Source Identification for the Time-Fractional Diffusion Equation With Robin Boundary Conditions北大核心CSCD

对Robin边界条件时间分数阶扩散方程的源项辨识问题进行了研究。这类问题是不适定的,因此提出了一种迭代型正则化方法,得到了源项辨识问题的正则近似解。给出了先验和后验参数选取规则下正则近似解和精确解之间的误差估计,数值算例验证了该方法的有效性。     

二维Helmholtz方程不适定问题的一种算子软化正则法

本文研究带有混合边界的二维Helmholtz方程不适定问题.为了获得稳定的数值解，利用基于de la ValléePoussin算子的软化正则方法，得到了正则近似解，给出正则近似解与精确解之间在先验参数选取规则之下的误差估计，并通过数值实验检验了数据有噪声扰动时方法的有效性和稳定性.     

一种新的正则化方法求解热传导方程的侧边值问题

考虑了四分之一平面内的热传导方程的侧边值问题,这类问题是严重不适定的.采用传统拟逆方法得到该问题的一个近似解,但发现它并不是一个正则化解.有趣的是,对解的分母项加以修正便可以得到侧边值问题的一个正则化解,进而提出了一种新的正则化方法,并分别给出先验和后验两种正则化参数选取规则下的Hlder型误差估计.数值实验验证了所提方法的可行性和有效性.     

一类非线性反向热传导问题的Fourier正则化方法

考虑了非线性抛物方程反向热传导问题,这类问题是不适定的,即问题的解不连续依赖于测量数据.利用Fourier截断正则化方法恢复其不适定性,得到问题的一个正则近似解,并且给出正则解和精确解之间具有Hlder型的误差估计.     

Poisson方程热源识别反问题

探讨了半带状区域上二维Poisson方程只含有一个空间变量的热源识别反问题.这类问题是不适定的,即问题的解(如果存在的话)不连续依赖于测量数据.利用Carasso-Tikhonov正则化方法,得到了问题的一个正则近似解,并且给出了正则解和精确解之间具有Holder型误差估计.数值实验表明Carasso-Tikhonov正则化方法对于这种热源识别是非常有效的.     

求解奇异摄动抛物型对流扩散问题的再生核方法（英文）

基于域分解方法和再生核方法,文章提出了一种求解一维奇异摄动抛物型对流扩散问题的数值方法.原问题被分解成边界层区域问题和正则区域问题,正则区域问题的近似解通过原问题对应的退化问题的解进行近似,边界层区域问题的近似解通过构造合适的再生核,并利用再生核理论给出.三个数值算例的实验结果表明所提出的数值方法是有效的.     

修正的Helmholtz方程未知源识别的Fourier截断正则化方法

探讨半无界区域上二维修正的Helmholtz方程只含有一个空间变量的未知源识别反问题.这类问题是不适定的,即问题的解(如果存在的话)不连续依赖于测量数据.利用Fourier截断正则化方法,得到问题的一个正则近似解,并且给出正则解和精确解之间收敛的误差估计.数值例子表明Fourier截断正则化方法对于这种未知源识别非常有效.     

非线性不适定问题的正则化方法

本文考虑非线性不适定问题Tx=y的近似求解,利用Тихоноь正则化方法来逼近问题的x-极小模解,当算子和右端都近似已知时,给出一种决定正则化参数的方法,并给出正则解的收效性和渐近收敛阶估计。     

Variational Regularization of the Inverse Problem of a Class of Nonlinear Time-Fractional Diffusion Equations北大核心CSCD

考虑了一类二维非线性时间分数阶扩散方程，并从最终位置获取的测量数据来反演物质在u(0, y, t)处的物理信息。这个问题是严重不适定的，即问题的解并不连续依赖于测量数据，因此提出了变分型正则化方法来稳定求解该问题。给出了精确解与正则近似解之间的误差估计，数值算例验证了该方法的有效性。     

求解隐式迭代方程的多尺度配置法

用多尺度快速配置法求解病态积分方程的隐式迭代方程.在积分算子是扇形紧算子时,该方法得到了离散隐式迭代方程的近似解.采用Morozov偏差原理作为停止准则,并证明了在该准则下隐式迭代正则化方法所得近似解的收敛率.最后,用数值实验证实理论结果和说明数值方法的有效性.     

