In this paper we propose a method to compute the solution to the fractional diffusion equation on directed networks, which can be expressed in terms of the graph Laplacian $L$ as a product $f(L^T) \boldsymbol{b}$, where $f$ is a non-analytic function involving fractional powers and $\boldsymbol{b}$ is a given vector. The graph Laplacian is a singular matrix, causing Krylov methods for $f(L^T) \boldsymbol{b}$ to converge more slowly. In order to overcome this difficulty and achieve faster convergence, we use rational Krylov methods applied to a desingularized version of the graph Laplacian, obtained with either a rank-one shift or a projection on a subspace.
In this paper, we propose a new choice of poles to define reliable rational Krylov methods. These methods are used for approximating function of positive definite matrices. In particular, the fractional power and the fractional resolvent are considered because of their importance in the numerical solution of fractional partial differential equations. The results of the numerical experiments we have carried out on some fractional models confirm that the proposed approach is promising.
Conjugated gradients on the normal equation (CGNE) is a popular method to regularise linear inverse problems. The idea of the method can be summarised as minimising the residuum over a suitable Krylov subspace. It is shown that using the same idea for the shift-and-invert rational Krylov subspace yields an order-optimal regularisation scheme.
The accumulated quadrature weights of Gaussian quadrature formulae constitute bounds on the integral over the intervals between the quadrature nodes. Classical results in this concern date back to works of Chebyshev, Markov and Stieltjes and are referred to as Separation Theorem of Chebyshev-Markov-Stieltjes (CMS Theorem). Similar separation theorems hold true for some classes of rational Gaussian quadrature. The Krylov subspace for a given matrix and initial vector is closely related to orthogonal polynomials associated with the spectral distribution of the initial vector in the eigenbasis of…
We establish an equivalence between two classes of methods for solving fractional diffusion problems, namely, Reduced Basis Methods (RBM) and Rational Krylov Methods (RKM). In particular, we demonstrate that several recently proposed RBMs for fractional diffusion can be interpreted as RKMs. This changed point of view allows us to give convergence proofs for some methods where none were previously available. We also propose a new RKM for fractional diffusion problems with poles chosen using the best rational approximation of the function $x^{-s}$ in the spectral interval of the spatial discreti…
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…
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 …
The second-order cone linear complementarity problem (SOCLCP) is a generalization of the classical linear complementarity problem. It has been known that SOCLCP, with the globally uniquely solvable property, is essentially equivalent to a zero-finding problem in which the associated function bears much similarity to the transfer function in model reduction [{\em SIAM J. Sci. Comput.}, 37 (2015), pp.~A2046--A2075]. In this paper, we propose a new rational Krylov subspace method to solve the zero-finding problem for the symmetric and positive definite SOCLCP. The algorithm consists of two stages…
We consider a class of linear matrix equations involving semi-infinite matrices which have a quasi-Toeplitz structure. These equations arise in different settings, mostly connected with PDEs or the study of Markov chains such as random walks on bidimensional lattices. We present the theory justifying the existence in an appropriate Banach algebra which is computationally treatable, and we propose several methods for their solutions. We show how to adapt the ADI iteration to this particular infinite dimensional setting, and how to construct rational Krylov methods. Convergence theory is discuss…
Rational approximation recently emerged as an efficient numerical tool for the solution of exterior wave propagation problems. Currently, this technique is limited to wave media which are invariant along the main propagation direction. We propose a new model order reduction-based approach for compressing unbounded waveguides with layered inclusions. It is based on the solution of a nonlinear rational least squares problem using the RKFIT method. We show that approximants can be converted into an accurate finite difference representation within a rational Krylov framework. Numerical experiments…
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…
A general framework for oblique projections of nonhermitian matrices onto rational Krylov subspaces is developed. To obtain this framework we revisit the classical rational Krylov subspace algorithm and prove that the projected matrix can be written efficiently as a structured pencil, where the structure can take several forms, such as Hessenberg or inverse Hessenberg. One specific instance of the structures appearing in this framework for oblique projections is a tridiagonal pencil. This is a direct generalization of the classical biorthogonal Krylov subspace method where the projection becom…
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…
In this work we present a rational Krylov subspace method for solving real large-scale polynomial eigenvalue problems with T-even (that is, symmetric/skew-symmetric) structure. Our method is based on the Even-IRA algorithm. To preserve the structure, a sparse T-even linearization from the class of block minimal bases pencils is applied. Due to this linearization, the Krylov basis vectors can be computed in a cheap way. A rational decomposition is derived so that our method explicitly allows for changes of the shift during the iteration. This leads to a method that is able to compute parts of t…
Block Krylov subspace methods (KSMs) comprise building blocks in many state-of-the-art solvers for large-scale matrix equations as they arise, e.g., from the discretization of partial differential equations. While extended and rational block Krylov subspace methods provide a major reduction in iteration counts over polynomial block KSMs, they also require reliable solvers for the coefficient matrices, and these solvers are often iterative methods themselves. It is not hard to devise scenarios in which the available memory, and consequently the dimension of the Krylov subspace, is limited. In s…
Problems of the numerical solution of the Cauchy problem for a first-order differential-operator equation are discussed. A fundamental feature of the problem under study is that the equation includes a fractional power of the self-adjoint positive operator. In computational practice, rational approximations of the fractional power operator are widely used in various versions. The purpose of this work is to construct special approximations in time when the transition to a new level in time provided a set of standard problems for the operator and not for the fractional power operator. Stable spl…
Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains
Southern University of Sciences and Technology（南方科技大学）；Nanyang Technological University（南洋理工大学）；Hong Kong Baptist University（香港浸会大学）；Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学研究所）
Fractional differential equations (FDEs) describe subdiffusion behavior of dynamical systems. Its non-local structure requires taking into account the whole evolution history during the time integration, which then possibly causes additional memory use to store the history, growing in time. An alternative to a quadrature for the history integral is to approximate the fractional kernel with the sum of exponentials, which is equivalent to considering the FDE solution as a sum of solutions to a system of ODEs. One possibility to construct this system is to approximate the Laplace spectrum of the …
We present a simple discretization scheme for the hypersingular integral representation of the fractional Laplace operator and solver for the corresponding fractional Laplacian problem. Through singularity subtraction, we obtain a regularized integrand that is amenable to the trapezoidal rule with equispaced nodes, assuming a high degree of regularity in the underlying function (i.e., $u\in C^6(R^d)$). The resulting quadrature scheme gives a discrete operator on a regular grid that is translation-invariant and thus can be applied quickly with the fast Fourier transform. For discretizations of …
