: convergence and pole selection
Evaluating the action of a matrix function on a vector, that is x= f(M) v, is an ubiquitous task in applications. When M is large, one usually relies on Krylov projection methods. In this paper, we provide effective choices for the poles of the rational Krylov method for approximating x when f(z) is either Cauchy–Stieltjes or Laplace–Stieltjes (or, which is equivalent, completely monotonic) and M is a positive definite matrix. Relying on the same tools used to analyze the generic situation, we then focus on the case M= I⊗ A- BT⊗ I, and v obtained vectorizing a low-rank matrix; this finds application, for instance, in solving fractional diffusion equation on two-dimensional tensor grids. We see how to leverage tensorized Krylov subspaces to exploit the Kronecker structure and we introduce an error analysis for the numerical approximation of x. Pole selection strategies with explicit convergence bounds are given also in this case
Analysis of stochastic probing methods for estimating the trace of functions of sparse symmetric matrices
We consider the problem of estimating the trace of a matrix function .
In certain situations, in particular if cannot be well approximated by a
low-rank matrix, combining probing methods based on graph colorings with
stochastic trace estimation techniques can yield accurate approximations at
moderate cost. So far, such methods have not been thoroughly analyzed, though,
but were rather used as efficient heuristics by practitioners. In this
manuscript, we perform a detailed analysis of stochastic probing methods and,
in particular, expose conditions under which the expected approximation error
in the stochastic probing method scales more favorably with the dimension of
the matrix than the error in non-stochastic probing. Extending results from [E.
Aune, D. P. Simpson, J. Eidsvik, Parameter estimation in high dimensional
Gaussian distributions, Stat. Comput., 24, pp. 247--263, 2014], we also
characterize situations in which using just one stochastic vector is always --
not only in expectation -- better than the deterministic probing method.
Several numerical experiments illustrate our theory and compare with existing
methods
Sensitivity of matrix function based network communicability measures: Computational methods and a priori bounds
When analyzing complex networks, an important task is the identification of
those nodes which play a leading role for the overall communicability of the
network. In the context of modifying networks (or making them robust against
targeted attacks or outages), it is also relevant to know how sensitive the
network's communicability reacts to changes in certain nodes or edges.
Recently, the concept of total network sensitivity was introduced in [O. De la
Cruz Cabrera, J. Jin, S. Noschese, L. Reichel, Communication in complex
networks, Appl. Numer. Math., 172, pp. 186-205, 2022], which allows to measure
how sensitive the total communicability of a network is to the addition or
removal of certain edges. One shortcoming of this concept is that sensitivities
are extremely costly to compute when using a straight-forward approach (orders
of magnitude more expensive than the corresponding communicability measures).
In this work, we present computational procedures for estimating network
sensitivity with a cost that is essentially linear in the number of nodes for
many real-world complex networks. Additionally, we extend the sensitivity
concept such that it also covers sensitivity of subgraph centrality and the
Estrada index, and we discuss the case of node removal. We propose a priori
bounds for these sensitivities which capture the qualitative behavior well and
give insight into the general behavior of matrix function based network indices
under perturbations. These bounds are based on decay results for Fr\'echet
derivatives of matrix functions with structured, low-rank direction terms which
might be of independent interest also for other applications than network
analysis
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