分裂可行问题(SFP)的投影算法

本文探讨了分裂可行问题(SFP)的投影算法．我们先定义了(SFP)的反问题(ISFP), 然后利用正交投影性质,给出了(SFP)与(ISFP)及某些投影不动点问题之间的关系．随后, 给出了求解(SFP)的几种正交投影算法,其中包括精确和不精确投影格式．基于变分不等式中投影算法收敛性的证明思路和分裂可行问题的特有形式,证明了这几种算法的收敛性．最后通过几个算例对讨论的方法进行了初步比较．     

A viscosity method with no spectral radius requirements for the split common fixed point problem

This paper is concerned with an algorithmic solution to the split common fixed point problem in Hilbert spaces. Our method can be regarded as a variant of the “viscosity approximation method”. Under very classical assumptions, we establish a strong convergence theorem with regard to involved operators belonging to the wide class of quasi-nonexpansive operators. In contrast with other related processes, our algorithm does not require any estimate of some spectral radius. The technique of analysis developed in this work is new and can be applied to many other fixed point iterations. Numerical experiments are also performed with regard to an inverse heat problem.     

Lipschitz stability in an inverse problem for the Kuramoto–Sivashinsky equation

In this article, we present an inverse problem for the nonlinear 1D Kuramoto–Sivashinsky (KS) equation. More precisely, we study the nonlinear inverse problem of retrieving the anti-diffusion coefficient from the measurements of the solution on a part of the boundary and also at some positive time in the whole space domain. The Lipschitz stability for this inverse problem is our main result and it relies on the Bukhge?m–Klibanov method. The proof is indeed based on a global Carleman estimate for the linearized KS equation.     

Remarks on the Born approximation and the Factorization Method

In this paper, we compare the far-field operators for the full nonlinear inverse scattering problem with the Born approximation as its linearization. The Factorization Method shows that both operators share the same behavior with respect to illposedness of the inverse problem. The results are derived for acoustic and electromagnetic scattering problems and the corresponding problem in electrical impedance tomography.     

一个抛物型方程侧边值问题的正则逼近解在一类Sobolev空间中的最优误差界

逆热传导问题(IHCP)是严重不适定问题,即问题的解(如果存在)不连续依赖于数据．但目前关于逆热传导问题的已有结果主要是针对标准逆热传导问题．文中给出了出现在实际问题中的一个抛物型方程侧边值问题,即一个含有对流项的非标准型逆热传导问题的正则逼近解一类Sobolev空间中的最优误差界．     

Approximation of inverse operators by a new family of high‐order iterative methods

The main goal of this paper is to approximate inverse operators by high‐order Newton‐type methods with the important feature of not using inverse operators. We analyse the semilocal convergence, the speed of convergence, and the efficiency of these methods. We determine that Chebyshev's method is the most efficient method and test it on two problems: one associated to the heat equation and the other one to a boundary value problem. We consider examples with matrices that are close to be singular and/or are badly conditioned. We check the robustness and the stability of the methods by considering situations with many steps and noised data. Copyright © 2013 John Wiley & Sons, Ltd.     

Recovery of the unknown coefficients in a two-dimensional hyperbolic equation

In this paper, an inverse boundary value problem for a two-dimensional hyperbolic equation with overdetermination conditions is studied. To investigate the solvability of the original problem, we first consider an auxiliary inverse boundary value problem and prove its equivalence to the original problem in a certain sense. We then use the Fourier method to reduce such an equivalent problem to a system of integral equations. Furthermore, we prove the existence and uniqueness theorem for the auxiliary problem by the contraction mappings principle. Based on the equivalency of these problems, the existence and uniqueness theorem for the classical solution of the original inverse problem is proved. Some discussions on the numerical solutions for this inverse problem are presented including some numerical examples.     

Inverse spectral problem for integro-differential operators

In this paper, we study the inverse spectral problem on a finite interval for the integro-differential operator ? which is the perturbation of the Sturm-Liouville operator by the Volterra integral operator. The potential q belongs to L ₂[0, π] and the kernel of the integral perturbation is integrable in its domain of definition. We obtain a local solution of the inverse reconstruction problem for the potential q, given the kernel of the integral perturbation, and prove the stability of this solution. For the spectral data we take the spectra of two operators given by the expression for ? and by two pairs of boundary conditions coinciding at one of the finite points.     

Recovery of differential equations with nonlinear dependence on the spectral parameter

The inverse spectral problem of recovering pencils of second-order differential operators on the half-line is studied. We give a formulation of the inverse problem, prove the uniqueness theorem and provided a procedure for constructing the solution of the inverse problem. We also establishe connections with inverse problems for partial differential equations.     