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.     

Optimal control for systems described by hyperbolic equations with strong nonlinearity

An optimization control problem for a hyperbolic equation is considered. The system is nonlinear with respect to the state derivative. The regularization technique for the state equation is applied. The necessary conditions of optimality for the regularized control problem are proved. It uses the extended differentiability of the control-state mapping for the regularized equation. The convergence of the regularization method is proved. Thus the optimal control for the regularized problem with a small enough regularization parameter can be chosen as an approximate solution of the initial optimization problem.     

Approximate Solution of Singular Optimization Problems

We study the optimal control problem for systems described by nonlinear elliptic equations. We have no information about the existence and uniqueness of the solution for some particular control. The extremum problem may be unsolvable. We regularize the problem by using a combination of the penalty method and the Tikhonov method. For the regularized problem, we prove the existence of the solution and find necessary conditions for optimality in the form of variational inequalities. We show that the regularization method used in this paper allows one to find an approximate (in some sense) solution of the original problem.     

Identifying an unknown source term in a spherically symmetric parabolic equation

The aim of this work is to solve the inverse problem of determining an unknown source term in a spherically symmetric parabolic equation. The problem is ill-posed: the solution (if it exists) does not depend continuously on the final data. A spectral method is applied to formulate a regularized solution, and a Hölder type estimate of the error between the approximate solution and the exact solution is obtained with a suitable choice of regularization parameter.     

Approximate Solution of Optimization Problems for Infinite-Dimensional Singular Systems

An optimal control problem is considered for a system described by a singular equation of parabolic type. The study bases on a special regularization method. We establish existence of a solution to the regularized problem, as well as the corresponding necessary optimality conditions. The results enable us to find an approximate solution to the original problem even in the absence of solvability.     

A modified quasi-boundary value method for solving the radially symmetric inverse heat conduction problem

In this paper, we consider a radially symmetric inverse heat conduction problem of determining the surface heat flux distribution from a fixed location inside a cylinder. This problem is ill-posed in the Hadamard sense and a conditional stability estimate is given for it. A modified quasi-boundary value regularization method is applied to formulate a regularized solution, and a sharp error estimate between the approximate solution and the exact solution is established by choosing a suitable regularization parameter. A numerical example is presented to verify the efficiency of the regularization method.     

A modified Tikhonov regularization method for an axisymmetric backward heat equation

This paper deals with the inverse time problem for an axisymmetric heat equation. The problem is ill-posed. A modified Tikhonov regularization method is applied to formulate regularized solution which is stably convergent to the exact one. estimate between the approximate solution and exact technical inequality and improving a priori smoothness Meanwhile, a logarithmic-HSlder type error solution is obtained by introducing a rather assumption.     

A regularized projection method for complementarity problems with non-Lipschitzian functions

We consider complementarity problems involving functions which are not Lipschitz continuous at the origin. Such problems arise from the numerical solution for differential equations with non-Lipschitzian continuity, e.g. reaction and diffusion problems. We propose a regularized projection method to find an approximate solution with an estimation of the error for the non-Lipschitzian complementarity problems. We prove that the projection method globally and linearly converges to a solution of a regularized problem with any regularization parameter. Moreover, we give error bounds for a computed solution of the non-Lipschitzian problem. Numerical examples are presented to demonstrate the efficiency of the method and error bounds.

     

Reconstruction of controls and parameters by the Tikhonov method with non-smooth stabilizers

We consider the problem of the reconstruction of an a priori unknown control in a dynamic system based on approximate a posteriori observations of the motion of this system. We propose to solve this problem by the Tikhonov method with a stabilizer which contains the total variation of the control. This provides the piecewise uniform convergence of regularized approximations and thus enables one to numerically reconstruct the fine structure of the desired solution.