Stable methods for an inverse problem in acoustic scattering by an obstacle and an inhomogeneous medium

We consider (in two-dimensional Euclidean space) the scattering of a plane, time-harmonic acoustic wave by an inhomogeneous medium Ω with compact support and a bounded obstacle D lying completely outside of the inhomogeneous medium. We show that one may determine the shape of D and the local speed of sound in Ω from a knowledge of the asymptotic behavior of the scattered wave (i.e. the far field). This is done by considering a constrained optimization problem and employing integral equation and conformal mapping techniques. By assuming a priori that the functions which determine the shape of D and the local speed of sound in Ω lie in given compact sets, we show that the problem is stable, in the sense that the solution of the inverse scattering problem depends continuously on the far field data.     

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.     

The method of boundary integral equations for a mixed scattering problem

In this paper we consider the scattering of an electromagnetic time-harmonic plane wave by an infinite cylinder having an open arc and a bounded domain in R² as cross section. To this end, we solve a scattering problem for the Helmholtz equation in R² where the scattering object is a combination of a crack Γ and a bounded obstacle D, and we have Dirichlet-impedance type boundary condition on Γ and Dirichlet boundary condition on ∂D (∂D∈C²). Applying potential theory, the problem can be reformulated as a boundary integral system. We establish the existence and uniqueness of a solution to the system by using the Fredholm theory.     

On approximation in the fictitious domain method for a bidimensional transport equation

In this article a numerical method for solving a two‐dimensional transport equation in the stationary case is presented. Using the techniques of the variational calculus, we find the approximate solution for a homogeneous boundary‐value problem that corresponds to a square domain D₂. Then, using the method of the fictitious domain, we extend our algorithm to a boundary value problem for a set D that has an arbitrary shape. In this approach, the initial computation domain D (called physical domain) is immersed in a square domain D₂. We prove that the solution obtained by this method is a good approximation of the exact solution. The theoretical results are verified with the help of a numerical example. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

The inverse scattering problem for cavities with impedance boundary condition

We consider the inverse scattering problem of determining the shape of a cavity with impedance boundary condition from sources and measurements placed on a curve inside the cavity. It is shown that both the shape ?D\partial D of the cavity and the surface impedance λ are uniquely determined by the measured data and numerical methods are given for determining both ?D\partial D and λ where neither one is known a priori. Numerical examples are given showing the viability of our method.     

Inverse Problems for Obstacles in a Waveguide

In this paper we prove that a particular entry in the scattering matrix, if known for all energies, determines certain rotationally symmetric obstacles in a generalized waveguide. The generalized waveguide X can be of any dimension and we allow either Dirichlet or Neumann boundary conditions on the boundary of the obstacle and on ?X. In the case of a two-dimensional waveguide, two particular entries of the scattering matrix suffice to determine the obstacle, without the requirement of symmetry.     

The three dimensional inverse scattering problem for acoustic waves

We consider the inverse scattering problem for an acoustically soft obstacle in R³. By assuming a priori that the unknown scattering obstacle is starlike and has its boundary lying in a compact family of Hölder continuously differentiable surfaces, it is shown that an optimal solution can be constructed which depends continuously on the measured far field data. Remarks are made on the numerical approximation of the optimal solution.     

Pointwise Estimates for Laplace Equation. Applications to the Free Boundary of the Obstacle Problem with Dini Coefficients

In this paper we are interested in pointwise regularity of solutions to elliptic equations. In a first result, we prove that if the modulus of mean oscillation of Δu at the origin is Dini (in L ^p average), then the origin is a Lebesgue point of continuity (still in L ^p average) for the second derivatives D ² u. We extend this pointwise regularity result to the obstacle problem for the Laplace equation with Dini right hand side at the origin. Under these assumptions, we prove that the solution to the obstacle problem has a Taylor expansion up to the order 2 (in the L ^p average). Moreover we get a quantitative estimate of the error in this Taylor expansion for regular points of the free boundary. In the case where the right hand side is moreover double Dini at the origin, we also get a quantitative estimate of the error for singular points of the free boundary. Our method of proof is based on some decay estimates obtained by contradiction, using blow-up arguments and Liouville Theorems. In the case of singular points, our method uses moreover a refined monotonicity formula.      

Huygens’ principle and iterative methods in inverse obstacle scattering

The inverse problem we consider in this paper is to determine the shape of an obstacle from the knowledge of the far field pattern for scattering of time-harmonic plane waves. In the case of scattering from a sound-soft obstacle, we will interpret Huygens’ principle as a system of two integral equations, named data and field equation, for the unknown boundary of the scatterer and the induced surface flux, i.e., the unknown normal derivative of the total field on the boundary. Reflecting the ill-posedness of the inverse obstacle scattering problem these integral equations are ill-posed. They are linear with respect to the unknown flux and nonlinear with respect to the unknown boundary and offer, in principle, three immediate possibilities for their iterative solution via linearization and regularization. In addition to presenting new results on injectivity and dense range for the linearized operators, the main purpose of this paper is to establish and illuminate relations between these three solution methods based on Huygens’ principle in inverse obstacle scattering. Furthermore, we will exhibit connections and differences to the traditional regularized Newton type iterations as applied to the boundary to far field map, including alternatives for the implementation of these Newton iterations.     

Modified Rayleigh conjecture method and its applications

The Rayleigh conjecture about convergence up to the boundary of the series representing the scattered field in the exterior of an obstacle D$D$ is widely used by engineers in applications. However this conjecture is false for some obstacles. AGR introduced the Modified Rayleigh Conjecture (MRC), which is an exact mathematical result. In this paper we present the theoretical basis for the MRC method for 2D and 3D obstacle scattering problems, for static problems, and for scattering by periodic structures. We also present successful numerical algorithms based on the MRC for various scattering problems. The MRC method is easy to implement for both simple and complex geometries. It is shown to be a viable alternative for other obstacle scattering methods. Various direct and inverse scattering problems require finding global minima of functions of several variables. The Stability Index Method (SIM) combines stochastic and deterministic method to accomplish such a minimization.     

