On piecewise-uniform meshes for upwind- and central-difference operators for solving singularly perturbed problems

Numerical methods composed of upwind-difference operators onuniform meshes are shown analytically to be defective for solvingsingularly perturbed differential equations, in the sense that,as the mesh is refined, the error in the numerical approximationincreases until the mesh parameter has decreased to the sameorder of magnitude as the singular perturbation parameter. Itis also shown that the same is true for upwind-difference operatorson piecewise uniform meshes having a transition point that isdependent solely on the singular perturbation parameter. Itis then shown, with specific analytic examples, that both upwind-and central-difference operators on specially designed piecewise-uniformmeshes give numerical methods which do not suffer from thisdefect. Conditions are also given on the structure of a piecewiseuniform mesh that are necessary if the numerical method, composedof this mesh and an upwind-difference operator, is to be convergentuniformly with respect to the singular perturbation parameter.     

Alternating Direction Method with Self-Adaptive Penalty Parameters for Monotone Variational Inequalities

The alternating direction method is one of the attractive approaches for solving linearly constrained separate monotone variational inequalities. Experience on applications has shown that the number of iterations depends significantly on the penalty parameter for the system of linear constraint equations. While the penalty parameter is a constant in the original method, in this paper we present a modified alternating direction method that adjusts the penalty parameter per iteration based on the iterate message. Preliminary numerical tests show that the self-adaptive adjustment technique is effective in practice.     

A parameter-uniform numerical method for a Sobolev problem with initial layer

The present study is concerned with the numerical solution, using finite difference method of a one-dimensional initial-boundary value problem for a linear Sobolev or pseudo-parabolic equation with initial jump. In order to obtain an efficient method, to provide good approximations with independence of the perturbation parameter, we have developed a numerical method which combines a finite difference spatial discretization on uniform mesh and the implicit rule on Shishkin mesh(S-mesh) for the time variable. The fully discrete scheme is shown to be convergent of order two in space and of order one expect for a logarithmic factor in time, uniformly in the singular perturbation parameter. Some numerical results confirming the expected behavior of the method are shown.      

A simple strategy for varying the restart parameter in GMRES()

When solving a system of linear equations with the restarted GMRES method, a fixed restart parameter is typically chosen. We present numerical experiments that demonstrate the beneficial effects of changing the value of the restart parameter in each restart cycle on the total time to solution. We propose a simple strategy for varying the restart parameter and provide some heuristic explanations for its effectiveness based on analysis of the symmetric case.     

A numerical method for solving nonlinear ill-posed problems

A two-step iterative process for the numerical solution of nonlinear problems is suggested. In order to avoid the ill-posed inversion of the Fréchet derivative operator, some regularization parameter is introduced. A convergence theorem is proved. The proposed method is illustrated by a numerical example in which a nonlinear inverse problem of gravimetry is considered. Based on the results of the numerical experiments practical recommendations for the choice of the regularization parameter are given. Some other iterative schemes are considered.     

Uniform difference method for parameterized singularly perturbed delay differential equations

This paper deals with the singularly perturbed initial value problem for quasilinear first-order delay differential equation depending on a parameter. A numerical method is constructed for this problem which involves an appropriate piecewise-uniform meshes on each time subinterval. The difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter. Some numerical experiments illustrate in practice the result of convergence proved theoretically.     

Stability analysis for a fully discrete spectral scheme for Boussinesq systems

In this paper we perform a stability analysis of a fully discrete numerical method for the solution of a family of Boussinesq systems, consisting of a Fourier collocation spectral method for the spatial discretization and a explicit fourth order Runge–Kutta (RK4) scheme for time integration. Our goal is to determine the influence of the parameters, associated to this family of systems, on the efficiency and accuracy of the numerical method. This analysis allows us to identify which regions in the parameter space are most appropriate for obtaining an efficient and accurate numerical solution. We show several numerical examples in order to validate the accuracy, stability and applicability of our MATLAB implementation of the numerical method.     

