基于多体作用的原子/连续耦合方法的先验误差估计

本文研究理想晶体发生位错时如何发生形变,应用本地化拟连续方法(QCL)、基于能量的拟连续方法(QCE)、非本地化拟连续方法(QNL),分析了多体作用下Frenkel-Kontorova模型在一维情形中先验误差分析,推导了该误差估计与原子模型解的光滑性的关系,并且由于考虑的是一维原子链,该误差还具备超收敛性.本文将一致性误差分析分解为模型误差和粗粒化误差,并推导出基于负范数的误差估计,稳定性分析将均匀应变扩充为非线性应变.最后利用数值实验说明了本文的分析结果.     

土壤水流与溶质耦合运移问题的混合元-迎风广义差分法研究

本文研究了一维非饱和土壤水流与溶质耦合运移问题的数学模型, 建立了求其数值解的守恒混合元-迎风广义差分格式. 对非线性土壤水分入渗方程, 采用守恒混合元法进行离散模拟, 同时得到了土壤含水量和水分通量; 而对对流-扩散形式的溶质运移方程, 利用迎风的广义差分法离散求解. 且分析了解的存在唯一性, 并讨论了误差估计. 最后给出数值算例, 模拟结果表明利用本文格式来求解非饱和土壤水流与溶质耦合运移问题是可靠的, 且该格式具有稳定性和可实用性.     

Identification of an inverse source problem for time‐fractional diffusion equation with random noise

In this paper, we consider an inverse source problem for a time fractional diffusion equation. In general, this problem is ill posed, therefore we shall construct a regularized solution using the filter regularization method in the random noise case. We will provide appropriate conditions to guarantee the convergence of the approximate solution to the exact solution. Then, we provide examples of filters in order to obtain error estimates for their approximate solutions. Finally, we present a numerical example to show efficiency of the method.     

Busemann functions and equilibrium measures in last passage percolation models

The interplay between two-dimensional percolation growth models and one-dimensional particle processes has been a fruitful source of interesting mathematical phenomena. In this paper we develop a connection between the construction of Busemann functions in the Hammersley last-passage percolation model with i.i.d. random weights, and the existence, ergodicity and uniqueness of equilibrium (or time-invariant) measures for the related (multi-class) interacting fluid system. As we shall see, in the classical Hammersley model, where each point has weight one, this approach brings a new and rather geometrical solution of the longest increasing subsequence problem, as well as a central limit theorem for the Busemann function.     

Existence and uniqueness of stationary solutions to a one-dimensional bipolar hydrodynamic model of semiconductors

We consider a one-dimensional bipolar hydrodynamic model of semiconductors. Although some results exist for the bipolar case, almost their conditions (the boundary condition, the doping profile, etc.) are far from practical application. In the present paper, under a condition appropriate for engineering, we shall prove the existence and the uniqueness of classical solutions for the stationary problem. The most difficult point is to obtain the bounded estimate and the energy estimate.     

Global Weak Solutions of the Cauchy Problem to a Hydrodynamic Model for Semiconductors

In this paper we are concerned with the global existence of weak solutions of the Cauchy problem for a simplified one-dimensional hydrodynamic model for semiconductors. Convergence of approximate solutions derived by the fractional step Lax-Friedrichs scheme is established by using the compensated compactness method.     

Solving Continuous-Time Linear Programming Problems Based on the Piecewise Continuous Functions

The numerical method is proposed in this article to solve a general class of continuous-time linear programming problems in which the functions appeared in the coefficients of this problem are assumed to be piecewise continuous. In order to make sure that all the subintervals of time interval will not contain the discontinuities, a different methodology for not equally partitioning the time interval is proposed. The main issue of this article is to obtain an analytic formula of error upper bound. In this article, we shall propose two kinds of computational procedure to evaluate the error upper bounds. One needs to solve the dual problem of the discretized linear programming problem, and another one does not need to solve the dual problem. Finally, we present a numerical example to demonstrate the usefulness of the numerical method.     

Boundary integral formulations for the forward problem in magnetic induction tomography

In this paper we present two models for the forward problem of magnetic induction tomography. In particular, we describe the eddy current model, and a reduced simplified model. The error between the reduced and the full model is analyzed in dependence of parameters such as the frequency and the conductivity. In the case of a piecewise constant conductivity we derive a boundary integral formulation for the reduced model. Finally, we comment on numerical results for the forward problem and give a comparison of both models. Copyrightcopyright 2011 John Wiley & Sons, Ltd.     

