53 research outputs found

    KRYLOV SUBSPACE METHODS FOR SOLVING LARGE LYAPUNOV EQUATIONS

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    Factorized Solution of Generalized Stable Sylvester Equations Using Many-Core GPU Accelerators

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    Factorized solution of generalized stable Sylvester equations using many-core GPU accelerators

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    [EN] We investigate the factorized solution of generalized stable Sylvester equations such as those arising in model reduction, image restoration, and observer design. Our algorithms, based on the matrix sign function, take advantage of the current trend to integrate high performance graphics accelerators (also known as GPUs) in computer systems. As a result, our realisations provide a valuable tool to solve large-scale problems on a variety of platforms.We acknowledge support of the ANII - MPG Independent Research Group: "Efficient Hetergenous Computing" at UdelaR, a partner group of the Max Planck Institute in Magdeburg.Benner, P.; Dufrechou, E.; Ezzatti, P.; Gallardo, R.; Quintana-Ortí, ES. (2021). Factorized solution of generalized stable Sylvester equations using many-core GPU accelerators. The Journal of Supercomputing (Online). 77(9):10152-19164. https://doi.org/10.1007/s11227-021-03658-y101521916477

    A parallel Schur method for solving continuous-time algebraic Riccati equations

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    Numerical algorithms for solving the continuous-time algebraic Riccati matrix equation on a distributed memory parallel computer are considered. In particular, it is shown that the Schur method, based on computing the stable invariant subspace of a Hamiltonian matrix, can be parallelized in an efficient and scalable way. Our implementation employs the state-of-the-art library ScaLAPACK as well as recently developed parallel methods for reordering the eigenvalues in a real Schur form. Some experimental results are presented, confirming the scalability of our implementation and comparing it with an existing implementation of the matrix sign iteration from the PLiCOC library

    Krylov methods for large-scale modern problems in numerical linear algebra

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    Large-scale problems have attracted much attention in the last decades since they arise from different applications in several fields. Moreover, the matrices that are involved in those problems are often sparse, this is, the majority of their entries are zero. Around 40 years ago, the most common problems related to large-scale and sparse matrices consisted in solving linear systems, finding eigenvalues and/or eigenvectors, solving least square problems or computing singular value decompositions. However, in the last years, large-scale and sparse problems of different natures have appeared, motivating and challenging numerical linear algebra to develop effective and efficient algorithms to solve them. Common difficulties that appear during the development of algorithms for solving modern large-scale problems are related to computational costs, storage issues and CPU time, given the large size of the matrices, which indicate that direct methods can not be used. This suggests that projection methods based on Krylov subspaces are a good option to develop procedures for solving large-scale and sparse modern problems. In this PhD Thesis we develop novel and original algorithms for solving two large-scale modern problems in numerical linear algebra: first, we introduce the R-CORK method for solving rational eigenvalue problems and, second, we present projection methods to compute the solution of T-Sylvester matrix equations, both based on Krylov subspaces. The R-CORK method is an extension of the compact rational Krylov method (CORK) [104] introduced to solve a family of nonlinear eigenvalue problems that can be expressed and linearized in certain particular ways and which include arbitrary polynomial eigenvalue problems, but not arbitrary rational eigenvalue problems. The R-CORK method exploits the structure of the linearized problem by representing the Krylov vectors in a compact form in order to reduce the cost of storage, resulting in a method with two levels of orthogonalization. The first level of orthogonalization works with vectors of the same size as the original problem, and the second level works with vectors of size much smaller than the original problem. Since vectors of the size of the linearization are never stored or orthogonalized, R-CORK is more efficient from the point of view of memory and orthogonalization costs than the classical rational Krylov method applied to the linearization. Moreover, since the R-CORK method is based on a classical rational Krylov method, the implementation of implicit restarting is possible and we present an efficient way to do it, that preserves the compact representation of the Krylov vectors. We also introduce in this dissertation projection methods for solving the TSylvester equation, which has recently attracted considerable attention as a consequence of its close relation to palindromic eigenvalue problems and other applications. The theory concerning T-Sylvester equations is rather well understood, and before the work in this thesis, there were stable and efficient numerical algorithms to solve these matrix equations for small- to medium- sized matrices. However, developing numerical algorithms for solving large-scale T-Sylvester equations was a completely open problem. In this thesis, we introduce several projection methods based on block Krylov subspaces and extended block Krylov subspaces for solving the T-Sylvester equation when the right-hand side is a low-rank matrix. We also offer an intuition on the expected convergence of the algorithm based on block Krylov subspaces and a clear guidance on which algorithm is the most convenient to use in each situation. All the algorithms presented in this thesis have been extensively tested, and the reported numerical results show that they perform satisfactorily in practice.Adicionalmente se recibió ayuda parcial de los proyectos de investigación: “Structured Numerical Linear Algebra: Matrix Polynomials, Special Matrices, and Conditioning” (Ministerio de Economía y Competitividad de España, Número de proyecto: MTM2012-32542) y “Structured Numerical Linear Algebra for Constant, Polynomial and Rational Matrices” (Ministerio de Economía y Competitividad de España, Número de proyecto: MTM2015-65798-P), donde el investigador principal de ambos proyectos fue Froilán Martínez Dopico.Programa Oficial de Doctorado en Ingeniería MatemáticaPresidente: José Mas Marí.- Secretario: Fernando de Terán Vergara.- Vocal: José Enrique Román Molt

