Reconstruction of closely spaced small inhomogeneities via boundary measurements for the full time-dependent Maxwell’s equations

We consider for the full time-dependent Maxwell’s equations the inverse problem of identifying locations and certain properties of small electromagnetic inhomogeneities in a homogeneous background medium from dynamic boundary measurements on the boundary for a finite time interval.     

Identification of small inhomogeneities: Asymptotic factorization

We consider the boundary value problem of calculating the electrostatic potential for a homogeneous conductor containing finitely many small insulating inclusions. We give a new proof of the asymptotic expansion of the electrostatic potential in terms of the background potential, the location of the inhomogeneities and their geometry, as the size of the inhomogeneities tends to zero. Such asymptotic expansions have already been used to design direct (i.e. noniterative) reconstruction algorithms for the determination of the location of the small inclusions from electrostatic measurements on the boundary, e.g. MUSIC-type methods. Our derivation of the asymptotic formulas is based on integral equation methods. It demonstrates the strong relation between factorization methods and MUSIC-type methods for the solution of this inverse problem.

     

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.     

Inverse coefficient problem for a quasilinear hyperbolic equation with final overdetermination

The inverse problem of recovering a solution-dependent coefficient multiplying the lowest derivative in a hyperbolic equation is investigated. As overdetermination is required in the inverse problem, an additional condition is imposed on the solution to the equation with a fixed value of the timelike variable. Global uniqueness and local existence theorems are proved for the solution to the inverse problem. An iterative method is proposed for solving the inverse problem.     

Autoresonance excitation of a breather in weak ferromagnetics

A mathematical model of the magnetodynamics of a weak ferromagnetic in an external magnetic field with variable frequency is studied. Conditions for the appearance of autoresonance are found under which the amplitude of the local magnetic inhomogeneities is considerably increased. Mathematically, the problem is to analyze the solutions to the sine-Gordon equation subject to a specific nonautonomous perturbation. For the perturbed equation, asymptotic solutions in the form of a breather with a slowly varying amplitude and a phase shift are constructed. The solutions whose amplitude increases with time from small values to quantities of order one are associated with the autoresonance phenomenon.     

Tension of a compound piezoceramic plate with a hole intersecting the phase interface

We study the two-dimensional electroelastic problem for an unbounded compound plate with an arbitrary hole located in both components of the plate. The corresponding boundary-value problem is reduced to a system of singular integral equations of second kind, which for the case of an elliptic hole can be solved numerically by the method of quadratures. We give the data of computations that characterize the concentration of the electroelastic fields near the hole subject to action at infinity by fields of mechanical stresses and electric tension. It is noted that in the case of the inverse piezoelectric effect the influence of inhomogeneities of the plate on the stress concentrations is sharply expressed. Six figures. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 67–75.     

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.     

On a coefficient inverse problem for a parabolic equation in a domain with free boundary

The present paper deals with the inverse problem of determination of the coefficient of the first derivative of the unknown function with respect to a spatial variable for a one-dimensional parabolic equation in the domain whose boundary is determined by two unknown functions. The conditions of local existence and uniqueness of a solution to the inverse problem are established.     

Optimality conditions for a class of inverse optimal control problems with partial differential equations

ABSTRACT

We consider bilevel optimization problems which can be interpreted as inverse optimal control problems. The lower-level problem is an optimal control problem with a parametrized objective function. The upper-level problem is used to identify the parameters of the lower-level problem. Our main focus is the derivation of first-order necessary optimality conditions. We prove C-stationarity of local solutions of the inverse optimal control problem and give a counterexample to show that strong stationarity might be violated at a local minimizer.     

Numerical Simulation of the Reconstruction of the Velocity Parameters of Elastic Inhomogeneities by the Method of Diffraction Tomography

In order to solve the direct problem, the finite-difference method is used, which enables us to take diffraction into account by not weakly contrasting local inhomogeneities with simple or complicated geometry. The inverse problem is solved by the method of diffraction tomography with the use of the Born approximation. Examples of the recovery of parameters of inhomogeneities are given. Here we use wave fields (the 2-D PSV problem) excited by a source of the type of a center of pressure in a homogeneous space for three positions of a source and for three observation points located on a linear profile. The possibility of separately recovering the elastic parameters (,) and the mass density is shown, which enables us to find both the velocity perturbation and the ratio of the velocities of the shear and compressional waves. Bibliography: 14 titles.     

Stability estimate for a multidimensional inverse spectral problem with partial spectral data

In Bellassoued, Choulli and Yamamoto (2009) [4] we proved a log-log type stability estimate for a multidimensional inverse spectral problem with partial spectral data for a Schrödinger operator, provided that the potential is known in a small neighbourhood of the boundary of the domain. In the present paper we discuss the same inverse problem. We show a log type stability estimate under an additional condition on potentials in terms of their X-ray transform. In proving our result, we follow the same method as in Alessandrini and Sylvester (1990) [1] and Bellassoued, Choulli and Yamamoto (2009) [4]. That is we relate the stability estimate for our inverse spectral problem to a stability estimate for an inverse problem consisting in the determination of the potential in a wave equation from a local Dirichlet to Neumann map (DN map in short).     