对流扩散方程的QC3无网格法

对流扩散方程作为偏微分运动方程的分支,在流体力学、气体动力学等领域有着重要应用.为解决对流扩散方程难以通过解析法得到解析解的难题,采用二阶一致3点积分(Quadratically Consistent 3-Point Integration,简称QC3)提高无网格法的计算效率,通过对积分点上形函数导数的修正,改善无网格...     

Eigenfrequencies of a tube bundle immersed in a fluid

In this paper we study a simplified version of a mathematical model that describes the eigenfrequencies and eigenmotions of a coupled system consisting of a set of tubes (or a tube bundle) immersed in an incompressible perfect fluid. The fluid is assumed to be contained in a rectangular cavity, and the tubes are assumed to be identical, and periodically distributed in the cavity. The mathematical model that governs this physical problem is an elliptic differential eigenvalue problem consisting of the Laplace equation with a nonlocal boundary condition on the holes, and a homogeneous Neumann boundary condition on the boundary of the cavity. In the simplified model that we study in this paper, the Neumann condition is replaced by a periodic boundary condition. Our goal in studying this simple version is to derive some basic properties of the problem that could serve as a guide to envisage similar properties for the original model. In practical situations, this kind of problem needs to be solved for tube bundles containing a very large number of tubes. Then the numerical analysis of these problems is in practice very expensive. Several approaches to overcome this difficulty have been proposed in the last years using homogenization techniques. Alternatively, we propose in this paper an approach that consists in obtaining an explicit decomposition of the problem into a finite family of subproblems, which can be easily solved numerically. Our study is based on a generalized notion of periodic function, and on a decomposition theorem for periodic functions that we introduce in the paper. Our results rely on the theory of almost periodic functions, and they provide a simple numerical method for obtaining approximations of all the eigenvalues of the problem for any number of tubes in the cavity. We also discuss a numerical example.     

Approximation of differential eigenvalue problems

To solve a one-dimensional second-order differential eigenvalue problem, we use the finite-element method with numerical integration. We analyze the influence of the quadrature formulas used on the error in the approximate eigenvalues and eigenelements. Theoretical results are illustrated by experiments for a model problem.     

Error analysis of the numerical solution of split differential equations

The operator splitting method is a widely used approach for solving partial differential equations describing physical processes. Its application usually requires the use of certain numerical methods in order to solve the different split sub-problems. The error analysis of such a numerical approach is a complex task. In the present paper we show that an interaction error appears in the numerical solution when an operator splitting procedure is applied together with a lower-order numerical method. The effect of the interaction error is investigated by an analytical study and by numerical experiments made for a test problem.     

Interaction Model and Model Selection for Function-on-Function Regression

Regression models with interaction effects have been widely used in multivariate analysis to improve model flexibility and prediction accuracy. In functional data analysis, however, due to the challenges of estimating three-dimensional coefficient functions, interaction effects have not been considered for function-on-function linear regression. In this article, we propose function-on-function regression models with interaction and quadratic effects. For a model with specified main and interaction effects, we propose an efficient estimation method that enjoys a minimum prediction error property and has good predictive performance in practice. Moreover, converting the estimation of three-dimensional coefficient functions of the interaction effects to the estimation of two- and one-dimensional functions separately, our method is computationally efficient. We also propose adaptive penalties to account for varying magnitudes and roughness levels of coefficient functions. In practice, the forms of the models are usually unspecified. We propose a stepwise procedure for model selection based on a predictive criterion. This method is implemented in our R package FRegSigComp. Supplemental materials are available online.     

A posteriori error indicators for Maxwell''s equations

In this paper we shall analyze a class of a posteriori error indicators for an electromagnetic scattering problem for Maxwell's equations in the presence of a bounded, inhomogeneous and anisotropic scatterer. Problems of this type arise when computing the interaction of electromagnetic radiation with biological tissue. We briefly recall existence and uniqueness theory associated with this problem. Then we show how a posteriori error indicators can be derived using an adjoint equation approach. The error indicators use both the jump in normal and tangential components of the field across faces in the mesh.     

