Showing papers on "Numerical analysis published in 2009"
TL;DR: Thanks to the spectral accuracy in both space and time of the proposed method, the storage requirement due to the “global time dependence” can be considerably relaxed, and therefore calculation of the long-time solution becomes possible.
Abstract: In this paper, we consider the numerical solution of the time fractional diffusion equation. Essentially, the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term. In the former case, the first-order time derivative is replaced by a fractional derivative, making the problem global in time. We propose a spectral method in both temporal and spatial discretizations for this equation. The convergence of the method is proven by providing a priori error estimate. Numerical tests are carried out to confirm the theoretical results. Thanks to the spectral accuracy in both space and time of the proposed method, the storage requirement due to the “global time dependence” can be considerably relaxed, and therefore calculation of the long-time solution becomes possible.
599 citations
TL;DR: In this article, a powerful and flexible MCMC algorithm for stochastic simulation is introduced, based on a pseudo-marginal method originally introduced in [Genetics 164 (2003) 1139-1160], showing how algorithms which are approximations to an idealized marginal algorithm, can share the same marginal stationary distribution as the idealized method.
Abstract: We introduce a powerful and flexible MCMC algorithm for stochastic simulation. The method builds on a pseudo-marginal method originally introduced in [Genetics 164 (2003) 1139-1160], showing how algorithms which are approximations to an idealized marginal algorithm, can share the same marginal stationary distribution as the idealized method. Theoretical results are given describing the convergence properties of the proposed method, and simple numerical examples are given to illustrate the promising empirical characteristics of the technique. Interesting comparisons with a more obvious, but inexact, Monte Carlo approximation to the marginal algorithm, are also given.
519 citations
TL;DR: In this article, the authors describe an efficient algorithm for low-rank approximation of matrices that produces accuracy that is very close to the best possible accuracy, for matrices of arbitrary sizes.
Abstract: Principal component analysis (PCA) requires the computation of a low-rank approximation to a matrix containing the data being analyzed. In many applications of PCA, the best possible accuracy of any rank-deficient approximation is at most a few digits (measured in the spectral norm, relative to the spectral norm of the matrix being approximated). In such circumstances, efficient algorithms have not come with guarantees of good accuracy, unless one or both dimensions of the matrix being approximated are small. We describe an efficient algorithm for the low-rank approximation of matrices that produces accuracy that is very close to the best possible accuracy, for matrices of arbitrary sizes. We illustrate our theoretical results via several numerical examples.
389 citations
TL;DR: A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation.
387 citations
TL;DR: In this article, a face-based smoothed finite element method (FS-FEM) was proposed to improve the accuracy of the FEM for 3D problems using 4-node tetrahedral elements that can be generated automatically for complicated domains.
Abstract: This paper presents a novel face-based smoothed finite element method (FS-FEM) to improve the accuracy of the finite element method (FEM) for three-dimensional (3D) problems. The FS-FEM uses 4-node tetrahedral elements that can be generated automatically for complicated domains. In the FS-FEM, the system stiffness matrix is computed using strains smoothed over the smoothing domains associated with the faces of the tetrahedral elements. The results demonstrated that the FS-FEM is significantly more accurate than the FEM using tetrahedral elements for both linear and geometrically non-linear solid mechanics problems. In addition, a novel domain-based selective scheme is proposed leading to a combined FS/NS-FEM model that is immune from volumetric locking and hence works well for nearly incompressible materials. The implementation of the FS-FEM is straightforward and no penalty parameters or additional degrees of freedom are used. The computational efficiency of the FS-FEM is found better than that of the FEM. Copyright © 2008 John Wiley & Sons, Ltd.
304 citations
TL;DR: This method is an extension of a method of Collatz (1942) for calculating the spectral radius of an irreducible nonnegative matrix and applies the method to studying higher-order Markov chains.
Abstract: In this paper we propose an iterative method for calculating the largest eigenvalue of an irreducible nonnegative tensor. This method is an extension of a method of Collatz (1942) for calculating the spectral radius of an irreducible nonnegative matrix. Numerical results show that our proposed method is promising. We also apply the method to studying higher-order Markov chains.
300 citations
TL;DR: A stability test procedure is proposed for linear nonhomogeneous fractional order systems with a pure time delay and sufficient conditions of this kind of stability are derived for particular class of fractional time-delay systems.
287 citations
TL;DR: In this article, a numerical scheme to solve the one-dimensional nonlinear Klein-Gordon equation with quadratic and cubic nonlinearity is proposed, which uses the collocation points and approximates the solution using Thin Plate Splines (TPS) radial basis functions (RBF).
286 citations
TL;DR: In this paper, a single-elastic beam model was developed to analyze the thermal vibration of single-walled carbon nanotubes (SWCNT) based on thermal elasticity mechanics, and nonlocal elasticity theory.
