利用远场模式的不完全数据反演声波阻尼系数

从阻尼边界条件声波散射问题的散射场远场模式的部分数据信息出发给出了反演声波阻尼系数的一种新方法,该问题既是非线性的又是不适定的,这里利用Tikhonov正则化方法将问题转化为一个最优化问题,成功地处理了第一类算子方程的不适定性及该问题的非线性性,给出了具体的数值方法并对其收敛性进行了严格地证明,数值结果表明该方法是非常准确且简单易行的．     

海洋波导中三维可穿透目标的快速声学成像

研究了用声传播远场分布信息来成像海洋波导环境中三维可穿透目标的反问题.建立了求解这类反问题的远场方程,基于内透射边界值问题的分析,讨论了远场方程解的唯一性和可解性,证明了总能找到远场方程的一个在最小平方意义下的近似解,其模在可穿透目标内部的取值是小的,而在外部的取值是大的,进而发展了一种快速成像可穿透目标的一种指示器样本方法.数值试验表明了这种方法是有效的,即使在有限孔径测量方式的情况,也能够得到未知目标的一个理想成像,而且不需要先验知道可穿透目标的任何几何与物理信息.     

基于Linear Sampling方法的二维可穿透区域反演

提出用时间调和声散射远场信息来反演二维可穿透目标的一种Linear Sampling方法,通过提取包含可穿透目标的一个样本区域的支集的点列来实现反演的,因为其在区域内与区域外有显著的不同取值,由此而获得区域的逼近.这个算法特别吸引人之处是不需关于障碍物的任何先验信息.并且只需散射场在某个有限孔径中的部分远场信息,即可获得穿透区域的一个逼近.一些数值算例保证了这个反演算法是有效的和实用的.     

利用散射波的叠加重建声波散射区域

本文研究了声波散射区域的重建，给上散射波的叠加重建散射区域的一个方法，该方法利用散射波的叠加，将声波障碍反散射这个非一不适定问题分两步处理，第一步求解一个第一类线性积分方程。第二步求解一个非线性最优化问题，我们证明了该方法的收敛性。     

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.     

逆散射问题求解的正则化GAUSS-NEWTON法

本文研究了一类重要的反问题-逆散射问题,它是从远场散射数据识别散射体中的障碍物的形状问题.利用迭代正则化Gauss-Newton法,通过数值模拟比较分析,验证了本文所提出的方法在求解逆散射问题时是可行有效的.     

多目标最优化的一种积分型实现算法

在文[1]中给出了求解多目标最优化的一种积分总极值的概念性算法．本文利用数论中的一致分布佳点集列，较为简便的得出了多目标最优化的积分总极值的实现算法和算法终止准则．并经过有关函数数值计算表明该算法是有效的，可用来求解多目标最优化问题的有效解．     

一类逆时反问题的改进正则化方法的收敛性

1引 言 非线性反问题广泛地存在于许多科学和工程问题中,反问题求解的主要困难在于问题的不适定性,即待求函数或参量不连续依赖于观测数据.用来求解非线性不适定问题的方法主要有Tikhonov正则化方法和迭代正则化方法[1,2,3,4].Tikhonov正则化方法是通过引入正则化参数及稳定泛函,将目标泛函离散化,从而得到解的一个稳定近似,即正则化解.     

二维椭圆型方程反问题中优化算法的比较

近年来，决定椭圆型方程系数反问题在地磁、地球物理、冶金和生物等实际问题上有着广泛的应用．本文讨论了二维的决定椭圆型方程系数反问题的数值求解方法．由误差平方和最小原则，这个反问题可化为一个变分问题，并进一步离散化为一个最优化问题，其目标函数依赖于要决定的方程系数．本文着重考察非线性共轭梯度法在此最优化问题数值计算中的表现，并与拟牛顿法作为对比．为了提高算法的效率我们适当选择加快收敛速度的预处理矩阵．同时还考察了线搜索方法的不同对优化算法的影响．数值实验的结果表明，非线性共轭梯度法在这类大规模优化问题中相对于拟牛顿法更有效．     

从数值最优化方法到学习最优化方法

传统最优化问题的求解方法主要是以梯度法为基础的数值最优化方法,它是解析与数值计算相结合的迭代求解方法,是一种基于固定模式的最优化方法.算法的迭代过程实质上是对迭代点进行非线性变换的过程,该非线性变换是通过一系列方向和步长来实现.对于最优化问题的每一个实例,都需要从头到尾执行整个算法,计算复杂度是固定的.一旦算法被程序实现,算法的效率(计算精度和复杂度)就被固定.人工智能解决问题的方法都具有学习功能.随着人工智能,特别是深度学习的兴起,学习类方法在一些领域取得了巨大的成功,如图像识别(特别是人脸识别、车牌识别、手写字符识别等)、网络攻击防范、自然语言处理、自动驾驶、金融、医疗等.本文从新的视角研究传统的数值最优化方法和智能优化方法,分析其特点,由此引出学习最优化方法,并对它们进行了对比,提出了学习最优化方法的设计思路.最后,以组合最优化为例,对该类方法的设计原理进行阐述.     