Error analysis for a potential problem on locally refined grids

Summary. For the simulation of biomolecular systems in an aqueous solvent a continuum model is often used for the solvent. The accurate evaluation of the so-called solvation energy coming from the electrostatic interaction between the solute and the surrounding water molecules is the main issue in this paper. In these simulations, we deal with a potential problem with jumping coefficients and with a known boundary condition at infinity. One of the advanced ways to solve the problem is to use a multigrid method on locally refined grids around the solute molecule. In this paper, we focus on the error analysis of the numerical solution and the numerical solvation energy obtained on the locally refined grids. Based on a rigorous error analysis via a discrete approximation of the Greens function, we show how to construct the composite grid, to discretize the discontinuity of the diffusion coefficient and to interpolate the solutions at interfaces between the fine and coarse grids. The error analysis developed is confirmed by numerical experiments. Received June 25, 1998 / Revised version received July 14, 1999 / Published online June 8, 2000     

Finite Element Analysis for a Class of System of Nonlinear and Non-Self-Adjoint Schrödinger Euqations

Because the nonlinear Schrödinger equation is met in many physical problems and is applied widely, The research on well-posedness for its solution and numerical methods has aroused more and more interest. The self-adjoint case has been considered by many authors. For a class of system of nonlinear and non-self-adjoint Schrodinger equations which refers to excitons occurring in one dimensional molecular crystals and in a spiral biomolecules, Guo Boling studied in [6], the pure initial and periodic initial value problems of this system and obtained the existence and uniqueness of its solution. In [7] we discuss the difference solution of this system and obtained its error estimate. In this paper, we shall study the finite element method for the periodic initial value problem of this system. Just as Guo pointed out in [6], since it has a non-self-adjoint term, it not only brings about trouble in mathematics, but also creates more difficulty in numerical analysis. Our analysis will show that for this system, in theoretically we can obtain the same results as when it has no non-self-adjoint term.     

A bilinear approach to the pooling problem†

In this paper we present an algorithm for the pooling problem in refinery optimization based on a bilinear programming approach. The pooling problem occurs frequently in process optimization problems, especially refinery planning models. The main difficulty is that pooling causes an inherent nonlinearity in the otherwise linear models. We shall define the problem by formulating an aggregate mathematical model of a refinery, comment on solution methods for pooling problems that have been presented in the literature, and develop a new method based on convex approximations of the bilinear terms. The method is illustrated on numerical examples     

NUMERICAL APPROXIMATION OF THE SMOLUCHOWSKI EQUATION USING RADIAL BASIS FUNCTIONS

The goal of this paper is to present a numerical method for the Smoluchowski equation,a drift-diffusion equation on the sphere,arising in the modelling of particle dynamics.The numerical method uses radial basis functions(RBF).This is a relatively new approach,which has recently mainly been used for geophysical applications.For a simplified model problem we compare the RBF approach with a spectral method,i.e.the standard approach used in related physical applications.This comparison as well as our other accuracy studies show that RBF methods are an attractive alternative for these kind of models.     

Helmholtz问题的Robin型区域分解法

Helmholtz问题的数值模拟在科学工程计算领域有着广泛的应用,快速高效求解Helmholtz方程离散代数系统一直是科学计算的重要研究方向.本文简要回顾了Helmholtz方程的区域分解型求解器的发展历程,重点介绍了我们提出的Robin型区域分解算法,同时比较了各类算法的优劣和特点.近年来Helmholtz方程的求解效率有了极大的提升,然而仍有一些本质困难尚待突破,如何高效求解Helmholtz方程,仍是具有挑战意义的研究课题.     

One-step methods of any order for ordinary differential equations with discontinuous right-hand sides

Summary The right-hand sides of a system of ordinary differential equations may be discontinuous on a certain surface. If a trajectory crossing this surface shall be computed by a one-step method, then a particular numerical analysis is necessary in a neighbourhood of the point of intersection. Such an analysis is presented in this paper. It shows that one can obtain any desired order of convergence if the method has an adequate order of consistency. Moreover, an asymptotic error theory is developed to justify Richardson extrapolation. A general one-step method is constructed satisfying the conditions of the preceding theory. Finally, a simplified Newton iteration scheme is used to implement this method.