Abstract: The aim of this paper is the derivation of an a-posteriori error estimate for the numerical method based on an exponential scheme in time and spectral Galerkin methods in space. We obtain analytically a rigorous bound on the conditional mean square error, which is conditioned to the given realization of the data calculated by a numerical method. This bound is explicitly computable and uses only the computed numerical approximation. Thus one can check a-posteriori the error for a given numerical computation for a fixed discretization without relying on an asymptotic result. All estimates are only based on the numerical data and the structure of the equation, but they do not use any a-priori information of the solution, which makes the approach applicable to equations where global existence and uniqueness of solutions is not known. For simplicity of presentation, we develop the method here in a relatively simple situation of a stable one-dimensional Allen-Cahn equation with additive forcing. PubDate: 2019-03-22

Abstract: We prove convergence of a finite difference scheme to the unique entropy solution of a general form of the Ostrovsky–Hunter equation on a bounded domain with non-homogeneous Dirichlet boundary conditions. Our scheme is an extension of monotone schemes for conservation laws to the equation at hand. The convergence result at the center of this article also proves existence of entropy solutions for the initial-boundary value problem for the general Ostrovsky–Hunter equation. Additionally, we show uniqueness using Kružkov’s doubling of variables technique. We also include numerical examples to confirm the convergence results and determine rates of convergence experimentally. PubDate: 2019-03-13

Abstract: This paper presents a study of the approximation error corresponding to a symplectic scheme of weak order one for a stochastic autonomous Hamiltonian system. A backward error analysis is done at the level of the Kolmogorov equation associated with the initial stochastic Hamiltonian system. An expansion of the weak error and expansions of the ergodic averages and of the invariant measures associated with the numerical scheme are obtained in terms of powers of the discretization step size and the solutions of the modified Kolmogorov equation. PubDate: 2019-03-12

Abstract: We show how a recently developed multivariate data fitting technique enables to solve a variety of scientific computing problems in filtering, queueing, networks, metamodelling, computational finance, graphics, and more. We can capture linear as well as nonlinear phenomena because the method uses a generalized multivariate rational model. The technique is a refinement of the basic ideas developed in Salazar et al. (Numer Algorithms 45:375–388, 2007. https://doi.org/10.1007/s11075-007-9077-3) and interpolates interval data. Intervals allow to take the inherent data error in measurements and simulation into consideration, whilst guaranteeing an upper bound on the tolerated range of uncertainty. The latter is the main difference with a best approximation or least squares technique which does as well as it can, but without respecting an a priori imposed threshold on the approximation error. Compared to the best approximations, the interval interpolant is relatively easy to compute. In applications where industry standards need to be guaranteed, the interval interpolation technique may be a valuable alternative. PubDate: 2019-03-01

Abstract: For solving time-dependent one-dimensional spatial-fractional diffusion equations of variable coefficients, we establish a banded M-splitting iteration method applicable to compute approximate solutions for the corresponding discrete linear systems resulting from certain finite difference schemes at every temporal level, and demonstrate its asymptotic convergence without imposing any extra condition. Also, we provide a multistep variant for the banded M-splitting iteration method, and prove that the computed solutions of the discrete linear systems by employing this iteration method converge to the exact solutions of the spatial fractional diffusion equations. Numerical experiments show the accuracy and efficiency of the multistep banded M-splitting iteration method. PubDate: 2019-03-01

Abstract: When polynomial roots vary widely in order of magnitude, severe numerical instability problem may occur due to deflation schemes. Peters and Wilkinson (IMA J Appl Math 8(1):16–35, 1971) have proposed deflation schemes to prevent the numerical stability of the remaining approximate roots from being severely worse than the one of the deflated root. In this paper, from the viewpoint of backward error of approximate roots, we show that this root distribution can be utilized to help improve the backward stability of some remaining approximate roots when using the deflation schemes proposed by Peters and Wilkinson. PubDate: 2019-03-01

Abstract: This paper deals with an inverse problem of identifying a space dependent coefficient in a time-fractional diffusion equation on a finite domain with final observation. The existence and uniqueness of this inverse problem are proved. A numerical scheme is proposed to solve the problem. The main idea of the proposed scheme is approximating the time fractional derivative by Diethelm’s quadrature formula and use the local discontinuous Galerkin method in space variable. Also, an error estimate for this problem is presented. Finally, two numerical example is studied to demonstrate the accuracy and efficiency of the proposed method. PubDate: 2019-03-01

