Duality-based regularization in a linear convex mathematical programming problem

For a linear convex mathematical programming (MP) problem with equality and inequality constraints in a Hilbert space, a dual-type algorithm is constructed that is stable with respect to input data errors. In the algorithm, the dual of the original optimization problem is solved directly on the basis of Tikhonov regularization. It is shown that the necessary optimality conditions in the original MP problem are derived in a natural manner by using dual regularization in conjunction with the constructive generation of a minimizing sequence. An iterative regularization of the dual algorithm is considered. A stopping rule for the iteration process is presented in the case of a finite fixed error in the input data.     

SOURCE TERM IDENTIFICATION WITH DISCONTINUOUS DUAL RECIPROCITY APPROXIMATION AND QUASI-NEWTON METHOD FROM BOUNDARY OBSERVATIONS

This paper deals with discontinuous dual reciprocity boundary element method for solving an inverse source problem.The aim of this work is to determine the source term in elliptic equations for nonhomogenous anisotropic media,where some additional boundary measurements are required.An equivalent formulation to the primary inverse problem is established based on the minimization of a functional cost,where a regularization term is employed to eliminate the oscillations of the noisy data.Moreover,an efficient algorithm is presented and tested for some numerical examples.     

Solving an Inverse Problem for an Elliptic Equation by d.c. Programming

An inverse problem of determination of a coefficient in an elliptic equation is considered. This problem is ill-posed in the sense of Hadamard and Tikhonov's regularization method is used for solving it in a stable way. This method requires globally solving nonconvex optimization problems, the solution methods for which have been very little studied in the inverse problems community. It is proved that the objective function of the corresponding optimization problem for our inverse problem can be represented as the difference of two convex functions (d.c. functions), and the difference of convex functions algorithm (DCA) in combination with a branch-and-bound technique can be used to globally solve it. Numerical examples are presented which show the efficiency of the method.     

对流弥散方程的时间依赖反应系数反演及应用

探讨了一维对流弥散方程的时间依赖反应系数函数的反演问题及其在一个土柱渗流试验中的应用.借助一个积分恒等式,讨论了正问题单调解的存在条件及反问题的数据相容性.进一步考虑一个扰动土柱试验模型模拟问题,应用一种最佳摄动量正则化算法,对反应系数函数进行了数值反演模拟,并应用于实际试验数据的反分析,反演重建结果不仅与相容性分析一致,而且与实际观测数据基本吻合.     

Iterative proximal regularization method for finding a saddle point in the semicoercive Signorini problem

An algorithm for seeking a saddle point for the semicoercive variational Signorini inequality is studied. The algorithm is based on an iterative proximal regularization of a modified Lagrangian functional.     

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.     

空间-时间分数阶变系数对流扩散方程微分阶数的数值反演

考虑终值数据条件下一维空间-时间分数阶变系数对流扩散方程中同时确定空间微分阶数与时间微分阶数的反问题.基于对空间-时间分数阶导数的离散,给出求解正问题的一个隐式差分格式,通过对系数矩阵谱半径的估计,证明差分格式的无条件稳定性和收敛性.联合最佳摄动量算法和同伦方法引入同伦正则化算法,应用一种单调下降的Sigmoid型传输函数作为同伦参数,对所提微分阶数反问题进行精确数据与扰动数据情形下的数值反演.结果表明同伦正则化算法对于空间-时问分数阶反常扩散的参数反演问题是有效的.     

Regularization of backward heat conduction problem

We study the backward heat conduction problem in an unbounded region. The problem is ill-posed, in the sense that the solution if it exists, does not depend continuously on the data. Continuous dependence of the data is restored by cutting-off high frequencies in Fourier domain. The cut-off parameter acts as a regularization parameter. The discrepancy principle, for choosing the regularization parameter and double exponential transformation methods for numerical implementation of regularization method have been used. An example is presented to illustrate applicability and accuracy of the proposed method.     

Recovering a space-dependent source term in a time-fractional diffusion wave equation

This work is concerned with identifying a space-dependent source function from noisy final time measured data in a time-fractional diffusion wave equation by a variational regularization approach. We provide a regularity of direct problem as well as the existence and uniqueness of adjoint problem. The uniqueness of the inverse source problem is discussed. Using the Tikhonov regularization method, the inverse source problem is formulated into a variational problem and a conjugate gradient algorithm is proposed to solve it. The efficiency and robust of the proposed method are supported by some numerical experiments.     

A control type method for solving the Cauchy–Stokes problem

In this work, we propose an approximate optimal control formulation of the Cauchy problem for the Stokes system. Here the problem is converted into an optimization one. In order to handle the instability of the solution of this ill-posed problem, a regularization technique is developed. We add a term in the least square function which happens to vanish while the algorithm converges. The efficiency of the proposed method is illustrated by numerical experiments.     

Regularized dual method for nonlinear mathematical programming

For a nonlinear programming problem with equality constraints in a Hilbert space, a dual-type algorithm is constructed that is stable with respect to input data errors. The algorithm is based on a modified dual of the original problem that is solved directly by applying Tikhonov regularization. The algorithm is designed to determine a norm-bounded minimizing sequence of feasible elements. An iterative regularization of the dual algorithm is considered. A stopping rule for the iteration process is given in the case of a finite fixed error in the input data.     