253 citations
TL;DR: This paper focuses on a class called spectral methods in which, typically, the various functions are expanded in sets of orthogonal polynomials or functions, and presents results obtained by various groups in the field of general relativity by means of spectral methods.
Abstract: Equations arising in general relativity are usually too complicated to be solved analytically and one must rely on numerical methods to solve sets of coupled partial differential equations. Among the possible choices, this paper focuses on a class called spectral methods in which, typically, the various functions are expanded in sets of orthogonal polynomials or functions. First, a theoretical introduction of spectral expansion is given with a particular emphasis on the fast convergence of the spectral approximation. We then present different approaches to solving partial differential equations, first limiting ourselves to the one-dimensional case, with one or more domains. Generalization to more dimensions is then discussed. In particular, the case of time evolutions is carefully studied and the stability of such evolutions investigated. We then present results obtained by various groups in the field of general relativity by means of spectral methods. Work, which does not involve explicit time-evolutions, is discussed, going from rapidly-rotating strange stars to the computation of black-hole-binary initial data. Finally, the evolution of various systems of astrophysical interest are presented, from supernovae core collapse to black-hole-binary mergers.
TL;DR: In this article, an alternative approach that can be applied to a very large class of dynamical systems (autonomous or forced) with smooth equations is presented. But the main idea is to systematically recast the dynamical system in quadratic polynomial form before applying the harmonic balance method.
TL;DR: A fast direct solver for large discretized linear systems using the supernodal multifrontal method together with low-rank approximations, especially suitable for large sparse problems and also has natural adaptability to parallel computations and great potential to provide effective preconditioners.
Abstract: In this paper we develop a fast direct solver for large discretized linear systems using the supernodal multifrontal method together with low-rank approximations. For linear systems arising from certain partial differential equations such as elliptic equations, during the Gaussian elimination of the matrices with proper ordering, the fill-in has a low-rank property: all off-diagonal blocks have small numerical ranks with proper definition of off-diagonal blocks. Matrices with this low-rank property can be efficiently approximated with semiseparable structures called hierarchically semiseparable (HSS) representations. We reveal the above low-rank property by ordering the variables with nested dissection and eliminating them with the multifrontal method. All matrix operations in the multifrontal method are performed in HSS forms. We present efficient ways to organize the HSS structured operations along the elimination. Some fast HSS matrix operations using tree structures are proposed. This new structured multifrontal method has nearly linear complexity and a linear storage requirement. Thus, we call it a superfast multifrontal method. It is especially suitable for large sparse problems and also has natural adaptability to parallel computations and great potential to provide effective preconditioners. Numerical results demonstrate the efficiency.
TL;DR: This work couple the incompressible steady Navier-Stokes equations with the Darcy equations, by means of the Beaver-Joseph-Saffman's condition on the interface, to prove existence of a weak solution as well as some a priori estimates.
Abstract: In this work, we couple the incompressible steady Navier-Stokes equations with the Darcy equations, by means of the Beaver-Joseph-Saffman's condition on the interface. Under suitable smallness conditions on the data, we prove existence of a weak solution as well as some a priori estimates. We establish local uniqueness when the data satisfy additional smallness restrictions. Then we propose a discontinuous Galerkin scheme for discretizing the equations and do its numerical analysis.
TL;DR: In this paper, the authors examined some practical numerical methods to solve a class of initial-boundary value problems for the fractional Fokker-Planck equation on a finite domain.
TL;DR: In this article, a matrix-based system reliability (MSR) method is proposed to compute the probabilities of general system events efficiently by simple matrix operations, which is uniformly applicable to any type of system events including series, parallel, cut-set and link-set systems.
TL;DR: In this article, the problem of bridging the gap between two scales in neuronal modeling is addressed, where neurons are considered individually and their behavior described by stochastic differential equations that govern the time variations of their membrane potentials.
Abstract: We deal with the problem of bridging the gap between two scales in neuronal modeling. At the first (microscopic) scale, neurons are considered individually and their behavior described by stochastic differential equations that govern the time variations of their membrane potentials. They are coupled by synaptic connections acting on their resulting activity, a nonlinear function of their membrane potential. At the second (mesoscopic) scale, interacting populations of neurons are described individually by similar equations. The equations describing the dynamical and the stationary mean field behaviors are considered as functional equations on a set of stochastic processes. Using this new point of view allows us to prove that these equations are well-posed on any finite time interval and to provide, by a fixed point method, a constructive method for effectively computing their unique solution. This method is proved to converge to the unique solution and we characterize its complexity and convergence rate. We also provide partial results for the stationary problem on infinite time intervals. These results shed some new light on such neural mass models as the one of Jansen and Rit (Jansen and Rit 1995): their dynamics appears as a coarse approximation of the much richer dynamics that emerges from our analysis. Our numerical experiments confirm that the framework we propose and the numerical methods we derive from it provide a new and powerful tool for the exploration of neural behaviors at different scales.
