A New Energy Inversion for Parameter Identification in Saddle Point Problems with an Application to the Elasticity Imaging Inverse Problem of Predicting Tumor Location

The primary objective of this work is a detailed theoretical and computational study of the elasticity imaging inverse problem for tumor identification within the human body. Apart from this inverse problem's important and interesting application, it also poses noteworthy mathematical challenges since the underlying mathematical model is a system of elasticity involving incompressibility. This gives rise to the “locking” effect and special treatment is necessary for both the direct and inverse problems. To study the inverse problem in an optimization framework, we introduce a general computational scheme for handling parameter identification in saddle point problems along with the introduction and analysis of a new energy output least-squares objective functionals. We also present a treatment of the identification of discontinuous elasticity coefficients using the total variation regularization method. General formulas for the computation of the coefficient-to-solution map and a complete convergence analysis are given for the continuous problem as well as for its discrete analogue. Discrete formulas and implementation issues are discussed in detail and numerical examples for smooth and discontinuous coefficients are given.     

带W权Drazin逆的一种新算法

利用矩阵A的带W权Drazin逆的一个性质特征,对任意的矩阵A∈Cm×n,W∈Cn×m,建立了带W权的Drazin逆Ad,w的一种新的表示式,给出了具体的算法步骤,并且在文末给出了算例.     

Generalized Inversion and Its Application in Inverse Scattering Problems

In this paper, the generalized inversion theory and its application in inverse scattering problems are discussed. An iterative solution of joint inversion of parameters describing the earth structures and sources is given and a numerical example is also shown.     

Vector Potential Theory on Nonsmooth Domains in R3 and Applications to Electromagnetic Scattering

We study boundary value problems for the time-harmonic form of the Maxwell equations, as well as for other related systems of equations, on arbitrary Lipschitz domains in the three-dimensional Euclidean space. The main goal is to develop the corresponding theory for L^p-integrable bounday data for optimal values of p’s. We also discuss a number of relevant applications in electromagnetic scattering.     

Method With Nonzero Background Potential. Application to Inverse Scattering for the

We generalize a number of formulas of the method for the Schrōdinger equation to the case of nonzero backqround potential. As a corollary we have found the following reciprocity property ot the Faddeev function h: Where h is a non—analytic extension of the scattering amplitude to the complex domain. As an application we prove the uniqueness theorem and give a reconstruction procedure in the inverse scattering problem for the two—dimensional acoustic equation.     

双空间指示函数方法在三维柱面对称波导中电磁波的反散射问题的推广

给出了双空间指示函数方法在三维柱面对称波导中电磁波的反散射问题的推广.基于这个观察:当Green函数的点源在障碍物内部时,那么远域数据的赋权积分可以很好地近似估计Green函数,但是当Green函数的点源在障碍物外部时,那么远域数据的赋权积分就不能很好地近似估计Green函数.建立一个积分方程:它的右边是点源在所重构区域的Green函数,那么我们可以知道这个积分方程的解的范数在未知障碍物的内部有极大值,而这些取得极大值的点所围成的区域恰好就是所重构的障碍物区域.方法最显著的优势在于它不依赖于未知障碍物的边界条件.     

A Numerical Algorithm for an Inverse Problem of a Partial Differential Equation with Multi-Parameter to Be Determined

A formulation of an inverse problem of a partial differential equation with multi-parameter to be determined is introduced. The numerical algorithm, pulse-spectrum technique, is extended to solve this type of inverse problem. An example for remote sensing of the thermal conductivity and specific heat of a nonhomogeneous material is demonstrated. Numerical simulations are carried out to test the feasibility and to study the general characteristics of this technique without real measurement data. It is found that the extended pulse-spectrum technique gives excellent results.     

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.     

声学反问题的整体唯一性

It is proved that a sound-soft scatterer in R^N （N = 2, 3） is uniquely determined by a finite number of acoustic far-field measurements. The admissible scatterer possibly consists of finitely many solid obstacles and subsets of （N - 1）- dimensional hyperplanes.     

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.     

On an Application of the Lax-Phillips Scattering Approach in the Theory of Singular Perturbations

For a singular perturbation , n ≤ ∞, of a positive self-adjoint operator A ₀ with Lebesgue spectrum, the spectral analysis of the corresponding self-adjoint operator realizations A _T is carried out and the scattering matrix is calculated in terms of parameters t _ij under some additional restrictions on singular elements ψ _j . The results obtained enable one to apply the Lax-Phillips approach in scattering theory. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 5, pp. 679–688, May, 2005.     

周期三对角矩阵逆的一种新算法

给出了一类周期三对角矩阵逆的新的递归算法.新方法充分利用周期三对角矩阵的结构特点,采用递归方法将高阶周期三对角矩阵求逆转化为低阶周期三对角矩阵的求逆.并同时得到简化的计算方法,方法可以有效地减少运算量和存储量,计算精度也有明显的优势.数值实验表明此算法是有效的.     

Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part II: Application to Scattering by a Homogeneous Dielectric Obstacle

We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves by a penetrable bounded obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering problems, we study the shape differentiability properties of the standard electromagnetic boundary integral operators. The latter are typically bounded on the space of tangential vector fields of mixed regularity T H^-\frac12(div_G,G){\mathsf T \mathsf H^{-\frac{1}{2}}({\rm div}_{\Gamma},\Gamma)}. Using Helmholtz decomposition, we can base their analysis on the study of pseudo-differential integral operators in standard Sobolev spaces, but we then have to study the Gateaux differentiability of surface differential operators. We prove that the electromagnetic boundary integral operators are infinitely differentiable without loss of regularity. We also give a characterization of the first shape derivative of the solution of the dielectric scattering problem as a solution of a new electromagnetic scattering problem.     

A Method for Solving the Inverse Problem in Soft Acoustic Scattering

The inverse problem considered is to determine the shape ofan acoustically soft obstacle in R³ from a knowledge of thetime-harmonic incident plane wave and the far-field patternof the scattered wave. To solve this inverse Dirichlet problemin acoustic scattering without requiring the solution of integralequations, a parametric representation is introduced in whichthe parameters are determined by a method of optimization. Directscattering can also be handled by this technique. Comparisonsreveal that results are obtained more easily than, and justas accurately as, in other methods.     

Integral Equations with Modified Fundamental Solution in Time-Harmonic Electromagnetic Scattering

To overcome the non-uniqueness problem arising in integral equationsfor the exterior boundary-value problems for the scalar Helmholtzequation, Ursell and Jones suggested adding a series of outgoingwaves to the free-space fundamental solution. The present papergives an extension of this approach to the case of the reducedMaxwell equations of electromagnetic scattering. For the caseof a sphere an explicit choice of coefficients in the modificationis presented which is optimal with respect to various criteria.