Analytic extension and reconstruction of obstacles from few measurements for elliptic second order operators

We deal with an inverse obstacle problem for general second order scalar elliptic operators with real principal part and analytic coefficients near the obstacle. We assume that the boundary of the obstacle is a non-analytic hypersurface. We show that, when we put Dirichlet boundary conditions, one measurement is enough to reconstruct the obstacle. In the Neumann case, we have results only for n = 2, 3 in general. More precisely, we show that one measurement is enough for n = 2 and we need 3 linearly independent inputs for n = 3. However, in the case for the Helmholtz equation, we only need n ? 1 linearly independent inputs, for any n ≥ 2. Here n is the dimension of the space containing the obstacle. These are justified by investigating the analyticity properties of the zero set of a real analytic function. In addition, we give a reconstruction procedure for each case to recover the shape of obstacle. Although we state the results for the scattering problems, similar results are true for the associated boundary value problems.     

Combined far‐ield operators in electromagnetic inverse scattering theory

We consider the inverse scattering problem of determining the shape of a perfect conductor D from a knowledge of the scattered electromagnetic wave generated by a time‐harmonic plane wave incident upon D. By using polarization effects we establish the validity of the linear sampling method for solving this problem that is valid for all positive values of the wave number. We also show that it suffices to consider incident directions and observation angles that are restricted to a limited aperture. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

On the scattering frequencies of the laplace operator for exterior domains

In this paper we show that the so-called scattering frequencies of the Laplace operator over an exterior domain, subject to Robin or Dirichlet boundary condition, cannot lie in certain portions of the upper half-plane. The excluded sets depend only on the type of boundary condition and the radius of the smallest sphere containing the scattering obstacle.     

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.     

A conformal mapping algorithm for the Bernoulli free boundary value problem

We propose a new numerical method for the solution of the Bernoulli free boundary value problem for harmonic functions in a doubly connected domain D in where an unknown free boundary Γ₀ is determined by prescribed Cauchy data on Γ₀ in addition to a Dirichlet condition on the known boundary Γ₁. Our main idea is to involve the conformal mapping method as proposed and analyzed by Akduman, Haddar, and Kress for the solution of a related inverse boundary value problem. For this, we interpret the free boundary Γ₀ as the unknown boundary in the inverse problem to construct Γ₀ from the Dirichlet condition on Γ₀ and Cauchy data on the known boundary Γ₁. Our method for the Bernoulli problem iterates on the missing normal derivative on Γ₁ by alternating between the application of the conformal mapping method for the inverse problem and solving a mixed Dirichlet–Neumann boundary value problem in D. We present the mathematical foundations of our algorithm and prove a convergence result. Some numerical examples will serve as proof of concept of our approach. Copyright © 2015 John Wiley & Sons, Ltd.     

The Cauchy Boundary Value Problems on Closed Piecewise Smooth Manifolds in <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Suppose that D is a bounded domain with a piecewise C^1 smooth boundary in C^n. Let ψ∈C^1 α(δD). By using the Hadamard principal value of the higher order singular integral and solid angle coefficient method of points on the boundary, we give the Plemelj formula of the higher order singular integral with the Boehner-Martinelli kernel, which has integral density ψ. Moreover, by means of the Plemelj formula and methods of complex partial differential equations, we discuss the corresponding Cauehy boundary value problem with the Boehner-Martinelli kernel on a closed piecewise smooth manifold and obtain its unique branch complex harmonic solution.     

Boundary integral equations for the Helmholtz equation: The third boundary value problem

In this paper we study the application of boundary integral equation methods for the solution of the third, or Robin, boundary value problem for the exterior Helmholtz equation. In contrast to earlier work, the boundary value problem is interpreted here in a weak sense which allows data to be specified in L_∞ (?D), ?D being the boundary of the exterior domain which we assume to be Lyapunov of index 1. For this exterior boundary value problem, we employ Green's theorem to derive a pair of boundary integral equations which have a unique simultaneous solution. We then show that this solution yields a solution of the original exterior boundary value problem.     

Regularization of a Nonstandard Cauchy Problem for a Dynamic Lame System

We consider the Cauchy problem for dynamic Lame systems in the cylinder G_T = D × (0,T) constructed over a domain D in a three-dimensional space, where the initial data are given in some strip in the lateral surface of the cylinder. The strip has the form S × (0,T), where S is an open subset of the boundary surface of the domain D. This problem is ill-posed. Under certain requirements to the configuration of S, we derive an explicit formula for solutions to this problem.

     

Boundary integral solution of the two-dimensional heat equation

Here we consider initial boundary value problems for the heat equation by using the heat potential representation for the solution. Depending on the choice of the representation we are led to a solution of the various boundary integral equations. We discuss the solvability of these equations in anisotropic Sobolev spaces. It turns out that the double-layer heat potential D and its spatial adjoint D′ have smoothing properties similar to the single-layer heat operator. This yields compactness of the operators D and D′. In addition, for any constant c ≠ 0, cI + D′ and cI + D′ are isomorphisms. Based on the coercivity of the single-layer heat operator and the above compactness we establish the coerciveness of the hypersingular heat operator. Moreover, we show an equivalence between the weak solution and the various boundary integral solutions. As a further application we describe a coupling procedure for an exterior initial boundary value problem for the non-homogeneous heat equation.