刚性目标形状反演的一种非线性最优化方法

发展了从声散射场的远场分布的信息来再现声刚性目标形状反问题的一种非线性最优化方法,它是通过独立地求解一个不适定的线性系统和一个适定的非线性最小化问题来实现的。对反问题的非线性和不适定性的这种分离式数值处理,使所建立方法的数值实现是非常容易和快速的,因为在确定声刚性障碍物形状的非线性最优化步中,只需求解一个只有一个未知函数的小规模的最小平方问题。该方法的另一个特别的性质是,只需要远场分布的一个Fourier系数,即可对未知的刚性目标作物形设别。进而提出了数值实现该方法的一种两步调整迭代算法。对具有各种形状的二维刚性障碍物的数值试验保证了本算法是有效和实用的。     

An Iterative Way of Solving the Inverse Scattering Problem for an Acoustic System of Equations in an Absorptive Layered Nonhomogeneous Medium

An algorithm is considered for solving the inverse scattering problem of seismic waves in a layered medium. The algorithm is based on solving a nonclassical ordinary differential equation with respect to an acoustic impedance, which also contains an unknown function characterizing the dissipative properties of the medium. The uniqueness of determining of these functions and the functional dependence associating them is established by solving the inverse problem of ground seismics. Results are presented from a computing experiment on applying the proposed algorithm.     

The numerical solution of an inverse scattering problem for acoustic waves

In a previous paper, the authors presented a dual space methodfor the numerical solution of the two-dimensional inverse scatteringproblem for acoustic waves in an inhomogeneous medium. Here,by making major modifications to the dual space method, a dramaticimprovement in the numerical performance of this method is achievedfor solving the inverse scattering problem.     

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.     

声速及损耗系数的同时反演

引言 本文讨论有损层状介质的反演问题.有损介质的反演问题的研究,即使是一维情形,也未成熟,且无一般提法.G.Kristensson和R.J.Krueger的一系列文章,利用一般的二次双曲型方程,对有限厚度的介质进行反演,用到的响应信息是透射和反射数据,不     

A smoothing augmented Lagrangian method for solving simple bilevel programs

In this paper, we design a numerical algorithm for solving a simple bilevel program where the lower level program is a nonconvex minimization problem with a convex set constraint. We propose to solve a combined problem where the first order condition and the value function are both present in the constraints. Since the value function is in general nonsmooth, the combined problem is in general a nonsmooth and nonconvex optimization problem. We propose a smoothing augmented Lagrangian method for solving a general class of nonsmooth and nonconvex constrained optimization problems. We show that, if the sequence of penalty parameters is bounded, then any accumulation point is a Karush-Kuch-Tucker (KKT) point of the nonsmooth optimization problem. The smoothing augmented Lagrangian method is used to solve the combined problem. Numerical experiments show that the algorithm is efficient for solving the simple bilevel program.     

An approximation problem in inverse scattering theory

In [3] a new method was introduced for solving the inverse scattering problem for acoustic waves in an inhomogeneous medium. This method is based on the solution of a new class of boundary value problems for the reduced wave equation called interior transmission problems. In this paper it is shown that if there is absorption there exists at most one solution to the interior transmission problem and an approximate solution can be found such that the metaharmonic part is a Herglotz wave function. These results provide the necessary theoretical basis for the inverse scattering method introduced in [3]     

A Numerical Way of Determining the Boundaries of a System of Bodies in a Three-Dimensional Medium by Means of Integral Equations

The propagation of acoustic waves in a three-dimensional medium with several local inhomogeneities of different shapes is analyzed. Solving the inverse problem of determining boundaries of local inhomogeneities from measurements of a field in a bounded receivers location domain is reduced to a system of integral equations. An iteration approach to solving the inverse problem is proposed, and the results from numerical experiments are presented.     

A numerical way of solving the inverse problem for the wave equation in a medium with local inhomogeneity

The inverse problem of determining the boundary of local inhomogeneity for measuring a field in a bounded receivers location domain in a three-dimensional medium is considered for the wave equation. The problem is reduced to a system of integral equations. An iteration approach to solving the inverse problem is proposed, and the results from numerical experiments are presented.     

Numerical solution of the inverse electrocardiography problem with the use of the Tikhonov regularization method

The inverse electrocardiography problem related to medical diagnostics is considered in terms of potentials. Within the framework of the quasi-stationary model of the electric field of the heart, the solution of the problem is reduced to the solution of the Cauchy problem for the Laplace equation in R ³. A numerical algorithm based on the Tikhonov regularization method is proposed for the solution of this problem. The Cauchy problem for the Laplace equation is reduced to an operator equation of the first kind, which is solved via minimization of the Tikhonov functional with the regularization parameter chosen according to the discrepancy principle. In addition, an algorithm based on numerical solution of the corresponding Euler equation is proposed for minimization of the Tikhonov functional. The Euler equation is solved using an iteration method that involves solution of mixed boundary value problems for the Laplace equation. An individual mixed problem is solved by means of the method of boundary integral equations of the potential theory. In the study, the inverse electrocardiography problem is solved in region Ω close to the real geometry of the torso and heart.     

求解粘性系数的迭代方法及其收敛性

求解粘性系数的迭代方法及其收敛性丛文相（黑龙江大学，哈尔滨１５００８０）１９９１年８月３０日收到．１９９２年８月２５日收到第一次修改稿．１９９２年１２月４日收到第二次修改稿．一、引言研究地震正、反问题时，一般把地球假设为完全弹性体，而实际地球介质并非...     

