Galerkin Method for the Scattering Problem of a Slit

In this paper, the scattering problem of a slit is considered. By use of the exact Dirichlet-to-Neumann map and the continuity of the total field on the slit, we reformulate this problem to an operator equation. The well-posedness of the solution is proved. We apply Galerkin method to solve this operator equation numerically and prove the convergence of the numerical solution. Finally some computational examples are presented to show the effectiveness of our method.     

Numerical approximation of a time-fractional Black–Scholes equation

In this paper a time-fractional Black–Scholes equation is examined. We transform the initial value problem into an equivalent integral–differential equation with a weakly singular kernel and use an integral discretization scheme on an adapted mesh for the time discretization. A rigorous analysis about the convergence of the time discretization scheme is given by taking account of the possibly singular behavior of the exact solution and first-order convergence with respect to the time variable is proved. For overcoming the possibly nonphysical oscillation in the computed solution caused by the degeneracy of the Black–Scholes differential operator, we employ a central difference scheme on a piecewise uniform mesh for the spatial discretization. It is proved that the scheme is stable and second-order convergent with respect to the spatial variable. Numerical experiments support these theoretical results.     

一维热传导问题时变边界上热通量重构问题

具有Neumann边界条件的抛物方程的初边值问题是偏微分方程研究领域的一类经典问题.正问题是由已知的边界条件和初始条件来求区域温度场的问题.如果边界条件不足,但给出了区域内部的一些额外信息,这样便构成了一类热通量重构的反问题.本文讨论了一维热传导问题时动边界上的热通量重构问题,借助于位势理论方法,引入密度函数,将反问题本质上转化为一类关于密度函数的具有弱奇性核的第一类Volterra积分方程,采用了Tikhonov正则化,在正则化参数的选取上采用了后验的模型函数方法,数值结果验证了反演方法的有效性.     

A numerical study of variable coefficient elliptic Cauchy problem via projection method

In this paper, we investigate a numerical method for the solution of an inverse problem of recovering lacking data on some part of the boundary of a domain from the Cauchy data on other part for a variable coefficient elliptic Cauchy problem. In the process, the Cauchy problem is transformed into the problem of solving a compact linear operator equation. As a remedy to the ill-posedness of the problem, we use a projection method which allows regularization solely by discretization. The discretization level plays the role of regularization parameter in the case of projection method. The balancing principle is used for the choice of an appropriate discretization level. Several numerical examples show that the method produces a stable good approximate solution.     

全变差正则化在抛物型方程初始条件反问题的应用

当反问题反演的函数不连续时,一般的正则化算法反演效果不令人满意,用全变差正则化方法对抛物型方程初始条件反问题进行求解,并进行了数值分析和数值模拟,结果显示数值解与真解吻合较好,表明该方法对于不连续函数求解具有高效、稳定等优点.     

Finite element approximation of regularized solutions of the inverse potential problem of electrocardiography and applications to experimental data

Summary This paper investigates the problem of estimating the electric potential distribution in proximity of the heart from potential data given on the body and is here reformulated as a control problem in terms of a ?transfer? operator and stabilized by means of a suitable regularization operator. The numerical approximation by means of the finite element method of the regularized problem is investigated; convergence results and error estimates are established. The numerical procedures for the computation of the approximate ?transfer? operator and for the solution of the least square problem approximating the regularized control problem are described. The performance of several criteria for the best choice of the regularization parameter is analyzed. Finally the numerical inverse procedure, extensively tested using test functions and data fromin vitro experiments, is applied to human data. Dedicated to Professor S. Faedo on his 70th birthday This work was supported through grants by MPI 40%, the Special Program on Bioengineering of CNR and IAN of CNR.     