Carleman estimate for a strongly damped wave equation and applications to an inverse problem

In this paper, we establish a Carleman estimate for a strongly damped wave equation in order to solve a coefficient inverse problems of retrieving a stationary potential from a single time‐dependent Neumann boundary measurement on a suitable part of the boundary. This coefficient inverse problem is for a strongly damped wave equation. We prove the uniqueness and the local stability results for this inverse problem. The proof of the results relies on Carleman estimate and a certain energy estimates for hyperbolic equation with strongly damped term. Moreover, this method could be used for a similar inverse problem for an integro‐differential equation with hyperbolic memory kernel. Copyright © 2012 John Wiley & Sons, Ltd.     

Solvability and stability of the inverse Sturm–Liouville problem with analytical functions in the boundary condition

The paper deals with the Sturm–Liouville eigenvalue problem with the Dirichlet boundary condition at one end of the interval and with the boundary condition containing entire functions of the spectral parameter at the other end. We study the inverse problem, which consists in recovering the potential from a part of the spectrum. This inverse problem generalizes partial inverse problems on finite intervals and on graphs and also the inverse transmission eigenvalue problem. We obtain sufficient conditions for global solvability of the studied inverse problem, which prove its local solvability and stability. In addition, application of our main results to the partial inverse Sturm–Liouville problem on the star-shaped graph is provided.     

Two approaches to the solution of coefficient inverse problems for wave equations

Two approaches to solving coefficient inverse problems for wave equations are compared. One approach is based on integral representations obtained with the help of the Green’s function for the wave equation. In the other approach, the gradient of the error functional is directly computed in terms of the solution of the adjoint problem for a partial differential equation. The methods developed are intended for finding inhomogeneities in homogeneous media and can be applied in medicine diagnostics, acoustic and seismic near surface exploration, engineering seismics, etc.     

Formulation of the Material Force Approach for Finite Inelasticity

The material forces represent the thermodynamically driving forces on any kind of inhomogeneity and discontinuity. For comprehensive investigations, the balance of the inverse motion problem has to be formulated and evaluated to separate different portions of the material forces arising from different inhomogeneities and discontinuities. This paper presents a general approach for inelastic material models based on the multiplicative split of the deformation gradient in order to calculate the J -integral of an inelastic crack tip. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Unique Solvability of an Inverse Problem for Parabolic Equations under a Final Overdetermination Condition

We consider the unique solvability of the inverse problem of determining the right-hand side of a parabolic equation with the leading coefficient depending on time and space variables under a final overdetermination condition. We obtain two types of conditions that are sufficient for the local solvability of the inverse problem and also prove the so-called Fredholm solvability of the inverse problem under study.     

On the inverse problem of determining the right-hand side of a parabolic equation under an integral overdetermination condition

We study the unique solvability of the inverse problem of determining the righthand side of a parabolic equation whose leading coefficient depends on both the time and the spatial variable under an integral overdetermination condition with respect to time. We obtain two types of condition sufficient for the local solvability of the inverse problem as well as study the so-called Fredholm solvability of the inverse problem under consideration.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 522–534.Original Russian Text Copyright © 2005 by V. L. Kamynin.This revised version was published online in April 2005 with a corrected issue number.     

Controllability for a Nonlinear Abstract Evolution Equation

We prove a theorem on the local controllability of a system described by a nonlinear evolution equation in Banach space when the control is a multiplier on the right-hand side. We obtain sufficient conditions on the size of the neighborhood from which we can take the function from the overdetermination condition so that the inverse problem is uniquely solvable.     

Estimate of the precision of determination of elastic parameters by the diffraction tomography method with the use of the finite-difference method (the SV-problem)

In the present paper, the finite-difference method is used for the solution of the direct problem, which enables us to take into account the diffraction phenomenon on local inhomogeneities of not weak contrast. Examples that allow us to estimate the precision of restoration of the parameters of such inhomogeneities, depending on the degree of their contrast, are given. The possibility of restoring the parameters of a local inhomogeneity in a medium containing interfaces with stepwise change of elastic properties for the case where source and receiver points are located on the free surface is demonstrated. An example of the numerical estimation of the precision of calculation of a scattered field in the Born approximation, approximating the wave field by the zero term of the ray method, is offered. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 136–152. Translated by Yu. V. Kiselev     

Stress distribution in dissimilar materials containing inhomogeneities near the interface using a novel finite element

This study is concerned with the determination of the elastic behaviour of two dissimilar materials containing a single and two interacting inhomogeneities near the interface using a novel finite element. The general form of the element, which is constructed from a cell containing a single circular inhomogeneity in a surrounding matrix, is derived explicitly in a series form using the complex potentials of Muskhelishvili. The strength of the adopted eight-noded plane element lies in its ability to treat periodically and/or arbitrarily located multiple inhomogeneities in dissimilar materials accurately and efficiently.

Pertinent stress distributions are examined to illustrate the importance of the elastic properties of the dissimilar materials, the distance between their interface and the inhomogeneities and the direction of the externally applied load with respect to that interface upon the resulting stress field. The results of our work assist in defining the design parameters which govern the elastic behaviour of the examined problem. This problem relates to fibre reinforcement in advanced composite materials, second phase particles in traditional materials and plates stiffened with stringers commonly used in the aerospace and automotive fields.