Abstract: The aim of this paper is to contribute a new second-order pseudo-spectral method via a non-uniform distribution of the computational nodes for solving multi-asset option pricing problems. In such problems, the prices are required to be as accurately as possible around the strike price. Accordingly, the proposed modification of the Chebyshev–Gauss–Lobatto points would concentrate on this area. This adaptation is also fruitful for the non-smooth payoffs which cause discontinuities in the strike price. The proposed scheme competes well with the existing methods such as finite difference, meshfree, and adaptive finite difference methods on several benchmark problems. PubDate: 2019-03-01

Abstract: We introduce a stabilised finite element formulation for the Kirchhoff plate obstacle problem and derive both a priori and residual-based a posteriori error estimates using conforming \(C^1\) -continuous finite elements. We implement the method as a Nitsche-type scheme and give numerical evidence for its effectiveness in the case of an elastic and a rigid obstacle. PubDate: 2019-03-01

Abstract: The modified Hunter–Saxton equation models the propagation of short capillary-gravity waves. As the equation involves a mixed derivative, its initial value problem on the periodic domain is much more complicated than the standard evolutionary equations. Although its local well-posedness has recently been proved, the behavior of its solution is yet to be investigated. In this paper, a conservative finite difference scheme is derived as a reliable numerical method for this problem. Then, the stability of the numerical solution in the sense of the uniform norm, and the uniform convergence of the numerical solutions to sufficiently smooth exact solutions are rigorously proved. Discrete conservation laws are used to overcome the difficulty due to the mixed derivative. PubDate: 2019-03-01

Abstract: Quotients for eigenvalue problems (generalized or not) are considered. To have a quotient optimally approximating an eigenvalue, conditions are formulated to maximize the one-dimensional projection of the eigenvalue problem. Respective optimal quotient iterations are derived under the assumption that applying the inverse is affordable. Inexact methods are also considered if applying the inverse is not affordable. Then, to approximate an eigenvector, optimality conditions are formulated to minimize linear independency over a subspace. Equivalence transformations are performed for preconditioning iterations and steering the convergence. These ideas extend to subspaces in a natural way. For the standard eigenvalue problem, a new Arnoldi method arises as an alternative to the classical Arnoldi method. PubDate: 2019-03-01

Abstract: We consider Arnoldi-like processes to obtain symplectic subspaces for Hamiltonian systems. Large dimensional systems are locally approximated by ones living in low dimensional subspaces, and we especially consider Krylov subspaces and some of their extensions. These subspaces can be utilized in two ways: by solving numerically local small dimensional systems and then mapping back to the large dimension, or by using them for the approximation of necessary functions in exponential integrators applied to large dimensional systems. In the former case one can expect an excellent energy preservation and in the latter this is so for linear systems. We consider second order exponential integrators which solve linear systems exactly and for which these two approaches are in a certain sense equivalent. We also consider the time symmetry preservation properties of the integrators. In numerical experiments these methods combined with symplectic subspaces show promising behavior also when applied to nonlinear Hamiltonian problems. PubDate: 2019-03-01

Abstract: By applying the minimum residual technique to the Hermitian and skew-Hermitian splitting (HSS) iteration scheme, we introduce a non-stationary iteration method named minimum residual HSS (MRHSS) iteration method to solve non-Hermitian positive definite linear systems. The convergence property of the MRHSS iteration method together with the property of the iteration parameters are carefully studied. Numerical results verify the effectiveness and robustness of the MRHSS iteration method. PubDate: 2019-03-01

Abstract: Highly oscillatory integrals, having amplitudes with algebraic (or logarithmic) endpoint singularities, are considered. An integral of this kind is first transformed into a regular oscillatory integral over an unbounded interval. After applying the method of finite sections, a composite modified Filon–Clenshaw–Curtis rule, recently developed by the author, is applied on it. By this strategy the original integral can be computed in a more stable manner, while the convergence orders of the composite Filon–Clenshaw–Curtis rule are preserved. By introducing the concept of an oscillation subinterval, we propose algorithms, which employ composite Filon–Clenshaw–Curtis rules on rather small intervals. The integral outside the oscillation subinterval is non-oscillatory, so it can be computed by traditional quadrature rules for regular integrals, e.g. the Gaussian ones. We present several numerical examples, which illustrate the accuracy of the algorithms. PubDate: 2019-03-01

