Prior recent work, devoted to the study of polynomial Krylov techniques for the approximation of the action of the matrix exponential ${\rm e}^{tA}v$, is extended to the case of associated $φ$-functions (which occur within the class of exponential integrators). In particular, a~posteriori error bounds and estimates, based on the notion of the defect (residual) of the Krylov approximation are considered. Computable error bounds and estimates are discussed and analyzed. This includes a new error bound which favorably compares to existing error bounds in specific cases. The accuracy of various error bounds is characterized in relation to corresponding Ritz values of $A$. Ritz values yield properties of the spectrum of $A$ (specific properties are known a~priori, e.g. for Hermitian or skew-Hermitian matrices) in relation to the actual starting vector $v$ and can be computed. This gives theoretical results together with criteria to quantify the achieved accuracy on the run. For other existing error estimates the reliability and performance is studied by similar techniques. Effects of finite precision (floating point arithmetic) are also taken into account.
The paper is concerned with methods for computing the best low multilinear rank approximation of large and sparse tensors. Krylov-type methods have been used for this problem; here block versions are introduced. For the computation of partial eigenvalue and singular value decompositions of matrices the Krylov-Schur (restarted Arnoldi) method is used. We describe a generalization of this method to tensors, for computing the best low multilinear rank approximation of large and sparse tensors. In analogy to the matrix case, the large tensor is only accessed in multiplications between the tensor a…
The computation of f(A)b, the action of a matrix function on a vector, is a task arising in many areas of scientific computing. In many applications, the matrix A is sparse but so large that only a rather small number of Krylov basis vectors can be stored. Here we discuss a new approach to overcome these limitations by randomized sketching combined with an integral representation of f(A)b. Two different approximations are introduced, one based on sketched FOM and another based on sketched GMRES approximation. The convergence of the latter method is analyzed for Stieltjes functions of positive …
The Fréchet derivative $L_f(A,E)$ of the matrix function $f(A)$ plays an important role in many different applications, including condition number estimation and network analysis. We present several different Krylov subspace methods for computing low-rank approximations of $L_f(A,E)$ when the direction term $E$ is of rank one (which can easily be extended to general low-rank). We analyze the convergence of the resulting method for the important special case that $A$ is Hermitian and $f$ is either the exponential, the logarithm or a Stieltjes function. In a number of numerical tests, both inclu…
New algorithms are proposed for the Tucker approximation of a 3-tensor, that access it using only the tensor-by-vector-by-vector multiplication subroutine. In the matrix case, Krylov methods are methods of choice to approximate the dominant column and row subspaces of a sparse or structured matrix given through the matrix-by-vector multiplication subroutine. Using the Wedderburn rank reduction formula, we propose an algorithm of matrix approximation that computes Krylov subspaces and allows generalization to the tensor case. Several variants of proposed tensor algorithms differ by pivoting str…
In this paper, we propose different algorithms for the solution of a tensor linear discrete ill-posed problem arising in the application of the meshless method for solving PDEs in three-dimensional space using multiquadric radial basis functions. It is well known that the truncated singular value decomposition (TSVD) is the most common effective solver for ill-conditioned systems, but unfortunately the operation count for solving a linear system with the TSVD is computationally expensive for large-scale matrices. In the present work, we propose algorithms based on the use of the well known Ein…
Guangdong University of Technology（广东工业大学）；RIKEN（理化学研究所）
Efficient and accurate low-rank approximation (LRA) methods are of great significance for large-scale data analysis. Randomized tensor decompositions have emerged as powerful tools to meet this need, but most existing methods perform poorly in the presence of noise interference. Inspired by the remarkable performance of randomized block Krylov iteration (rBKI) in reducing the effect of tail singular values, this work designs an rBKI-based Tucker decomposition (rBKI-TK) for accurate approximation, together with a hierarchical tensor ring decomposition based on rBKI-TK for efficient compression …
For the large-scale linear discrete ill-posed problem $\min\|Ax-b\|$ or $Ax=b$ with $b$ contaminated by white noise, the Golub-Kahan bidiagonalization based LSQR method and its mathematically equivalent CGLS, the Conjugate Gradient (CG) method applied to $A^TAx=A^Tb$, are most commonly used. They have intrinsic regularizing effects, where the iteration number $k$ plays the role of regularization parameter. The long-standing fundamental question is: {\em Can LSQR and CGLS find 2-norm filtering best possible regularized solutions}? The author has given definitive answers to this question for sev…
Given a limited amount of memory and a target accuracy, we propose and compare several polynomial Krylov methods for the approximation of f(A)b, the action of a Stieltjes matrix function of a large Hermitian matrix on a vector. Using new error bounds and estimates, as well as existing results, we derive predictions of the practical performance of the methods, and rank them accordingly. As by-products, we derive new results on inexact Krylov iterations for matrix functions in order to allow for a fair comparison of rational Krylov methods with polynomial inner solves.
Many scientific applications require the solution of large initial-value problems, such as those produced by the method of lines after semi-discretization in space of partial differential equations. The computational cost of implicit time discretizations is dominated by the solution of nonlinear systems of equations at each time step. In order to decrease this cost, the recently developed Rosenbrock-Krylov (ROK) time integration methods extend the classical linearly-implicit Rosenbrock(-W) methods, and make use of a Krylov subspace approximation to the Jacobian computed via an Arnoldi process.…
We propose a time-exact Krylov-subspace-based method for solving linear ODE (ordinary differential equation) systems of the form $y'=-Ay + g(t)$, where $y(t)$ is the unknown function. The method consists of two stages. The first stage is an accurate polynomial approximation of the source term $g(t)$, constructed with the help of the truncated SVD (singular value decomposition). The second stage is a special residual-based block Krylov subspace method. The accuracy of the method is only restricted by the accuracy of the polynomial approximation and by the error of the block Krylov process. Sinc…
We present PNKH-B, a projected Newton-Krylov method for iteratively solving large-scale optimization problems with bound constraints. PNKH-B is geared toward situations in which function and gradient evaluations are expensive, and the (approximate) Hessian is only available through matrix-vector products. This is commonly the case in large-scale parameter estimation, machine learning, and image processing. In each iteration, PNKH-B uses a low-rank approximation of the (approximate) Hessian to determine the search direction and construct the metric used in a projected line search. The key featu…
We present a class of algorithms based on rational Krylov methods to compute the action of a generalized matrix function on a vector. These algorithms incorporate existing methods based on the Golub-Kahan bidiagonalization as a special case. By exploiting the quasiseparable structure of the projected matrices, we show that the basis vectors can be updated using a short recurrence, which can be seen as a generalization to the rational case of the Golub-Kahan bidiagonalization. We also prove error bounds that relate the error of these methods to uniform rational approximation. The effectiveness …
In this paper we extend the Residual Arnoldi method for calculating an extreme eigenvalue (e.g. largest real part, dominant,...) to the case where the matrices depend on parameters. The difference between this Arnoldi method and the classical Arnoldi algorithm is that in the former the residual is added to the subspace. We develop a Tensor-Krylov method that applies the Residual Arnoldi method (RA) for a grid of parameter points at the same time. The subspace contains an approximate Krylov space for all these points. Instead of adding the residuals for all parameter values to the subspace we c…
Evaluating the action of a matrix function on a vector, that is $x=f(\mathcal M)v$, is an ubiquitous task in applications. When $\mathcal M$ 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 $\mathcal M$ is a positive definite matrix. Relying on the same tools used to analyze the generic situation, we then focus on the case $\mathcal M=I \otimes A - B^T \otimes I$, a…