The Kirsch-Kress method for inverse scattering by infinite locally rough interfaces

This paper is concerned with the inverse problem of reconstructing an infinite, locally rough interface from the scattered field measured on line segments above and below the interface in two dimensions. We extend the Kirsch-Kress method originally developed for inverse obstacle scattering problems to the above inverse transmission problem with unbounded interfaces. To this end, we reformulate our inverse problem as a nonlinear optimization problem with a Tikhonov regularization term. We prove the convergence of the optimization problem when the regularization parameter tends to zero. Finally, numerical experiments are carried out to show the validity of the inversion algorithm.     

非线性二维导热反问题的混沌-正则化混合解法

考虑热传导系数随温度变化,建立了非线性二维稳态导热反问题数值计算模型。并把混沌优化方法和梯度正则化方法相结合,构成一种混沌-正则化混合算法求该计算模型的全局解。以热传导系数随温度线性变化为例,由布置在结构边界上的观测点温度信息确定了结构材料热传导系数及其随温度变化规律。结果表明混合算法计算结果与初值无关,具有很好的全局寻优性能,而且计算量远比经典遗传算法和单纯采用混沌优化方法小。     

The Levenberg-Marquardt method applied to a parameter estimation problem arising from electrical resistivity tomography

An efficient and robust electrical resistivity tomographic inversion algorithm based on the Levenberg-Marquardt method is considered to obtain quantities like grain size, spatial scale and particle size distribution of mineralized rocks. The corresponding model in two dimensions is based on the Maxwell equations and leads to a partial differential equation with mixed Dirichlet-Neumann boundary conditions. The forward problem is solved numerically with the finite-difference method. However, the inverse problem at hand is a classical nonlinear and ill-posed parameter estimation problem. Linearizing and applying the Tikhonov regularization method yields an iterative scheme, the Levenberg-Marquardt method. Several large systems of equations have to be solved efficiently in each iteration step which is accomplished by the conjugate gradient method without setting up the corresponding matrix. Instead fast matrix-vector multiplications are performed directly. Therefore, the derivative and its adjoint for the parameter-to-solution map are needed. Numerical results demonstrate the performance of our method as well as the possibility to reconstruct some of the desired parameters.     

On a radially symmetric inverse heat conduction problem

In this paper, we treat an inverse problem for a radially symmetric heat equation, which arises from non-destructive evaluation by thermal imaging. The problem can also be considered as an inverse heat conduction problem. Based on a weighted energy method, we give a conditional stability estimate. A feasible regularization method is provided for numerical simulation. The reconstruction experiment is done for verifying the efficiency of the regularization method.     

Convergence analysis of Tikhonov-type regularization algorithms for multiobjective optimization problems

In this paper, we introduce a vector-valued Tikhonov-type regularization algorithm for an extended-valued multiobjective optimization problem. Under some mild conditions, we prove that any sequence generated by this algorithm converges to a weak Pareto optimal solution of the multiobjective optimization problem. Our results improve and generalize some known results.     

LOCAL REGULARIZATION METHODS FOR THE STABILIZATION OF LINEAR ILL-POSED EQUATIONS OF VOLTERRA TYPE

Inverse problems based on first-kind Volterra integral equations appear naturally in the study of many applications, from geophysical problems to the inverse heat conduction problem. The ill-posedness of such problems means that a regularization technique is required, but classical regularization schemes like Tikhonov regularization destroy the causal nature of the underlying Volterra problem and, in general, can produce oversmoothed results. In this paper we investigate a class of local regularization methods in which the original (unstable) problem is approximated by a parameterized family of well-posed, second-kind Volterra equations. Being Volterra, these approximating second-kind equations retain the causality of the original problem and allow for quick sequential solution techniques. In addition, the regularizing method we develop is based on the use of a regularization parameter which is a function (rather than a single constant), allowing for more or less smoothing at localized points in the domain. We take this approach even further by adopting the flexibility of an additional penalty term (with variable penalty function) and illustrate the sequential selection of the penalty function in a numerical example.     

Modified Tikhonov regularization method for the Cauchy problem of the Helmholtz equation

In this paper, the Cauchy problem for the Helmholtz equation is investigated. By Green’s formulation, the problem can be transformed into a moment problem. Then we propose a modified Tikhonov regularization algorithm for obtaining an approximate solution to the Neumann data on the unspecified boundary. Error estimation and convergence analysis have been given. Finally, we present numerical results for several examples and show the effectiveness of the proposed method.     

Numerical solution of the linear inverse problem for the Euler-Darboux equation

An inverse problem of the reconstruction of the right-hand side of the Euler-Darboux equation is studied. This problem is equivalent to the Volterra integral equation of the third kind with the operator of multiplication by a smooth nonincreasing function. Numerical solution of this problem is constructed using an integral representation of the solution of the inverse problem, the regularization method, and the method of quadratures. The convergence and stability of the numerical method is proved.     

An inverse vibration problem in estimating the spatial and temporal-dependent external forces for cutting tools

An inverse forced vibration problem, based on the conjugate gradient method (CGM), (or the iterative regularization method), is examined in this study to estimate the unknown spatial and temporal-dependent external forces for the cutting tools by utilizing the simulated beam displacement measurements. The tool is represented by an Euler–Bernoulli beam. The accuracy of the inverse analysis is examined by using the simulated exact and inexact displacement measurements. The numerical experiments are performed to test the validity of the present algorithm by using different types of external forces, sensor arrangements and measurement errors. Results show that excellent estimations on the external forces can be obtained with any arbitrary initial guesses.