On the stability function of functionally-fitted Runge--Kutta methods

Classical collocation Runge--Kutta methods are polynomially fitted in the sense that they integrate an ODE problem exactly if its solution is an algebraic polynomial up to some degree. Functionally fitted Runge--Kutta methods are collocation techniques that generalize this principle to solve an ODE problem exactly if its solution is a linear combination of a chosen set of arbitrary basis functions. Given for example a periodic or oscillatory ODE problem with a known frequency, it might be advantageous to tune a trigonometric functionally fitted Runge--Kutta method targeted at such a problem. However, functionally fitted Runge--Kutta methods lead to variable coefficients that depend on the parameters of the problem, the time, the step size, and the basis functions in a non-trivial manner that inhibits any in-depth analysis of the behavior of the methods in general. We present the class of so-called separable basis functions and show that it is possible to characterize the stability region of some special methods in this particular class. Explicit stability functions are given for some representative examples

Evaluation of the performance of inexact GMRES

AbstractThe inexact GMRES algorithm is a variant of the GMRES algorithm where matrix–vector products are performed inexactly, either out of necessity or deliberately, as part of a trading of accuracy for speed. Recent studies have shown that relaxing matrix–vector products in this way can be justified theoretically and experimentally. Research, so far, has focused on decreasing the workload per iteration without significantly affecting the accuracy. But relaxing the accuracy per iteration is liable to increase the number of iterations, thereby increasing the overall runtime, which could potentially end up being greater than that of the exact GMRES if there were not enough savings in the matrix–vector products. In this paper, we assess the benefit of the inexact approach in terms of actual CPU time derived from realistic problems, and we provide cases that provide instructive insights into results affected by the build-up of the inexactness. Such information is of vital importance to practitioners who need to decide whether switching their workflow to the inexact approach is worth the effort and the risk that might come with it. Our assessment is drawn from extensive numerical experiments that gauge the effectiveness of the inexact scheme and its suitability for use in addressing certain problems, depending on how much inexactness is allowed in the matrix–vector products

An Implementation of The Method of Moments on Chemical Systems with Constant and Time-dependent Rates

Among numerical techniques used to facilitate the analysis of biochemical reactions, we can use the method of moments to directly approximate statistics such as the mean numbers of molecules. The method is computationally viable in time and memory, compared to solving the chemical master equation (CME) which is notoriously expensive. In this study, we apply the method of moments to a chemical system with a constant rate representing a vascular endothelial growth factor (VEGF) model, as well as another system with time-dependent propensities representing the susceptible, infected, and recovered (SIR) model with periodic contact rate. We assess the accuracy of the method using comparisons with approximations obtained by the stochastic simulation algorithm (SSA) and the chemical Langevin equation (CLE). The VEGF model is of interest because of the role of VEGF in the growth of cancer and other inflammatory diseases and the potential use of anti-VEGF therapies in the treatment of cancer. The SIR model is a popular epidemiological model used in studying the spread of various infectious diseases in a population

Transient solutions of Markov processes by Krylov subspaces

In this note we exploit the knowledge embodied in infinitesimal generators of Markov processes to compute efficiently and economically the transient solution of continuous time Markov processes. We consider the Krylov subspace approximation method which has been analysed by Y. Saad for solving linear differential equations. We place special emphasis on error bounds and stepsize control. We discuss the computation of the exponential of the Hessenberg matrix involved in the approximation and an economic evaluation of the Pade method is presented. We illustrate the usefulness of the approach by providing some application examples

Exponential integrators for coupled self-adjoint non-autonomous partial differential systems

We consider the numerical integration of coupled self-adjoint non-autonomous partial differential systems. Under convergence conditions, the solution can be written as a series expansion where each of its terms correspond to solutions of linear time dependent matrix differential equations with oscillatory solutions that must be solved numerically. In this work, we analyze second order of Magnus integrators whose numerical error grows with the number of terms considered in the truncated series, n, at a rate that still allows us to guarantee convergence of the numerical series. In addition, the integrator can be implemented with a recursive algorithm such that the computational cost of the method grows only linearly with the number of terms of the series. Higher order Magnus integrators are also analyzed. Commutator-free Magnus integrators can be used with a similar recursive algorithm and can provide highly accurate results, but they show a faster error growth with n, and some caution must be taken if these methods are used. Numerical experiments confirm the performance of the proposed algorithm. (C) 2014 Elsevier Inc. All rights reserved.The work of S. Blanes has been partially supported by Ministerio de Ciencia e Innovacion (Spain) under project MTM2010-18246-C03. The work of E. Ponsoda has also been partially supported by the Universitat Politecnica de Valencia under project 2087.Ponsoda Miralles, E.; Blanes Zamora, S. (2014). Exponential integrators for coupled self-adjoint non-autonomous partial differential systems. Applied Mathematics and Computation. 243:1-11. https://doi.org/10.1016/j.amc.2014.05.050S11124

