1,720,988 research outputs found
The restarted shift-and-invert Krylov method for matrix functions
No full textIn this paper, the numerical evaluation of matrix functions expressed in partial fraction form is addressed.
The shift-and-invert Krylov method is analyzed, with special attention to error estimates. Such estimates
give insights into the selection of the shift parameter and lead to a simple and effective restart
procedure. Applications to the class of Mittag–Leffler functions are presented
A computationally efficient strategy for time-fractional diffusion-reaction equations
No full textAn efficient strategy for the numerical solution of time-fractional diffusion-reaction problems is devised. A standard finite difference discretization of the space derivative is initially applied which results in a linear stiff term. Then a rectangular product-integration (PI) rule is implemented in an implicit-explicit (IMEX) framework in order to implicitly treat this linear stiff term and handle in an explicit way the non-linear, and usually non-stiff, term. The kernel compression scheme (KCS) is successively adopted to reduce the overload of computation and storage need for the persistent memory term. To reduce the computational effort the semidiscretized problem is described in a matrix-form, so as to require the solution of Sylvester equations only with small matrices. Theoretical results on the accuracy, together with strategies for the optimal selection of the main parameters of the whole method, are derived and verified by means of numerical experiments carried out in two-dimensional domains. The computational advantages with respect to other approaches are also shown and some applications to the detection of pattern formation are illustrated at the end of the paper
Efficient approximation of functions of some large matrices by partial fraction expansions
No full textSome important applicative problems require the evaluation of functions of large and sparse and/or localized matrices A. Popular and interesting techniques for computing (A) and (A)v, where v is a vector, are based on partial fraction expansions. However, some of these techniques require solving several linear systems whose matrices differ from A by a complex multiple of the identity matrix I for computing (A)v or require inverting sequences of matrices with the same characteristics for computing (A). Here we study the use and the convergence of a recent technique for generating sequences of incomplete factorizations of matrices in order to face with both these issues. The solution of the sequences of linear systems and approximate matrix inversions above can be computed efficiently provided that A−1 shows certain decay properties. These strategies have good parallel potentialities. Our claims are confirmed by numerical tests
On the Numerical Computation of Transition Matrix pth Root
No full textTransition matrices arise in a wide class of engineering problems especially in
mathematical models using Markov chains.
Usually they describe the transition probabilities of a vector state at time to
the same state vector at time , where is the shortest period over
which a transition matrix can be estimated. If a short term transition matrix is needed
it can be obtained by computing a th root.
For example in risk management of portfolio
the company's credit ratings are recorded yearly, thus to define, at the end of the year,
a transition matrix with all the recorded information. However, investment horizon is shorter
than a year, thus to require the computation of the matrix th root.
In this paper we consider the numerical computation of the th root of a transition
matrix. The aim of the paper
is to highlight the properties of some numerical methods preserving the geometric peculiarities of the
pth root of a transition matrix
Evaluation of fractional integrals and derivatives of elementary functions: Overview and tutorial
Get PDFSeveral fractional-order operators are available and an in-depth knowledge of the selected operator is necessary for the evaluation of fractional integrals and derivatives of even simple functions. In this paper, we reviewed some of the most commonly used operators and illustrated two approaches to generalize integer-order derivatives to fractional order; the aim was to provide a tool for a full understanding of the specific features of each fractional derivative and to better highlight their differences. We hence provided a guide to the evaluation of fractional integrals and derivatives of some elementary functions and studied the action of different derivatives on the same function. In particular, we observed how Riemann-Liouville and Caputo's derivatives converge, on long times, to the Grunwald-Letnikov derivative which appears as an ideal generalization of standard integer-order derivatives although not always useful for practical applications
Exponential Integrators for Fractional Differential Equations
No full textExponential integrators are a well-established class of effective methods for the numerical integration of systems of differential equations of large dimension, especially in the presence of stiffness. Problems of this kind are common in several applications and usually come from semi-discretization of partial differential equations. The main idea behind exponential integrators is to solve in an exact way the stiff term of the problem and hence apply a time-step integration to the non-stiff term, thus to allow to use less expensive explicit schemes and obtain, at the same time, good stability properties. Recently, the investigation of exponential integrators has been generalized to differential equations of fractional order (FDEs). In this context exponential integrators turn out to be very effective since they overcome some of the typical weak points of numerical methods for FDEs: indeed, it is possible to derive methods with excellent stability properties and, when working with linear problems, to surmount some limitations in accuracy and order of convergence which represent a severe constriction for common methods for FDEs. In this Chapter we illustrate the main aspects concerning the derivation of some families of exponential integrators for FDEs and we study the convergence properties. Moreover, we address their numerical implementation with special attention to some peculiar aspects as, for example, the evaluation of Mittag-Leffler functions, with scalar and/or matrix arguments; this is a central feature in the implementation of exponential integrators for fractional order problems. The application to some time- fractional partial differential equations is also presented by means of some numerical examples
Exponential Quadrature Rules for Linear Fractional Differential Equations
Get PDFThis paper focuses on the numerical solution of initial value problems for fractional differential equations of linear type. The approach we propose grounds on expressing the solution in terms of some integral weighted by a generalized Mittag-Leffler function. Then, suitable quadrature rules are devised and order conditions of algebraic type are derived. Theoretical findings are validated by means of numerical experiments and the effectiveness of the proposed approach is illustrated by means of comparisons with other standard methods
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