    The Dynamical Functional Particle Method for Multi-Term Linear Matrix Equations

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    Recent years have seen a renewal of interest in multi-term linear matrix equations, as these have come to play a role in a number of important applications. Here, we consider the solution of such equations by means of the dynamical functional particle method, an iterative technique that relies on the numerical integration of a damped second order dynamical system. We develop a new algorithm for the solution of a large class of these equations, a class that includes, among others, all linear matrix equations with Hermitian positive definite or negative definite coefficients. In numerical experiments, our MATLAB implementation outperforms existing methods for the solution of multi-term Sylvester equations. For the Sylvester equation AX + XB = C, in particular, it can be faster and more accurate than the built-in implementation of the Bartels–Stewart algorithm, when A and B are well conditioned and have very different size

    The automatic solution of partial differential equations using a global spectral method

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    A spectral method for solving linear partial differential equations (PDEs) with variable coefficients and general boundary conditions defined on rectangular domains is described, based on separable representations of partial differential operators and the one-dimensional ultraspherical spectral method. If a partial differential operator is of splitting rank 22, such as the operator associated with Poisson or Helmholtz, the corresponding PDE is solved via a generalized Sylvester matrix equation, and a bivariate polynomial approximation of the solution of degree (nx,ny)(n_x,n_y) is computed in O((nxny)3/2)\mathcal{O}((n_x n_y)^{3/2}) operations. Partial differential operators of splitting rank 3\geq 3 are solved via a linear system involving a block-banded matrix in O(min(nx3ny,nxny3))\mathcal{O}(\min(n_x^{3} n_y,n_x n_y^{3})) operations. Numerical examples demonstrate the applicability of our 2D spectral method to a broad class of PDEs, which includes elliptic and dispersive time-evolution equations. The resulting PDE solver is written in MATLAB and is publicly available as part of CHEBFUN. It can resolve solutions requiring over a million degrees of freedom in under 6060 seconds. An experimental implementation in the Julia language can currently perform the same solve in 1010 seconds.Comment: 22 page

    Fast linear algebra is stable

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    In an earlier paper, we showed that a large class of fast recursive matrix multiplication algorithms is stable in a normwise sense, and that in fact if multiplication of nn-by-nn matrices can be done by any algorithm in O(nω+η)O(n^{\omega + \eta}) operations for any η>0\eta > 0, then it can be done stably in O(nω+η)O(n^{\omega + \eta}) operations for any η>0\eta > 0. Here we extend this result to show that essentially all standard linear algebra operations, including LU decomposition, QR decomposition, linear equation solving, matrix inversion, solving least squares problems, (generalized) eigenvalue problems and the singular value decomposition can also be done stably (in a normwise sense) in O(nω+η)O(n^{\omega + \eta}) operations.Comment: 26 pages; final version; to appear in Numerische Mathemati
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