Source and coefficient inverse problems for an elliptic equation in a rectangle

The inverse problems of determining the source and coefficient of an elliptic equation in a rectangle are studied. Additional information on the solution to the direct problem (overdetermination) is the trace of its solution on an interval inside the rectangle. Sufficient existence and uniqueness conditions (global) are derived for the inverse problems. The study is performed in the class of continuously differentiable functions whose derivatives satisfy a Hölder condition.     

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.     

Zeros of the Bessel and spherical Bessel functions and their applications for uniqueness in inverse acoustic obstacle scattering

Some novel interlacing properties of the zeros for the Besseland spherical Bessel functions are first presented and thenapplied to prove an interesting uniqueness result in inverseacoustic obstacle scattering. It is shown that in the resonanceregion, the shape of a sound-soft/sound-hard ball in R³ or asound-soft/sound-hard disc in R² is uniquely determined by asingle far-field datum measured at some fixed spot correspondingto a single incident plane wave.     

On the far-field operator in elastic obstacle scattering

We investigate the far-field operator for the scattering oftime-harmonic elastic plane waves by either a rigid body, acavity, or an absorbing obstacle. Extending results of Colton& Kress for acoustic obstacle scattering, for the spectrumof the far-field operator we show that there exist an infinitenumber of eigenvalues and determine disks in the complex planewhere these eigenvalues lie. In addition, as counterpart ofan identity in acoustic scattering due to Kress & Päivärinta,we will establish a factorization for the difference of thefar-field operators for two different scatterers. Finally, extendinga sampling method for the approximate solution of the acousticinverse obstacle scattering problem suggested by Kirsch to elasticity,this factorization is used for a characterization of a rigidscatterer in terms of the eigenvalues and eigenelements of thefar-field operator.     

A Carleman estimate for the linear magnetoelastic waves system and an inverse source problem in a bounded conductive medium

ABSTRACT

In this paper, we consider an inverse problem for the simultaneous diffusion process of elastic and electromagnetic waves in an isotropic heterogeneous elastic body which is identified with an open bounded domain. From the mathematical point of view, the system under consideration can be viewed as the coupling between the hyperbolic system of elastic waves and a parabolic system for the magnetic field. We study an inverse problem of determining the external source terms by observations data in a neighborhood of the boundary and we prove the Hölder stability. For the proof, we show a Carleman estimate for the displacement and the magnetic field of the magnetoelastic system.     

Direct and inverse problems for electromagnetic scattering by a doubly periodic structure with a partially coated dielectric

Consider the problem of scattering of electromagnetic waves by a doubly periodic Lipschitz structure. The medium above the structure is assumed to be homogenous and lossless with a positive dielectric coefficient. Below the structure there is a perfect conductor with a partially coated dielectric boundary. We first establish the well‐posedness of the direct problem in a proper function space and then obtain a uniqueness result for the inverse problem by extending Isakov's method. Copyright © 2009 John Wiley & Sons, Ltd.     

The linear sampling method for the inverse electromagnetic scattering by a partially coated bi‐periodic structure

In this paper, we consider the inverse problem of recovering a doubly periodic Lipschitz structure through the measurement of the scattered field above the structure produced by point sources lying above the structure. The medium above the structure is assumed to be homogeneous and lossless with a positive dielectric coefficient. Below the structure is a perfect conductor partially coated with a dielectric. A periodic version of the linear sampling method is developed to reconstruct the doubly periodic structure using the near field data. In this case, the far field equation defined on the unit ball of ?³ is replaced by the near field equation which is a linear integral equation of the first kind defined on a plane above the periodic surface. Copyright © 2010 John Wiley & Sons, Ltd.     

On nonlinear ill-posed inverse problems with applications to pricing of defaultable bonds and option pricing

This paper considers the estimation of an unknown function h that can be characterized as a solution to a nonlinear operator equation mapping between two infinite dimensional Hilbert spaces. The nonlinear operator is unknown but can be consistently estimated, and its inverse is discontinuous, rendering the problem ill-posed. We establish the consistency for the class of estimators that are regularized using general lower semicompact penalty functions. We derive the optimal convergence rates of the estimators under the Hilbert scale norms. We apply our results to two important problems in economics and finance: (1) estimating the parameters of the pricing kernel of defaultable bonds; (2) recovering the volatility surface implied by option prices allowing for measurement error in the option prices and numerical error in the computation of the operator. The first anther was supported by US National Science Foundation (Grant No. SES-0631613) and the Cowles Foundation for Research in Economics     

Implicit and efficient schemes for a parabolic equation in a spherical layer

An implicit and an efficient three-level scheme for a parabolic equation in spherical coordinates is constructed in a spherical layer. No axial symmetry is assumed. The convergence rates of the schemes are estimated under minimum requirements on the initial data. The estimates are uniform with respect to the inner diameter of the domain. The order of convergence is τ^α/2 + h ^α, α = 1, 2, depending on the smoothness of the data. The results remain valid for a domain without a hole.     