A wavelet-Galerkin method for high order numerical differentiation

Numerical differentiation is a classical ill-posed problem. In this paper, we propose a wavelet-Galerkin method for high order numerical differentiation. By an appropriate choice of the regularization parameter an order optimal stability estimate of Hölder type is obtained. Some numerical examples show that the method is effective and stable.     

Practical estimation of a splitting parameter for a spectral method for the ternary Cahn–Hilliard system with a logarithmic free energy

We propose a practical estimation of a splitting parameter for a spectral method for the ternary Cahn–Hilliard system with a logarithmic free energy. We use Eyre's convex splitting scheme for the time discretization and a Fourier spectral method for the space variables. Given an absolute temperature, we find composition values that make the total free energy be minimum. Then, we find the splitting parameter value that makes the two split homogeneous free energies be convex on the neighborhood of the local minimum concentrations. For general use, we also propose a sixth‐order polynomial approximation of the minimum concentration and derive a useful formula for the practical estimation of the splitting parameter in terms of the absolute temperature. The numerical tests are phase separation and total energy decrease with different temperature values. The linear stability analysis shows a good agreement between the exact and numerical solutions with an optimal value s. Various computational experiments confirm that the proposed splitting parameter estimation gives stable numerical results. Copyright © 2016 John Wiley & Sons, Ltd.     

An Efficient Monte Carlo Method for Optimal Control Problems with Uncertainty

A general framework is proposed for what we call the sensitivity derivative Monte Carlo (SDMC) solution of optimal control problems with a stochastic parameter. This method employs the residual in the first-order Taylor series expansion of the cost functional in terms of the stochastic parameter rather than the cost functional itself. A rigorous estimate is derived for the variance of the residual, and it is verified by numerical experiments involving the generalized steady-state Burgers equation with a stochastic coefficient of viscosity. Specifically, the numerical results show that for a given number of samples, the present method yields an order of magnitude higher accuracy than a conventional Monte Carlo method. In other words, the proposed variance reduction method based on sensitivity derivatives is shown to accelerate convergence of the Monte Carlo method. As the sensitivity derivatives are computed only at the mean values of the relevant parameters, the related extra cost of the proposed method is a fraction of the total time of the Monte Carlo method.     

A parameter‐uniform scheme for the parabolic singularly perturbed problem with a delay in time

In this paper, a parameter‐uniform numerical scheme for the solution of singularly perturbed parabolic convection–diffusion problems with a delay in time defined on a rectangular domain is suggested. The presence of the small diffusion parameter ? leads to a parabolic right boundary layer. A collocation method consisting of cubic B ‐spline basis functions on an appropriate piecewise‐uniform mesh is used to discretize the system of ordinary differential equations obtained by using Rothe's method on an equidistant mesh in the temporal direction. The parameter‐uniform convergence of the method is shown by establishing the theoretical error bounds. The numerical results of the test problems validate the theoretical error bounds.     

Convergence of the Swartztrauber-Sweet Method for Parabolic Problems on a Disk with Singularities

In this article, we study the convergence analysis for the initial and boundary value problem of parabolic equations on a disk with singular solutions. It is assumed that the exact solution performs singular properties that its derivatives go to infinity at the boundary of the disk. We propose a fully implicit time-stepping numerical scheme. A stretching polynomial-like function with a parameter is used to construct a local grid refinement. Over the nonuniform partition, we combine the Swartztrauber-Sweet scheme and the backward Euler method in spatial and temporal discretization, respectively. We carry out convergence analysis and analyze the effects of the parameter. It is shown that our numerical scheme is of first order accuracy for temporal discretization and of almost second order accuracy for spatial discretization. Numerical experiments are performed to illustrate our analysis results and show that there exists an optimal value for the parameter to obtain a best approximate solution.     

A single-step HSS method for non-Hermitian positive definite linear systems

In this paper, based on the Hermitian and skew-Hermitian splitting (HSS) iteration method, a single-step HSS (SHSS) iteration method is introduced to solve the non-Hermitian positive definite linear systems. Theoretical analysis shows that, under a loose restriction on the iteration parameter, the SHSS method is convergent to the unique solution of the linear system. Furthermore, we derive an upper bound for the spectral radius of the SHSS iteration matrix, and the quasi-optimal parameter is obtained to minimize the above upper bound. Numerical experiments are reported to the efficiency of the SHSS method; numerical comparisons show that the proposed SHSS method is superior to the HSS method under certain conditions.     