具有未知源的某些非线性伪抛物型方程的反问题

考 虑具有未 知源项的 某些非 线性伪 抛物 型方程 的反演 问题． 首先 将伪抛 物型 方程初 边值问 题化为非线 性发展方 程 Ｃｏｕｃｈ ｙ 问题,然 后,利用半 群理论,论 证发展 方程反问 题解的存 在唯一 性,最后, 利用不 动点方法得到 伪抛物型方程反 问题的可解性     

具有特征参数多项式边界条件的Sturm-Liouville 方程的逆结点问题

逆结点问题是通过特征函数的零点重构算子. 本文主要讨论具有特征参数多项式边界条件的 Sturm-Liouville 方程的逆结点问题. 20世纪50年代以后,人们发现在许多工程领域, Sturm-Liouville 问题的谱参数不仅出现在方程中, 而且也出现在边界条件中,因此带参数边界条件的逆结点问题对数学物理方面的研究有重要意义. 本文讨论区间 $[0,1]$ 上边界条件为参数多项式的 Sturm-Liouville 方程的逆结点问题, 并证明在 $[0,b]$ \big($ b\in \big(\frac{1}{2},1\big]$\big) 上结点的稠密子集可唯一确定 $[0,1]$ 上的势函数和边界条件中多项式的未知系数.     

An inverse problem for Sturm–Liouville differential operators with deviating argument

We consider second-order functional differential operators with a constant delay. Properties of their spectral characteristics are obtained and a nonlinear inverse problem is studied, which consists in recovering the operators from their spectra. We establish the uniqueness and develop a constructive algorithm for solution of the inverse problem.     

Full discretization of a nonlinear parabolic problem containing volterra operators and an unknown dirichlet boundary condition

The reconstruction of an unknown solely time‐dependent Dirichlet boundary condition in a nonlinear parabolic problem containing a linear and a nonlinear Volterra operator is considered. The inverse problem is converted into a variational problem in which the unknown Dirichlet condition is eliminated using a given integral overdetermination. A time‐discrete recurrent approximation scheme is designed, using Backward Euler's method. The convergence of the approximations towards a solution of the variational problem is proved under appropriate assumptions on the data and on the Volterra operators. The uniqueness of this solution is shown in the case that the nonlinear Volterra operator satisfies a particular inequality. Moreover, the Finite Element Method is used to discretize the time‐discrete approximation scheme in space. Finally, full‐discrete error estimates are derived for a particular choice of the finite elements. The corresponding convergence rates are supported by a numerical experiment. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1444–1460, 2015     

Identification of a time-dependent source term for a time fractional diffusion problem

In this paper, we study an inverse problem of identifying a time-dependent term of an unknown source for a time fractional diffusion equation using nonlocal measurement data. Firstly, we establish the conditional stability for this inverse problem. Then two regularization methods are proposed to for reconstructing the time-dependent source term from noisy measurements. The first method is an integral equation method which formulates the inverse source problem into an integral equation of the second kind; and a prior convergence rate of regularized solutions is derived with a suitable choice strategy of regularization parameters. The second method is a standard Tikhonov regularization method and formulates the inverse source problem as a minimizing problem of the Tikhonov functional. Based on the superposition principle and the technique of finite-element interpolation, a numerical scheme is proposed to implement the second regularization method. One- and two-dimensional examples are carried out to verify efficiency and stability of the second regularization method.     

一类Laplace双曲型方程广义Cauchy问题的反问题

本文讨论了确定Ｌａｐｌｉｔｃｅ双曲型方程u_xy(x,y)+a(x,y)u_x(x,y)+b(x,y)+u_y(x,y)+q(x)u(x,y)=f(x,y)的广义Ｃａｕｃｈｙ问题中系数ｑ（ｘ）的反问题。文中利用特征法线及不动点理论，导出了与反问题等价的非线性积分方程组，证明了反问题局部解的存在唯一性，最后给出了反问题整体用的唯一性定理。     

一类非线性色散型发展方程反问题的整体解

将一类非线性色散型发展方程反问题转化为抽象空间非线性发展方程Cauchy问题。利用半群方法和赋等价范数技巧,建立了该类抽象发展方程整体解的存在唯一性定理,并应用于所论反问题,得到了该类非线性色散型发展方程反问题整体解的存在唯一性定理,本质地改进了袁忠信得出的解的局部存在唯一性结果。     

An Inverse Heat Conduction Problem

In this paper, an inverse heat conduction problem will be considered. By reducing this inverse problem and using an overspecified condition, it is shown that the solution to the problem exists, and this solution is unique.AMS Subject Classification (2000): 45D05, 34A55     

Isoenergy Spectral Problem for Multidimensional Difference Operators

In this paper, the direct and inverse isoenergy spectral problems are solved for a class of multidimensional periodic difference operators. It is proved that the inverse spectral problem is solvable in terms of theta functions of curves added to the spectral variety under compactification, and multidimensional analogs of the Veselov–Novikov relations are found.     

一非线性发展方程的反问题

研究一非线性发展方程的未知源项的反演问题.首先,把所考虑的初边值问题化成一等价非线性发展方程的Cauchy问题;然后,利用半群理论,论证反问题解的存在性和唯一性;最后,利用压缩映射不动点方法,得到反问题的可解性.