Solving an inverse problem for a generalized time-delayed Burgers-Fisher equation by Haar wavelet method

In this paper, a numerical method consists of combining Haar wavelet method and Tikhonov regularization method to determine unknown boundary condition and unknown nonlinear source term for the generalized time-delayed Burgers-Fisher equation using noisy data is presented. A stable numerical solution is determined for the problem. We also show that the rate of convergence of the method is as exponential $\Bigl(O\left(\frac{1}{2^{J+1}}\right)\Bigr)$, where $J$ is maximal level of resolution of wavelet. Some numerical results are reported to show the efficiency and robustness of the proposed approach for solving the inverse problems.     

On the finite element method for elliptic problems with degenerate and singular coefficients

We consider Dirichlet boundary value problems for second order elliptic equations over polygonal domains. The coefficients of the equations under consideration degenerate at an inner point of the domain, or behave singularly in the neighborhood of that point. This behavior may cause singularities in the solution. The solvability of the problems is proved in weighted Sobolev spaces, and their approximation by finite elements is studied. This study includes regularity results, graded meshes, and inverse estimates. Applications of the theory to some problems appearing in quantum mechanics are given. Numerical results are provided which illustrate the theory and confirm the predicted rates of convergence of the finite element approximations for quasi-uniform meshes.

     

Some remarks on a class of inverse problems related to the parabolic approximation to the Maxwell equations: a controllability approach

We consider a class of inverse source problems for the parabolic approximation to the Maxwell equations.We relate this to an exact controllability problem; the regularisation of the considered source problems is studied with an optimal control method. Copyright © 2014 John Wiley & Sons, Ltd.     

On a conjecture by Ghahramani-Lau and related problems concerning topological centres

Let A be a Banach algebra, and consider A^** equipped with the first Arens product. We establish a general criterion which ensures that A is left strongly Arens irregular, i.e., the first topological centre of A^** is reduced to A itself. Using this criterion, we prove that for a very large class of locally compact groups, Ghahramani-Lau's conjecture (cf. [Lau 94] and [Gha-Lau 95]) stating the left strong Arens irregularity of the measure algebra M(G), holds true. (Our methods obviously yield as well the right strong Arens irregularity in the situation considered.)Furthermore, the same condition used above implies that every linear left A^**-module homomorphism on A^* is automatically bounded and w^*-continuous. We finally show that our criterion also yields a partial answer to a question raised by Lau-Ülger (Trans. Amer. Math. Soc. 348 (3) (1996) 1191) on the topological centre of the algebra (A^*⊙A)^*, for A having a right approximate identity bounded by 1.     

Solving the 2D and 3D nonlinear inverse source problems of elliptic type partial differential equations by a homogenization function method

In this article, the inverse source problems of 2D and 3D elliptic type nonlinear partial differential equations are resolved. For this purpose, a family of single-parameter homogenization functions that automatically meet the given boundary conditions are deduced and employed as the bases to expand the solution. We solve a linear algebraic equations system which satisfies the over-specified Neumann boundary condition to obtain the unspecified coefficients, and then the solution in the entire domain is permitted. Taking the solution into the governing equation, the unknown source function can be determined quickly. The present novel method is verified to be an accurate, effective, and robust scheme which is without solving nonlinear equations and iterations, and the additional data used are quite economical.     

On mistaken papers by Gouzheng Yan et al and related papers,and on a paper by Weibing Wang and Xuxin Yang

The purpose of the present note is to point out mathematical mistakes in some published papers on boundary value problems to prevent usage of mistaken results and methods by mathematicians. Copyright © 2013 John Wiley & Sons, Ltd.     

On the inverse of the generator of a bounded <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>-semigroup

Let A be the generator of a uniformly bounded C ₀-semigroup in a Banach space B, and let A have a densely defined inverse A ^-1 . We present sufficient conditions on the resolvent (A-I ^-1, Re > 0, under which A ^-1 is also the generator of a uniformly bounded C ₀-semigroup.Dedicated to V. B. Lidskii on the occasion of his eightieth birthdayTranslated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 6–12, 2004Original Russian Text Copyright © by A. M. Gomilko