Abstract: In this paper, we establish Razumikhin-type theorems on \(\alpha \) th moment polynomial stability of exact solution for the stochastic pantograph differential equations, which improves the existing stochastic Razumikhin-type theorems. By using discrete Razumikhin-type technique, we construct conditions for the stability of general numerical scheme of the stochastic pantograph differential equations (SPDEs). The stabilities mainly conclude the global \(\alpha \) th moment asymptotically stability and \(\alpha \) th moment polynomial stability. Using the conditions constructed for the stability of the numerical solutions, we discuss the stability of two special numerical methods, namely the Euler–Maruyama method and the backward Euler–Maruyama method. Finally, an example is given to illustrate the consistence with the theoretical results on \(\alpha \) th moment polynomial stability. PubDate: 2019-03-01

Abstract: The stability properties of the Newton interpolation formula depend on the order of the nodes and can be measured through a condition number. Increasing and Leja orderings have been previously considered (Carnicer et al. in J Approx Theory, 2017. https://doi.org/10.1016/j.jat.2017.07.005; Reichel in BIT 30:332–346, 1990). We analyze central orderings for equidistant nodes on a bounded real interval. A bound for conditioning is given. We demonstrate in particular that this ordering provides a more stable Newton formula than the natural increasing order. We also analyze of a central ordering with respect to the evaluation point, which provides low bounds for the conditioning. Numerical examples are included. PubDate: 2019-01-02

Abstract: Matrix functions have a major role in science and engineering. One of the fundamental matrix functions, which is particularly important due to its connections with certain matrix differential equations and other special matrix functions, is the matrix gamma function. This research article focus on the numerical computation of this function. Well-known techniques for the scalar gamma function, such as Lanczos, Spouge and Stirling approximations, are extended to the matrix case. This extension raises many challenging issues and several strategies used in the computation of matrix functions, like Schur decomposition and block Parlett recurrences, need to be incorporated to make the methods more effective. We also propose a fourth technique based on the reciprocal gamma function that is shown to be competitive with the other three methods in terms of accuracy, with the advantage of being rich in matrix multiplications. Strengths and weaknesses of the proposed methods are illustrated with a set of numerical examples. Bounds for truncation errors and other bounds related with the matrix gamma function will be discussed as well. PubDate: 2019-01-02

Abstract: In this work, we investigate the spectra of “flipped” Toeplitz sequences, i.e., the asymptotic spectral behaviour of \(\{Y_nT_n(f)\}_n\) , where \(T_n(f)\in \mathbb {R}^{n\times n}\) is a real Toeplitz matrix generated by a function \(f\in L^1([-\pi ,\pi ])\) , and \(Y_n\) is the exchange matrix, with 1s on the main anti-diagonal. We show that the eigenvalues of \(Y_nT_n(f)\) are asymptotically described by a \(2\times 2\) matrix-valued function, whose eigenvalue functions are \(\pm \, f \) . It turns out that roughly half of the eigenvalues of \(Y_nT_n(f)\) are well approximated by a uniform sampling of f over \([-\,\pi ,\pi ]\) , while the remaining are well approximated by a uniform sampling of \(-\, f \) over the same interval. When f vanishes only on a set of measure zero, this motivates that the spectrum is virtually half positive and half negative. Some insights on the spectral distribution of related preconditioned sequences are provided as well. Finally, a wide number of numerical results illustrate our theoretical findings. PubDate: 2018-12-18

Abstract: Some Krylov subspace methods for approximating the action of matrix functions are presented in this paper. The main idea of these techniques is to project the approximation problem onto a subspace of much smaller dimension. Then the matrix function operation is performed with a much smaller matrix. These methods are projection methods that use the Hessenberg process to generate bases of the approximation spaces. We also use the introduced methods to solve shifted linear systems. Some numerical experiments are presented in order to show the efficiency of the proposed methods. PubDate: 2018-12-18

Abstract: Two-by-two block matrices with square matrix blocks arise in many important applications. Since the problems are of large scale, iterative solution methods must be used. Thereby the choice of an efficient and robust preconditioner is crucial. This paper presents two earlier used such preconditioners followed by a novel preconditioner based on transforming the given matrix to a proper form. Sharp eigenvalue estimates are derived. The condition numbers of each of the three methods are robust with respect to all parameters involved, including the mesh parameter. Therefore, the preconditioners are suitable for a variety of problems where such matrix structures arise. The performance of the methods are also compared numerically on a set of test problems. PubDate: 2018-12-13