Numerical Integration of the Master Equation in Some Models of Stochastic Epidemiology

The processes by which disease spreads in a population of individuals are inherently stochastic. The master equation has proven to be a useful tool for modeling such processes. Unfortunately, solving the master equation analytically is possible only in limited cases (e.g., when the model is linear), and thus numerical procedures or approximation methods must be employed. Available approximation methods, such as the system size expansion method of van Kampen, may fail to provide reliable solutions, whereas current numerical approaches can induce appreciable computational cost. In this paper, we propose a new numerical technique for solving the master equation. Our method is based on a more informative stochastic process than the population process commonly used in the literature. By exploiting the structure of the master equation governing this process, we develop a novel technique for calculating the exact solution of the master equation – up to a desired precision – in certain models of stochastic epidemiology. We demonstrate the potential of our method by solving the master equation associated with the stochastic SIR epidemic model. MATLAB software that implements the methods discussed in this paper is freely available as Supporting Information S1

A piecewise-linearized algorithm based on the Krylov subspace for solving stiff ODEs

Numerical methods for solving Ordinary Differential Equations (ODEs) have received considerable attention in recent years. In this paper a piecewise-linearized algorithm based on Krylov subspaces for solving Initial Value Problems (IVPs) is proposed. MATLAB versions for autonomous and non-autonomous ODEs of this algorithm have been implemented. These implementations have been compared with other piecewise-linearized algorithms based on Pad approximants, recently developed by the authors of this paper, comparing both precisions and computational costs in equal conditions. Four case studies have been used in the tests that come from stiff biology and chemical kinetics problems. Experimental results show the advantages of the proposed algorithms, especially when the dimension is increased in stiff problems. © 2009 Elsevier B.V. All rights reserved.This work was supported by the Spanish CICYT project CGL2007-66440-C04-03.Ibáñez González, JJ.; Hernández García, V.; Ruiz Martínez, PA.; Arias, E. (2011). A piecewise-linearized algorithm based on the Krylov subspace for solving stiff ODEs. Journal of Computational and Applied Mathematics. 235(7):1798-1804. https://doi.org/10.1016/j.cam.2010.07.012S17981804235

Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear evolution equations of parabolic type

[EN] The main objective of this work is to provide a stability and error analysis of high-order commutator-free quasi-Magnus (CFQM) exponential integrators. These time integration methods for nonautonomous linear evolution equations are formed by products of exponentials involving linear combinations of the defining operator evaluated at certain times. In comparison with other classes of time integration methods, such as Magnus integrators, an inherent advantage of CFQM exponential integrators is that structural properties of the operator are well preserved by the arising linear combinations. Employing the analytical framework of sectorial operators in Banach spaces, evolution equations of parabolic type and dissipative quantum systems are included in the scope of applications. In this context, however, numerical experiments show that CFQM exponential integrators of nonstiff order five or higher proposed in the literature suffer from poor stability properties. The given analysis delivers insight that CFQM exponential integrators are well defined and stable only if the coefficients occurring in the linear combinations satisfy a positivity condition and that an alternative approach for the design of stable high-order schemes relies on the consideration of complex coefficients. Together with suitable local error expansions, this implies that a high-order CFQM exponential integrator retains its nonstiff order of convergence under appropriate regularity and compatibility requirements on the exact solution. Numerical examples confirm the theoretical result and illustrate the favourable behaviour of novel schemes involving complex coefficients in stability and accuracy.Ministerio de Economia y Competitividad (Spain) through projects MTM2013-46553-C3 and MTM2016-77660-P (AEI/FEDER, UE) to S.B. and F.C.Blanes Zamora, S.; Casas, F.; Mechthild Thalhammer (2018). Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear evolution equations of parabolic type. IMA Journal of Numerical Analysis. 38(2):743-778. https://doi.org/10.1093/imanum/drx012S74377838