TL;DR: This paper presents a generalization of the Cholesky factor ADI method for Sylvester equations and demonstrates that Galerkin projection via ADI subspaces often produces much more accurate solutions than ADI solutions.
TL;DR: A new implicit closest point method for surface PDEs that allows for large, stable time steps while retaining the principal benefits of the original method and works on sharply defined bands without degrading the accuracy of the method.
Abstract: Many applications in the natural and applied sciences require the solutions of partial differential equations (PDEs) on surfaces or more general manifolds. The closest point method is a simple and accurate embedding method for numerically approximating PDEs on rather general smooth surfaces. However, the original formulation is designed to use explicit time stepping. This may lead to a strict time-step restriction for some important PDEs such as those involving the Laplace-Beltrami operator or higher-order derivative operators. To achieve improved stability and efficiency, we introduce a new implicit closest point method for surface PDEs. The method allows for large, stable time steps while retaining the principal benefits of the original method. In particular, it maintains the order of accuracy of the discretization of the underlying embedding PDE, it works on sharply defined bands without degrading the accuracy of the method, and it applies to general smooth surfaces. It also is very simple and may be applied to a rather general class of surface PDEs. Convergence studies for the in-surface heat equation and a fourth-order biharmonic problem are given to illustrate the accuracy of the method. We demonstrate the flexibility and generality of the method by treating flows involving diffusion, reaction-diffusion, and fourth-order spatial derivatives on a variety of interesting surfaces including surfaces of mixed codimension.
TL;DR: A new method for the evolution of inextensible vesicles immersed in a Stokesian fluid is presented and two semi-implicit schemes are presented that circumvent the severe stability constraints on the time step and whose computational cost per time step is comparable to that of an explicit scheme.
TL;DR: A modified anomalous subdiffusion equation with a nonlinear source term for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term is considered.
TL;DR: In this paper, an approach for the solution of these relativistic resistive MHD equations exploiting the properties of implicit-explicit (IMEX) Runge-Kutta methods is presented.
Abstract: Many astrophysical processes involving magnetic fields and quasi-stationary processes are well described when assuming the fluid as a perfect conductor. For these systems, the idealmagnetohydrodynamics (MHD) description captures the dynamics effectively and a number of well-tested techniques exist for its numerical solution. Yet, there are several astrophysical processes involving magnetic fields which are highly dynamical and for which resistive effects can play an important role. The numerical modelling of such non-ideal MHD flows is significantly more challenging as the resistivity is expected to change of several orders of magnitude across the flow and the equations are then either of hyperbolic‐parabolic nature or hyperbolic with stiff terms. We here present a novel approach for the solution of these relativistic resistive MHD equations exploiting the properties of implicit‐explicit (IMEX) Runge‐Kutta methods. By examining a number of tests, we illustrate the accuracy of our approach under a variety of conditions and highlight its robustness when compared with alternative methods, such as the Strang splitting. Most importantly, we show that our approach allows one to treat, within a unified framework, those regions of the flow which are both fluid-pressure dominated (such as in the interior of compact objects) and instead magnetic-pressure dominated (such as in their magnetospheres). In view of this, the approach presented here could find a number of applications and serve as a first step towards a more realistic modelling of relativistic astrophysical plasmas.
TL;DR: A contour integral method is proposed to solve nonlinear eigenvalue problems numerically by reducing the original problem to a linear eigen value problem that has identical eigenvalues in the domain.
Abstract: A contour integral method is proposed to solve nonlinear eigenvalue problems numerically. The target equation is F (λ)x = 0, where the matrix F (λ) is an analytic matrix function of λ. The method can extract only the eigenvalues λ in a domain defined by the integral path, by reducing the original problem to a linear eigenvalue problem that has identical eigenvalues in the domain. Theoretical aspects of the method are discussed, and we illustrate how to apply of the method with some numerical examples.
TL;DR: In this article, a finite element approximation for time-harmonic wave propagation governed by the Helmholtz equation has been proposed, which requires a minimal resolution of the mesh beyond what it takes to resolve the wavelength.
Abstract: We are concerned with a finite element approximation for time-harmonic wave propagation governed by the Helmholtz equation. The usually oscillatory behavior of solutions, along with numerical dispersion, render standard finite element methods grossly inefficient already in medium-frequency regimes. As an alternative, methods that incorporate information about the solution in the form of plane waves have been proposed. We focus on a class of Trefftz-type discontinuous Galerkin methods that employs trial and test spaces spanned by local plane waves. In this paper we give ap riori convergence estimates for the h-version of these plane wave discontinuous Galerkin methods in two dimensions. To that end, we develop new inverse and approximation estimates for plane waves and use these in the context of duality techniques. Asymptotic optimality of the method in a mesh dependent norm can be established. However, the estimates require a minimal resolution of the mesh beyond what it takes to resolve the wavelength. We give numerical evidence that this requirement cannot be dispensed with. It reflects the presence of numerical dispersion.