求解TSP问题的一种改进的遗传算法

TSP问题是典型的NP完全问题,遗传算法是求解NP完全问题的一种理想方法。文章针对解决TSP问题,提出使用改进的遗传算法,即用浓度控制选择策略以保证群体的多样性,用贪婪交叉算子和启发式倒位变异算子来提高算法的收敛速度,较好地解决了群体的多样性和收敛速度的矛盾。算法的分析和测试表明,该文算法的改进是有效的。     

On a non-smooth version of Newton's method based on Hölderian assumptions

In this study we are concerned with the problem of approximating a locally unique solution of an operator equation in Banach space using Newton's method. The differentiability of the operator involved is not assumed, and we use Hölderian assumptions. We provide a semilocal convergence analysis based on a suitable point-based approximation. Our approach is justified by numerical examples where our results apply, whereas earlier ones cannot be applied in this way.     

Convexification of a 3-D coefficient inverse scattering problem

A version of the so-called “convexification” numerical method for a coefficient inverse scattering problem for the 3D Helmholtz equation is developed analytically and tested numerically. Backscattering data are used, which result from a single direction of the propagation of the incident plane wave on an interval of frequencies. The method converges globally. The idea is to construct a weighted Tikhonov-like functional. The key element of this functional is the presence of the so-called Carleman Weight Function (CWF). This is the function which is involved in the Carleman estimate for the Laplace operator. This functional is strictly convex on any appropriate ball in a Hilbert space for an appropriate choice of the parameters of the CWF. Thus, both the absence of local minima and convergence of minimizers to the exact solution are guaranteed. Numerical tests demonstrate a good performance of the resulting algorithm. Unlikeprevious the so-called tail functions globally convergent method, we neither do not impose the smallness assumption of the interval of wavenumbers, nor we do not iterate with respect to the so-called tail functions.     

Constrained optimal control of Navier–Stokes flow by semismooth Newton methods

We propose and analyze a semismooth Newton-type method for the solution of a pointwise constrained optimal control problem governed by the time-dependent incompressible Navier–Stokes equations. The method is based on a reformulation of the optimality system as an equivalent nonsmooth operator equation. We analyze the flow control problem and prove q-superlinear convergence of the method. In the numerical implementation, adjoint techniques are combined with a truncated conjugate gradient method. Numerical results are presented that support our theoretical results and confirm the viability of the approach.     

The backward problem for a time-fractional diffusion-wave equation in a bounded domain

This paper is devoted to solve the backward problem for a time-fractional diffusion-wave equation in a bounded domain. Based on the series expression of the solution for the direct problem, the backward problem for searching the initial data is converted into solving the Fredholm integral equation of the first kind. The existence, uniqueness and conditional stability for the backward problem are investigated. We use the Tikhonov regularization method to deal with the integral equation and obtain the series expression of the regularized solution for the backward problem. Furthermore, the convergence rate for the regularized solution can be proved by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Numerical results for five examples in one-dimensional case and two-dimensional case show that the proposed method is efficient and stable.     

求解0-1背包问题的二进制狮群算法

针对传统二进制群智能算法求解0-1背包问题易陷入局部最优、收敛速度慢的缺点,提出一种新的解决离散空间问题的二进制狮群算法BLSO。二进制狮群算法对狮王、母狮和幼狮的位置重新定义,引入反置运算、移动算子和学习算子建立全新的位置转移方式和局部搜索规则;加入贪心策略进行解的可行化处理和充分利用,增强局部搜索能力,进一步提高收敛速度。对9个典型的0-1背包算例进行仿真实验,实验结果表明,该算法不仅可以有效求解0-1背包问题,而且还能够以较快的速度搜索到精度较高的次优解甚至全局最优解,具有较好的稳定性;同时,对高维背包问题的求解与参考算法相比,在寻优时间和精度上更具优势。     

区间粒子算法与线源反问题求解

目前国内外对线源反问题数值求解尚没有一种成熟有效的算法。本文在研究区间搜索算法基础上,提出了一种新的求解算法—区间粒子算法(Range Particle Algorithm)来求解线源反问题。首先简要介绍了线源反问题的求解特点,并根据线源方程建立了反问题求解的目标函数;其次基于该目标函数,设计了区间粒子算法来求解,探讨了算法实现的基本步骤和参数调整问题;最后通过模拟数据和实测数据分别检验了该算法求解的效果,结果表明区间粒子算法求解精度高、收敛速度快和计算稳定,在线源反问题数值求解中是适用的。     

