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.     

一类具有转移条件的Sturm-Liouville方程的谱性质

研究一类具有转移条件和特征参数相关边界条件的不连续的Sturm-Liouville方程.构造了一个新的算子,并且在新的Hilbert空间中证明了其自伴性.构造了基本解,给出了特征值和特征函数的一些性质,以及渐近估计式,证明了特征函数系的完备性,并且得到了问题的格林函数和预解算子.     

广义算子半群与广义分布参数系统的适定性

首先,针对广义分布参数系统的求解问题,提出了由Hilbert空间中有界线性算子所引导的广义算子半群和广义积分半群;其次,讨论了广义预解算子的性质、广义算子半群与广义积分半群的性质;最后,研究了广义分布参数系统的适定性问题.     

Hilbert空间中的不可微最优化问题的基路径增量目标水平算法

主要研究了Hilbert空间中的一类不可微最优化问题的基路径增量目标水平算法.证明了当约束集合为有界集时,问题的最优解集非空,且这时通过算法生成的迭代点列是弱收敛于最优解的..     

一类非线性 Dirichlet 边值问题的正径向解

通过构造适当的锥并利用锥拉伸与锥压缩型的不动点定理研究了单位球上一类椭圆 Dirichlet 边值问题的正径向解的存在性, 其中非线性项可以是奇异的. 主要结论表明正径向解的存在性仅依赖于非线性项在其定义域的某个有界子集上的性质, 而与非线性项在此集合以外的性质无关.     

平衡问题变分包含问题及不动点问题的二次极小化

借助预解式技巧,寻求二次极小化问题minx∈Ω‖x‖2的解,其中Ω是Hilbert空间中某一广义平衡问题的解集,与一无穷族非扩张映像的公共不动点的集合,以及某一变分包含的解集的交集．在适当的条件下,逼近上述极小化问题的解的一新的强收敛定理被证明．     

多目标决策问题解集映射的下半连续性

多目标决策问题的研究中,关于解的概念有不同的提法,Yu提出了目标空间以凸锥定义优劣结构的概念,后来Bergstresser,charnes,Yu等人推广了这一概念,提出以一般的凸集来定义优劣结构,他们利用这些概念,将多目标决策问题归纳为在目标空间集合中寻求全部非劣解的问题,而此目标空间集合是某个问量值函数在决策空间集合上的象集。于是,当目标     

复Swift-Hohenberg方程在一些Banach空间内的指数吸引子

本文研究了N-维(N≤3)复Swift-Hohenberg方程在一些Banach空间x~α中解的渐近行为.运用Cholewa等人的技巧,证明了整体解的存在性以及整体吸引子A的存在性.最后,作为本文的另—个主要结果,证明了指数吸引子M的存在性,从而得到A有有限的分形维数.由于应用于Hilbert空间中所谓的挤压性质在我们的框架下不能成立,为了构造M,没有应用Hilbert空间中的标准的方法,而是应用Efendiev,Miranville,和Zelik最近的结果.     

带有单模糊映射的一般变分不等式解的存在性

众所周知，在各种变分不等式问题中，解的存在性是最基础也是最重要的问题之一．本文的目的是在Hilbert空间中研究带有单模糊映射的一般变分不等式解的存在性．此外，还讨论了几种特殊情况．     

广义非线性变分包含的带误差的近似点算法

引入和研究了一类新的广义非线性变分包含．在Hilbert空间中利用与极大η-单调映象相联系的预解算子的性质，对新的广义非线性变分包含建立了一个新的寻求近似解的带误差的近似点算法，并证明了求近似解序列强收敛于精确解．其所得结果是近期相关结果的改进和推广．     

Reciprocity relations and completeness of far-field pattern vectors for obstacle scattering of acoustic wave in a stratified medium

Under a generalized Sommerfeld radiation condition, we proved the uniqueness and existence of the direct obstacle scattering problem of time-harmonic acoustic waves in a stratified medium [8]. In this paper, we study the asymptotic behaviour of the scattered waves and prove three reciprocity relations among the free-wave far-field patterns and the guided-wave far-field pattern vectors corresponding to incident distorted plane waves and normal mode waves. Then we prove conditions under which a set of far-field patterns is complete in a Hilbert space based on the reciprocity relation. These properties are important in investigating the inverse obstacle scattering problems.     