TL;DR: In this paper, a cell-centered finite volume scheme on general polygonal meshes fitting the interfaces is derived to solve the set of equations with the additional differential transmission conditions linking both pressure and normal velocity jumps through the interfaces.
Abstract: This study concerns some asymptotic models used to compute the flow outside and inside fractures in a bidimensional porous medium. The flow is governed by the Darcy law both in the fractures and in the porous matrix with large discontinuities in the permeability tensor. These fractures are supposed to have a small thickness with respect to the macroscopic length scale, so that we can asymptotically reduce them to immersed polygonal fault interfaces and the model finally consists in a coupling between a 2D elliptic problem and a 1D equation on the sharp interfaces modelling the fractures. A cell-centered finite volume scheme on general polygonal meshes fitting the interfaces is derived to solve the set of equations with the additional differential transmission conditions linking both pressure and normal velocity jumps through the interfaces. We prove the convergence of the FV scheme for any set of data and parameters of the models and derive existence and uniqueness of the solution to the asymptotic models proposed. The models are then numerically experimented for highly or partially immersed fractures. Some numerical results are reported showing different kinds of flows in the case of impermeable or partially/highly permeable fractures. The influence of the variation of the aperture of the fractures is also investigated. The numerical solutions of the asymptotic models are validated by comparing them to the solutions of the global Darcy model or to some analytic solutions.
TL;DR: A novel discontinuous Galerkin method for convection-diffusion-reaction problems, characterized by three main properties: it is hybridizable, efficiently implementable and competitive with the main existing methods for these problems, and it exhibits superconvergence properties of the approximation to the scalar variable.
Abstract: In this article, we propose a novel discontinuous Galerkin method for convection-diffusion-reaction problems, characterized by three main properties. The first is that the method is hybridizable; t...
TL;DR: In this article, a modified numerical scheme for a class of Frac- tional Optimal Control Problems (FOCPs) formulated in Agrawal (2004) where a fractional derivative (FD) is defined in the Riemann-Liouville sense is presented.
Abstract: This paper presents a modified numerical scheme for a class of Frac- tional Optimal Control Problems (FOCPs) formulated in Agrawal (2004) where a Fractional Derivative (FD) is defined in the Riemann-Liouville sense. In this scheme, the entire time domain is divided into several sub- domains, and a fractional derivative (FDs) at a time node point is approx- imated using a modified Grunwald-Letnikov approach. For the first order derivative, the proposed modified Grunwald-Letnikov definition leads to a central difference scheme. When the approximations are substituted into the Fractional Optimal Control (FCO) equations, it leads to a set of alge- braic equations which are solved using a direct numerical technique. Two examples, one time-invariant and the other time-variant, are considered to study the performance of the numerical scheme. Results show that 1) as the order of the derivative approaches an integer value, these formulations lead to solutions for integer order system, and 2) as the sizes of the sub- domains are reduced, the solutions converge. It is hoped that the present scheme would lead to stable numerical methods for fractional differential equations and optimal control problems.
TL;DR: A decoupling approach based on interface approximation via temporal extrapolation is proposed for devising decoupled marching algorithms for the mixed model that models coupled fluid flow and porous media flow.
Abstract: We study numerical methods for solving a non-stationary mixed Stokes-Darcy problem that models coupled fluid flow and porous media flow. A decoupling approach based on interface approximation via temporal extrapolation is proposed for devising decoupled marching algorithms for the mixed model. Error estimates are derived and numerical experiments are conducted to demonstrate the computational effectiveness of the decoupling approach.
TL;DR: In this article, a numerical method for the solution of a distributed-order differential equation of the general form @!"0^mA(r,D"*^ru(t))dr=f(t) where m is a positive real number and the derivative D" *^r is taken to be a fractional derivative of Caputo type of order r.
TL;DR: This paper describes how the Faber transform applied to the field of values of A can be used to determine improved error bounds for popular polynomial approximation methods based on the Arnoldi process.
Abstract: The need to evaluate expressions of the form $f(A)$ or $f(A)b$, where $f$ is a nonlinear function, $A$ is a large sparse $n\times n$ matrix, and $b$ is an $n$-vector, arises in many applications This paper describes how the Faber transform applied to the field of values of $A$ can be used to determine improved error bounds for popular polynomial approximation methods based on the Arnoldi process Applications of the Faber transform to rational approximation methods and, in particular, to the rational Arnoldi process also are discussed