Structured matrix algorithms for inverse scattering on the line

Abstract In this article the Marchenko integral equations leading to the solution of the inverse scattering problem for the 1-D Schr?dinger equation on the line are solved numerically. The linear system obtained by discretization has a structured matrix which allows one to apply FFT based techniques to solve the inverse scattering problem with minimal computational complexity. The numerical results agree with exact solutions when available. A proof of the convergence of the discretization scheme is given. Keywords Structured matrix systems, 1-D inverse scattering, Marchenko integral equation     

基于遗传算法的机械臂逆运动学问题解决方案

提出应用遗传算法求解机械臂的逆运动学问题,将种群定义于机械臂的关节角轨迹层面,利用连续性函数实现算法的初始化算子,交叉算子和变异算子。算法仅使用表现型数据表示方式,克服了传统遗传算法在数据的基因型和表现型之间频繁地进行编码和解码操作。通过和传统遗传算法进行对比分析,验证了所提出的方法能够避免传统遗传算法求解逆运动学问题时存在的多重切换点现象,能够获得更平滑的关节角轨迹,缩短了算法的收敛时间,生成的笛卡尔轨迹具有更高的精度。     

基于改进混合蜂群算法的非线性电路谐波平衡分析

针对射频电路非线性分析中谐波平衡方程求解问题,提出一种基于高斯扰动、锦标赛选择策略以及拟牛顿局部寻优算子的改进混合蜂群算法,该算法在搜索方程中引入基于当前全局最优解的高斯扰动,能有效防止算法陷入局部最优并加快算法收敛;跟随蜂采用锦标赛选择策略在一定程度上避免了算法的早熟现象;采用拟牛顿算子进行局部寻优,可使算法快速收敛。实验结果表明,改进混合蜂群算法成功应用于谐波平衡方程求解,与其他求解算法对比,收敛时间较短,性能较优。     

Propagation of non-linear waves in a non-ideal relaxing gas

We have derived an evolution equation governing the far-field behaviour of small amplitude waves in a non-ideal relaxing gas for planar and converging flow. Asymptotic expansions of the flow variables for small amplitude waves have been used to derive the evolution equation. This equation turns out to be a generalized Burger's equation. The numerical solution of this equation is obtained by using the homotopy analysis method (HAM) proposed by Liao with two different initial conditions. Using the HAM, we have studied the effect of relaxation and nonlinearity. The convergence control parameter enables us to find a good approximate solution for such a complex flow problem. This method also confirms the capabilities and usefulness of convergence control parameter and HAM for complex and highly non-linear problems.     

一种求解图着色问题的优化组合遗传算法①

图着色算法是一种典型的NP-完全问题。在逆序算子、对偶算子和矩阵遗传算子的性能研究基础上,采用自然数与二进制相互转换的编码方案,应用图着色问题的约束条件建立适应度评价函数,将具有良好局部搜索性能的矩阵遗传算子与具有良好局部搜索性能的逆序与对偶组合算子优化组合应用,构造了一种用于求解图着色问题的优化组合遗传算法,保证了算法的全局收敛性。与基本遗传算法相比较,实验结果表明,该算法对图着色问题有较好的求解性能。     

Asymptotic output tracking for a class of semilinear parabolic equations: A semianalytical approach

This paper addresses the problem of asymptotic output tracking of a class of semilinear parabolic equations with pointwise in‐domain actuation. First, the assessment of the well‐posedness of the considered systems is performed, and then, the stability of boundary controlled systems is analyzed via Chaffee‐Infante equation and Fisher's equation. The application of the zero dynamics inverse design results in a dynamic control scheme that is implemented by using the technique of trajectory planning for flat systems and the Adomian decomposition method. The convergence of the solution of the original systems to that of the corresponding zero dynamics and the convergence of the solution expressed by an Adomian series are also analyzed. Numerical simulations are carried out to illustrate the effectiveness of the developed approach.