Comparison of software for computing the action of the matrix exponential

The implementation of exponential integrators requires the action of the matrix exponential and related functions of a large matrix. There are various methods in the literature for carrying out this task. In this paper we describe a new implementation of a method based on interpolation at Leja points. We numerically compare this method with others from the literature. As we are interested in exponential intergrators we choose the test examples from spatial discretization of time dependent partial differential equations in two and three space dimensions. The test matrices thus have large eigenvalues and can be nonnormal

On "marcov" inequalities

As colleagues and friends we wish to dedicate these pages to Marco Vianello on the occasion of his 60th birthday, which is on October 26, 2021. Marco has made many important contributions to approximation theory and beyond. Here we briefly summarize some of them in the spirit of the occasion

A massively parallel exponential integrator for advection-diffusion models

This work considers the Real Leja Points Method (ReLPM) for the exponential integration of large-scale sparse systems of ODEs, generated by Finite Element or Finite Difference discretizations of 3-D advection-diffusion models. We present an efficient parallel implementation of ReLPM for polynomial interpolation of the matrix exponential propagators. A scalability analysis of the most important computational kernel inside the code, the parallel sparse matrix\u2013vector product, has been performed, as well as an experimental study of the communication overhead. As a result of this study an optimized parallel sparse matrix\u2013vector product routine has been implemented. The resulting code shows good scaling behavior even when using more than one thousand processors. The numerical results presented on a number of very large test cases gives experimental evidence that ReLPM is a reliable and efficient tool for the simulation of complex hydrodynamic processes on parallel architectures

Backward error analysis of polynomial approximations for computing the action of the matrix exponential

We describe how to perform the backward error analysis for the approximation of exp(A)v by p(s 121 A) s v, for any given polynomial p(x). The result of this analysis is an optimal choice of the scaling parameter s which assures a bound on the backward error, i.e. the equivalence of the approximation with the exponential of a slightly perturbed matrix. Thanks to the SageMath package expbea we have developed, one can optimize the performance of the given polynomial approximation. On the other hand, we employ the package for the analysis of polynomials interpolating the exponential function at so called Leja\u2013Hermite points. The resulting method for the action of the matrix exponential can be considered an extension of both Taylor series approximation and Leja point interpolation. We illustrate the behavior of the new approximation with several numerical examples

Efficient approximation of the exponential operator for discrete 2D advection-diffusion problems

In this paper we compare Krylov subspace methods with Faber series expansion for approximating the matrix exponential operator on large, sparse, non-symmetric matrices. We consider in particular the case of Chebyshev series, corresponding to an initial estimate of the spectrum of the matrix by a suitable ellipse. Experimental results upon matrices with large size, arising from space discretization of 2D advection-diffusion problems, demonstrate that the Chebyshev method can be an effective alternative to Krylov techniques

Approximation of the Neutron Diffusion Equation on Hexagonal Geometries

La ecuación de la difusión neutrónica describe la población de neutrones de un reactor nuclear. Este trabajo trata con este modelo para reactores nucleares con geometría hexagonal. En primer lugar se estudia la ecuación de la difusión neutrónica. Este es un problema diferencial de valores propios, llamado problema de los modos Lambda. Para resolver el problema de los modos Lambda se han comparado diferentes métodos en geometrías unidimensionales, resultando como el mejor el método de elementos espectrales. Usando este método discretizamos los operadores en geometrías bidimensiones y tridimensionales, resolviendo el problema algebraica de valores propios resultante con el método de Arnoldi. La distribución de neutrones estado estacionario se utiliza como condición inicial para la integración de la ecuación de la difusión neutrónica dependiente del tiempo. Se utiliza un método de Euler implícito para integrar en el tiempo. Cuando un nodo está parcialmente insertado aparece un comportamiento no físico de la solución, el efecto ``rod cusping'', que se corrige mediante la ponderación de las secciones eficaces con el flujo del paso de tiempo anterior. Cuando la solución de los sistemas algebraicos que surgen en el método hacia atrás, un método de Krylov se utiliza para resolver los sistemas resultantes, y diferentes estrategias de precondicionamiento se evalúan se. La primera consiste en el uso de la estructura de bloque obtenido por los grupos de energía para resolver el sistema por bloques, y diferentes técnicas de aceleración para el esquema iterativo de bloques y un precondicionador utilizando esta estructura de bloque se proponen. Además se estudia un precondicionador espectral, que hace uso de la información en un subespacio de Krylov para precondicionar el siguiente sistema. También se proponen métodos exponenciales de segundo y cuarto orden integrar la ecuación de difusión neutrónica dependiente del tiempo, donde la exponencial de la matriz del sistema tiene quGonzález Pintor, S. (2012). Approximation of the Neutron Diffusion Equation on Hexagonal Geometries [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/17829Palanci