Self-adaptive operator splitting methods for monotone variational inequalities

Summary. Solving a variational inequality problem VI(Ω,F) is equivalent to finding a solution of a system of nonsmooth equations (a hard problem). The Peaceman-Rachford and /or Douglas-Rachford operator splitting methods are advantageous when they are applied to solve variational inequality problems, because they solve the original problem via solving a series of systems of nonlinear smooth equations (a series of easy problems). Although the solution of VI(Ω,F) is invariant under multiplying F by some positive scalar β, yet the numerical experiment has shown that the number of iterations depends significantly on the positive parameter β which is a constant in the original operator splitting methods. In general, it is difficult to choose a proper parameter β for individual problems. In this paper, we present a modified operator splitting method which adjusts the scalar parameter automatically per iteration based on the message of the iterates. Exact and inexact forms of the modified method with self-adaptive variable parameter are suggested and proved to be convergent under mild assumptions. Finally, preliminary numerical tests show that the self-adaptive adjustment rule is proper and necessary in practice.     

Decomposition Method with a Variable Parameter for a Class of Monotone Variational Inequality Problems

In this paper, we focus on a useful modification of the decomposition method by He et al. (Ref. 1). Experience on applications has shown that the number of iterations of the original method depends significantly on the penalty parameter. The main contribution of our method is that we allow the penalty parameter to vary automatically according to some self-adaptive rules. As our numerical simulations indicate, the modified method is more flexible and efficient in practice. A detailed convergence analysis of our method is also included.     

Continuous-discontinuous finite element method for convection-diffusion problems with characteristic layers

We study convergence properties of a numerical method for convection-diffusion problems with characteristic layers on a layer-adapted mesh. The method couples standard Galerkin with an h-version of the nonsymmetric discontinuous Galerkin finite element method with bilinear elements. In an associated norm, we derive the error estimate as well as the supercloseness result that are uniform in the perturbation parameter. Applying a post-processing operator for the discontinuous Galerkin method, we construct a new numerical solution with enhanced convergence properties.     

A posteriori error analysis for defect correction method for two parameter singular perturbation problems

Defect correction method is used for two parameter singular perturbation problem on Bakhvalov-Shishkin mesh. Use of defect correction method on Bakhvalov-Shishkin mesh gives a second order convergence. A posteriori error estimate is obtained. The numerical examples are given to establish the second order convergence in practice.     

On adaptive mesh for the initial boundary value singularly perturbed delay Sobolev problems

A uniform finite difference method on a B-mesh is applied to solve the initial-boundary value problem for singularly perturbed delay Sobolev equations. To solve the foresold problem, finite difference scheme on a special nonuniform mesh, whose solution converges point-wise independently of the singular perturbation parameter is constructed and analyzed. The present paper also aims at discussing the stability and convergence analysis of the method. An error analysis shows that the method is of second order convergent in the discrete maximum norm independent of the perturbation parameter. A numerical example and the simulation results show the effectiveness of our theoretical results.     

An algorithm to construct Monte Carlo confidence intervals for an arbitrary function of probability distribution parameters

We derive a new algorithm for calculating an exact confidence interval for a parameter of location or scale family, based on a two-sided hypothesis test on the parameter of interest, using some pivotal quantities. We use this algorithm to calculate approximate confidence intervals for the parameter or a function of the parameter of one-parameter continuous distributions. After appropriate heuristic modifications of the algorithm we use it to obtain approximate confidence intervals for a parameter or a function of parameters for multi-parameter continuous distributions. The advantage of the algorithm is that it is general and gives a fast approximation of an exact confidence interval. Some asymptotic (analytical) results are shown which validate the use of the method under certain regularity conditions. In addition, numerical results of the method compare well with those obtained by other known methods of the literature on the exponential, the normal, the gamma and the Weibull distribution.