A Riemannian variant of the Fletcher–Reeves conjugate gradient method for stochastic inverse eigenvalue problems with partial eigendata

In this paper, we focus on the stochastic inverse eigenvalue problem with partial eigendata of constructing a stochastic matrix from the prescribed partial eigendata. A Riemannian variant of the Fletcher–Reeves conjugate gradient method is proposed for solving a general unconstrained minimization problem on a Riemannian manifold, and the corresponding global convergence is established under some assumptions. Then, we reformulate the inverse problem as a nonlinear least squares problem over a matrix oblique manifold, and the application of the proposed geometric method to the nonlinear least squares problem is investigated. The proposed geometric method is also applied to the case of prescribed entries and the case of column stochastic matrix. Finally, some numerical tests are reported to illustrate that the proposed geometric method is effective for solving the inverse problem.     

On the reconstruction of media inhomogeneity by inverse wave scattering model

Consider the reconstruction of the complex refraction index of an object, which is immersed in a known homogeneous background, from the knowledge of scattered waves of the point sources outside of the object. We firstly establish the uniqueness for this inverse problem, which provides the theoretical basis for the reconstruction scheme. Then based on the contrast source inversion(CSI) method, we propose an algorithm determining the refraction index and the artificial wave sources alternately by a dynamic iterative scheme. The algorithm defines the iterates by solving a series of minimization problems with uniformly convex penalty terms, which are allowed to be non-smooth to include L1 and total variation like functionals, ensuring the reconstruction quality when the unknown refraction index has the special features such as sparsity and discontinuity. By choosing the regularizing parameter automatically, the algorithm is terminated in terms of discrepancy principle. The convergence property of the iterative sequence is rigorously proven. Numerical implementations demonstrate the validity of the proposed algorithm.     

A numerical method for solving an inverse thermoacoustic problem

In this paper, we consider an inverse problem of determining the initial condition of an initial boundary value problem for the wave equation with some additional information about solving a direct initial boundary value problem. The information is obtained from measurements at the boundary of the solution domain. The purpose of our paper is to construct a numerical algorithm for solving the inverse problem by an iterative method called a method of simple iteration (MSI) and to study the resolution quality of the inverse problem as a function of the number and location of measurement points. Three two-dimensional inverse problem formulations are considered. The results of our numerical calculations are presented. It is shown that the MSI decreases the objective functional at each iteration step. However, due to the ill-posedness of the inverse problem the difference between the exact and approximate solutions decreases up to some fixed number k _min, and then monotonically increases. This shows the regularizing properties of the MSI, and the iteration number can be considered a regularization parameter.     

扩散系数反问题正则化数值方法的收敛性分析

扩散系数反问题正则化数值方法的收敛性分析朝红阳（中山大学计算机科学系）ＣＯＮＶＥＲＧＥＮＣＥＡＮＡＬＹＳＩＳＯＦＡＲＥＧＵＬＡＲＩＺＥＤＮＵＭＥＲＩＣＡＬＭＥＴＨＯＤＦＯＲＳＯＬＶＩＮＧＴＨＥＤＩＦＦＵＳＩＶＥＣＯＥＦＦＩＣＩＥＮＴＩＮＶＥＲＳＥＰＲ...     

Numerical methods for some inverse problems of heart electrophysiology

We consider numerical methods for solving inverse problems that arise in heart electrophysiology. The first inverse problem is the Cauchy problem for the Laplace equation. Its solution algorithm is based on the Tikhonov regularization method and the method of boundary integral equations. The second inverse problem is the problem of finding the discontinuity surface of the coefficient of conductivity of a medium on the basis of the potential and its normal derivative given on the exterior surface. For its numerical solution, we suggest a method based on the method of boundary integral equations and the assumption on a special representation of the unknown surface.     

用正则化方法求解声波散射反问题

研究了从声波散射场的远场模式的信息来再现散射物边界形状的反问题.首先构造表达散射物特征的指示函数,然后利用该函数之特性,建立求解该类反问题的基本方程,从而确定散射物的边界形状.在这个算法中,不需预先知道散射物的边界类型和形状等知识,从T ikhonov正则化方法进行的数值计算结果表明了该方法是有效的和实用的.     

Coefficient identification in Euler-Bernoulli equation from over-posed data

A method for solving the inverse problem for coefficient identification in the Euler-Bernoulli equation from over-posed data is presented. The original inverse problem is replaced by a minimization problem. The method is applied to the problem for identifying the coefficient in the case when it is a piece-wise polynomial function. Several examples are elaborated and the numerical results confirm that the solution of the imbedding problem coincides with the direct simulation of the original problem within the second order of approximation.     

An Inverse Problem in Elastostatics

This paper assembles a variety of methods which have been devisedfor acoustic and elastic wave propagation inverse problems andadapts them to the problem of determining the shear modulusprofile of an elastic half-space from a knowledge of the torsionaldeflection and shear stress distributions on the surface. Methodsinclude the reduction to a Gel'fand-Levitan integral equation,for which a fast numerical algorithm is presented; a methodbased on modelling the half-space as a layered medium; identificationof the medium as a member of a family for which a closed formsolution is possible; a Green's function approach for a mediumwith small variations from uniformity.     

Time domain scattering and inverse scattering problems in a locally perturbed half-plane

This paper is concerned with efficient numerical methods for solving the time-dependent scattering and inverse scattering problems of acoustic waves in a locally perturbed half-plane. By symmetric continuation, the scattering problem is reformulated as an equivalent symmetric problem defined in the whole plane. The retarded potential boundary integral equation method is modified to solve the forward problem. Then we consider the inverse scattering problem of determinating the local perturbation from the measured scattered data. The time domain linear sampling method is employed to deal with the inverse problem. The computation schemes proposed in this paper are relatively simple and easy to implement. Several numerical examples are presented to show the effectiveness of the proposed methods.