An a posteriori estimate for the error of a standard Krylov approximation to the matrix exponential is derived. The estimate is based on the defect (residual) of the Krylov approximation and is proven to constitute a rigorous upper bound on the error, in contrast to existing asymptotical approximations. It can be computed economically in the underlying Krylov space. In view of time-stepping applications, assuming that the given matrix is scaled by a time step, it is shown that the bound is asymptotically correct (with an order related to the dimension of the Krylov space) for the time step tending to zero. This means that the deviation of the error estimate from the true error tends to zero faster than the error itself. Furthermore, this result is extended to Krylov approximations of $φ$-functions and to improved versions of such approximations. The accuracy of the derived bounds is demonstrated by examples and compared with different variants known from the literature, which are also investigated more closely. Alternative error bounds are tested on examples, in particular a version based on the concept of effective order. For the case where the matrix exponential is used in time integration algorithms, a step size selection strategy is proposed and illustrated by experiments.
Krylov-based algorithms have long been preferred to compute the matrix exponential and exponential-like functions appearing in exponential integrators. Of late, direct polynomial interpolation of the action of these exponential-like functions have been shown to be competitive with the Krylov methods. We analyse the performance of the state-of-the-art Krylov algorithm, KIOPS, and the method of polynomial interpolation at Leja points for a number of exponential integrators for various test problems and with varying amounts of stiffness. Additionally, we investigate the performance of an iterativ…
This paper presents a new algorithm KIOPS for computing linear combinations of $φ$-functions that appear in exponential integrators. This algorithm is suitable for large-scale problems in computational physics where little or no information about the spectrum or norm of the Jacobian matrix is known a priori. We first show that such problems can be solved efficiently by computing a single exponential of a modified matrix. Then our approach is to compute an appropriate basis for the Krylov subspace using the incomplete orthogonalization procedure and project the matrix exponential on this subspa…
Matrix exponential discriminant analysis (EDA) is a generalized discriminant analysis method based on matrix exponential. It can essentially overcome the intrinsic difficulty of small sample size problem that exists in the classical linear discriminant analysis (LDA). However, for data with high dimension, one has to solve a large matrix exponential eigenproblem in this method, and the time complexity is dominated by the computation of exponential of large matrices. In this paper, we propose two inexact Krylov subspace algorithms for solving the large matrix exponential eigenproblem effectivel…
In this paper a new restarting method for Krylov subspace matrix exponential evaluations is proposed. Since our restarting technique essentially employs the residual, some convergence results for the residual are given. We also discuss how the restart length can be adjusted after each restart cycle, which leads to an adaptive restarting procedure. Numerical tests are presented to compare our restarting with three other restarting methods. Some of the algorithms described in this paper are a part of the Octave/Matlab package expmARPACK available at http://team.kiam.ru/botchev/expm/.
We propose two methods to find a proper shift parameter in the shift-and-invert method for computing matrix exponential matrix-vector products. These methods are useful in the case of matrix exponential action has to be computed for a number of vectors. The first method is based on the zero-order optimization of the mean residual norm for a given number of initial vectors. The second method processes the vectors one-by-one and estimates, for each vector, the derivative of the residual norm as a function of the shift parameter. The estimated derivative value is then used to update the shift val…
An accurate residual--time (AccuRT) restarting for computing matrix exponential actions of nonsymmetric matrices by the shift-and-invert (SAI) Krylov subspace method is proposed. The proposed restarting method is an extension of the recently proposed RT (residual--time) restarting and it is designed to avoid a possible accuracy loss in the conventional RT restarting. An expensive part of the SAI Krylov method is solution of linear systems with the shifted matrix. Since the AccuRT algorithm adjusts the shift value, we discuss how the proposed restarting can be implemented with just a single LU~…
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…
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 er…
We present a unified framework to efficiently approximate solutions to fractional diffusion problems of stationary and parabolic type. After discretization, we can take the point of view that the solution is obtained by a matrix-vector product of the form $f^{\boldsymbolτ}(L)\mathbf{b}$, where $L$ is the discretization matrix of the spatial operator, $\mathbf{b}$ a prescribed vector, and $f^{\boldsymbolτ}$ a parametric function, such as the fractional power or the Mittag-Leffler function. In the abstract framework of Stieltjes and complete Bernstein functions, to which the functions we are int…