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.     

Locating an obstacle in a 3D finite depth ocean using the convex scattering support

We consider an inverse scattering problem in a 3D homogeneous shallow ocean. Specifically, we describe a simple and efficient inverse method which can compute an approximation of the vertical projection of an immersed obstacle. This reconstruction is obtained from the far-field patterns generated by illuminating the obstacle with a single incident wave at a given fixed frequency. The technique is based on an implementation of the theory of the convex scattering support [S. Kusiak, J. Sylvester, The scattering support, Commun. Pure Appl. Math. (2003) 1525–1548]. A few examples are presented to show the feasibility of the method.     

A Transmission problem for bi-isotropic media

In this work we study the solvability of the transmission problem describing the scattering of a plane time-harmonic electromagnetic wave by a penetrable bi-isotropic obstacle. Existence and uniqueness of the solution to this problem is established by a weak solutions approach, with the use of the Dirichlet-to-Neumann operator. Moreover, in the low frequency case, we reduce the problem to a sequence of potential problems which can be solved iteratively.     

Radiation condition and scattering problem for time-harmonic acoustic waves in a stratified medium with a nonstratified inhomogeneity

In this paper, a generalized Sommerfeld radiation conditionis presented for the scattering waves in a stratified mediumwith a nonstratified inhomogeneity. Using integral equationmethods, the uniqueness and existence of the direct scatteringproblem are proved. Relations between the scattered acousticwaves in the far field and the sound profile of the inhomogeneityare obtained. Using these relations, the author proves threereciprocity relations between the free-wave far-field patternsand the guided-wave far-field pattern vectors correspondingto incident distorted plane waves and normal mode waves. Thenconditions under which a set of far-field patterns is completein a Hilbert space are determined using the reciprocity relation.These properties are important in investigating inverse scatteringproblems.     

The inverse scattering problem for partially penetrable obstacles

We consider the direct and inverse problems for the scattering of a partially penetrable obstacle. Here ‘partially penetrable obstacle’ means that the waves transmit into the obstacle just from partial boundary of the obstacle with the rest of the boundary touching a known perfect and thin scatterer. The solvability of the direct scattering problem is presented using the classical boundary integral equation method. An interesting interior transmission problem is investigated for the purpose of solving the inverse obstacle scattering problem. Then the linear sampling method is proposed to reconstruct the shape and location of the obstacle from near field measurements. We note that the inversion algorithm can be implemented by avoiding the use of background Green function as a test function due to a mixed reciprocal principle.     

Inverse problem for wave propagation in a perturbed layered half-space

This paper is concerned with the inverse medium scattering problem in a perturbed, layered, half-space, which is a problem related to the seismologial investigation of inclusions inside the earth’s crust. A wave penetrable object is located in a layer where the refraction index is different from the other part of the half-space. Wave propagation in such a layered half-space is different from that in a homogeneous half-space. In a layered half-space, a scattered wave consists of a free wave and a guided wave. In many cases, only the free-wave far-field or only the guided-wave far-field can be measured.We establish mathematical formulas for relations between the object, the incident wave and the scattered wave. In the ideal condition where exact data are given, we prove the uniqueness of the inverse problem. A numerical example is presented for the reconstruction of a penetrable object from simulated noise data.     

3D elastic scattering theorems for point-generated dyadic fields

The problem of scattering of elastic spherical waves by a rigid body, cavity or a penetrable obstacle in 3D linear elasticity is considered. For two point sources, dyadic far-field pattern generators are defined, which are used for the formulation of a general scattering theorem. The main reciprocity principle and mixed scattering relations are also established. We provide the necessary energy considerations, presenting relative energy functionals and expressions for the differential and the scattering cross section due to point-source dyadic incidence. Finally, an application of the general scattering theorem for appropriate locations of the point source leads to an optical theorem. Copyright © 2007 John Wiley & Sons, Ltd.     

Reconstruction of a penetrable obstacle by complex spherical waves

We consider an inverse acoustic scattering problem for identifying a non-convex penetrable obstacle in three dimensions in a homogeneous medium. We apply the complex geometrical optics solutions with logarithmic phase, which is called complex spherical waves, to reconstruction problem. The reconstruction schemes will be demonstrated in the last section.