A hybrid method for sound-hard obstacle reconstruction

We are interested in solving the inverse problem of acoustic wave scattering to reconstruct the position and the shape of sound-hard obstacles from a given incident field and the corresponding far field pattern of the scattered field. The method we suggest is an extension of the hybrid method for the reconstruction of sound-soft cracks as presented in [R. Kress, P. Serranho, A hybrid method for two-dimensional crack reconstruction, Inverse Problems 21 (2005) 773–784] to the case of sound-hard obstacles. The designation of the method is justified by the fact that it can be interpreted as a hybrid between a regularized Newton method applied to a nonlinear operator equation with the operator that maps the unknown boundary onto the solution of the direct scattering problem and a decomposition method in the spirit of the potential method as described in [A. Kirsch, R. Kress, On an integral equation of the first kind in inverse acoustic scattering, in: Cannon, Hornung (Eds.), Inverse Problems, ISNM, vol. 77, 1986, pp. 93–102. Since the method does not require a forward solver for each Newton step its computational costs are reduced. By some numerical examples we illustrate the feasibility of the method.     

Continuous Frechet differentiability with respect to a Lipschitz domain and a stability estimate for direct acoustic scattering problems

We consider direct acoustic scattering problems with eithera sound-soft or sound-hard obstacle, or lossy boundary conditions,and establish continuous Fréchet differentiability withrespect to the shape of the scatterer of the scattered fieldand its corresponding far-field pattern. Our proof is basedon the implicit function theorem, and assumes that the boundaryof the scatterer as well as the deformation are only Lipschitzcontinuous. From continuous Fréchet differentiability,we deduce a stability estimate governing the variation of thefar-field pattern with respect to the shape of the scatterer.We illustrate this estimate with numerical results obtainedfor a two-dimensional high-frequency acoustic scattering problem.     

Inverse source problem for a diffusion equation involving the fractional spectral Laplacian

In this paper, we consider an inverse problem related to a fractional diffusion equation. The model problem is governed by a nonlinear partial differential equation involving the fractional spectral Laplacian. This study is focused on the reconstruction of an unknown source term from a partial internal measured data. The considered ill‐posed inverse problem is formulated as a minimization one. The existence, uniqueness, and stability of the solution are discussed. Some theoretical results are established. The numerical reconstruction of the unknown source term is investigated using an iterative process. The proposed method involves a denoising procedure at each iteration step and provides a sequence of source term approximations converging in norm to the actual solution of the minimization problem. Some numerical results are presented to show the efficiency and the accuracy of the proposed approach.     

Numerical estimation of the piecewise constant Robin coefficient by identifying its discontinuous points

We propose a new numerical method for estimating the piecewise constant Robin coefficient in two-dimensional elliptic equation from boundary measurements. The Robin inverse problem is recast into a minimization of an output least-square formulation. A technique based on determining the discontinuous points of the unknown coefficient is suggested, and we investigate the differentiability of the solution and the objective functional with respect to the discontinuous points. Then we apply the Gauss-Newton method for reconstructing the shape of the unknown Robin coefficient. Numerical examples illustrate its efficiency and stability.     

A Regularized Newton Method in Inverse Scattering Problem from Cavities

We consider the inverse problem to determine the shape of a open cavity embedded in the infinite ground plane from knowledge of the far-field pattern of the scattering of TM polarization.For its approximate solution we propose a regularized Newton iteration scheme.For a foundation of Newton type methods we establish the Fréchet differentiability of solution to the scattering problem with respect to the boundary of the cavity.Some numerical examples of the feasibility of the method are presented.     

解洞穴逆散射问题的一种正则化Newton方法

We consider the inverse problem to determine the shape of a open cavity embedded in the infinite ground plane from knowledge of the far-field pattern of the scattering of TM polarization.For its approximate solution we propose a regularized Newton iteration scheme.For a foundation of Newton type methods we establish the Fréchet differentiability of solution to the scattering problem with respect to the boundary of the cavity.Some numerical examples of the feasibility of the method are presented.     

涡流检测中识别缺陷形状的一种新方法

涡流检测反演技术是一种非常重要的反演缺陷形状尺寸的无损检测方法．运用Dirichlet边界条件下涡流检测反演的远场区域导数,构造了反演缺陷形状的一种新算法,并且给出了二维及三维的算例,数值反演的结果与实际缺陷吻合得较好．从而说明了:对较小的波数,即使用较少的入射和观测方向的远场测量信息,亦可得到未知缺陷形状的一个合理的重构,算法是可行的、正确的．     

Inverse scattering from an open arc

A Newton method is presented for the approximate solution of the inverse problem to determine the shape of a sound-soft or perfectly conducting arc from a knowledge of the far-field pattern for the scattering of time-harmonic plane waves. Fréchet differentiability with respect to the boundary is shown for the far-field operator, which for a fixed incident wave maps the boundary arc onto the far-field pattern of the scattered wave. For the sake of completeness, the first part of the paper gives a short outline on the corresponding direct problem via an integral equation method including the numerical solution.     

Regularized Newton iteration method for a penetrable cavity with internal measurements in inverse scattering problem

In this paper, we consider the inverse scattering problem of determining the shape of a cavity with a penetrable inhomogeneous medium of compact support from one source and a knowledge of measurements placed on a curve inside the cavity. First, the boundary value problem of the partial differential equations can be transformed into an equivalent system of nonlinear and ill-posed integral equations for the unknown boundary. Then, we apply the regularized Newton iterative method to reconstruct the boundary and prove the injectivity for the linearized system. Finally, we present some numerical examples to show the feasibility of our method.