This work develops novel rational Krylov methods for updating a large-scale matrix function f(A) when A is subject to low-rank modifications. It extends our previous work in this context on polynomial Krylov methods, for which we present a simplified convergence analysis. For the rational case, our convergence analysis is based on an exactness result that is connected to work by Bernstein and Van Loan on rank-one updates of rational matrix functions. We demonstrate the usefulness of the derived error bounds for guiding the choice of poles in the rational Krylov method for the exponential funct…
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…
Field-of-values analysis of augmented Krylov methods for matrix $φ$-function actions
Max Planck Institute for Dynamics of Complex Technical Systems（马克斯·普朗克复杂技术系统动力学研究所）；School of Mathematics and Natural Sciences, Bergische Universität Wuppertal（伍珀塔尔大学数学与自然科学学院）
An efficient Krylov subspace algorithm for computing actions of the $φ$ matrix function for large matrices is proposed. This matrix function is widely used in exponential time integration, Markov chains and network analysis and many other applications. Our algorithm is based on a reliable residual based stopping criterion and a new efficient restarting procedure. For matrices with numerical range in the stable complex half plane, we analyze residual convergence and prove that the restarted method is guaranteed to converge for any Krylov subspace dimension. Numerical tests demonstrate efficienc…
High order exponential integrators require computing linear combination of exponential like $φ$-functions of large matrices $A$ times a vector $v$. Krylov projection methods are the most general and remain an efficient choice for computing the matrix-function-vector-product evaluation when the matrix is $A$ is large and unable to be explicitly stored, or when obtaining information about the spectrum is expensive. The Krylov approximation relies on the Gram-Schmidt (GS) orthogonalization procedure to produce the orthonormal basis $V_m$. In parallel, GS orthogonalization requires \textit{global …
A novel numerical approach to solving the shallow-water equations on the sphere using high-order numerical discretizations in both space and time is proposed. A space-time tensor formalism is used to express the equations of motion covariantly and to describe the geometry of the rotated cubed-sphere grid. The spatial discretization is done with the direct flux reconstruction method, which is an alternative formulation to the discontinuous Galerkin approach. The equations of motion are solved in differential form and the resulting discretization is free from quadrature rules. It is well known t…
We present $TimeEvolver$, a program for computing time evolution in a generic quantum system. It relies on well-known Krylov subspace techniques to tackle the problem of multiplying the exponential of a large sparse matrix $i H$, where $H$ is the Hamiltonian, with an initial vector $v$. The fact that $H$ is Hermitian makes it possible to provide an easily computable bound on the accuracy of the Krylov approximation. Apart from effects of numerical roundoff, the resulting a posteriori error bound is rigorous, which represents a crucial novelty as compared to existing software packages such as $…
A Monte Carlo method for computing the action of a matrix exponential for a certain class of matrices on a vector is proposed. The method is based on generating random paths, which evolve through the indices of the matrix, governed by a given continuous-time Markov chain. The vector solution is computed probabilistically by averaging over a suitable multiplicative functional. This representation extends the existing linear algebra Monte Carlo-based methods, and was used in practice to develop an efficient algorithm capable of computing both, a single entry or the full vector solution. Finally,…
Simulation Algorithms with Exponential Integration for Time-Domain Analysis of Large-Scale Power Delivery Networks
University of California, San Diego（加利福尼亚大学圣迭戈分校）；Tsinghua National Laboratory for Information Science and Technology, the Department of Computer Science and Technology, Tsinghua University（清华大学计算机科学与技术系、清华大学信息科学技术国家重点实验室）；Facebook, Inc.（脸书公司）
A novel algorithm for computing the action of a matrix exponential over a vector is proposed. The algorithm is based on a multilevel Monte Carlo method, and the vector solution is computed probabilistically generating suitable random paths which evolve through the indices of the matrix according to a suitable probability law. The computational complexity is proved in this paper to be significantly better than the classical Monte Carlo method, which allows the computation of much more accurate solutions. Furthermore, the positive features of the algorithm in terms of parallelism were exploited …
We propose an efficient algorithmic framework for time domain circuit simulation using exponential integrator. This work addresses several critical issues exposed by previous matrix exponential based circuit simulation research, and makes it capable of simulating stiff nonlinear circuit system at a large scale. In this framework, the system's nonlinearity is treated with exponential Rosenbrock-Euler formulation. The matrix exponential and vector product is computed using invert Krylov subspace method. Our proposed method has several distinguished advantages over conventional formulations (e.g.…
The time-ordered exponential is defined as the function that solves a system of coupled first-order linear differential equations with generally non-constant coefficients. In spite of being at the heart of much system dynamics, control theory, and model reduction problems, the time-ordered exponential function remains elusively difficult to evaluate. The *-Lanczos algorithm is a (symbolic) algorithm capable of evaluating it by producing a tridiagonalization of the original differential system. In this paper, we explain how the *-Lanczos algorithm is built from a generalization of Krylov subspa…
