Iterative Methods for Solving the Cauchy Problem for the Helmholtz Equation

Abstract

The inverse problem of reconstructing the acoustic, or electromagnetic, field from inexact measurements on a part of the boundary of a domain is important in applications, for instance for detecting the source of acoustic noise. The governing equation for the applications we consider is the Helmholtz equation. More precisely, in this thesis we study the case where Cauchy data is available on a part of the boundary and we seek to recover the solution in the whole domain. The problem is ill-posed in the sense that small errors in the Cauchy data may lead to large errors in the recovered solution. Thus special regularization methods that restore the stability with respect to measurements errors are used.In the thesis, we focus on iterative methods for solving the Cauchy problem. The methods are based on solving a sequence of well-posed boundary value problems. The specific choices for the boundary conditions used are selected in such a way that the sequence of solutions converges to the solution for the original Cauchy problem. For the iterative methods to converge, it is important that a certain bilinear form, associated with the boundary value problem, is positive definite. This is sometimes not the case for problems with a high wave number.The main focus of our research is to study certain modifications to the problem that restore positive definiteness to the associated bilinear form. First we add an artificial interior boundary inside the domain together with a jump condition that includes a parameter µ. We have shown by selecting an appropriate interior boundary and sufficiently large value for µ, we get a convergent iterative regularization method. We have proved the convergence of this method. This method converges slowly. We have therefore developed two conjugate gradient type methods and achieved much faster convergence. Finally, we have attempted to reduce the size of the computational domain by solving well-posed problems only in a strip between the outer and inner boundaries. We demonstrate that by alternating between Robin and Dirichlet conditions on the interior boundary, we can get a convergent iterative regularization method. Numerical experiments are used to illustrate the performance of the methods suggested. Populärvetenskaplig sammanfattningDet inversa problemet att rekonstruera akustiska eller elektromagnetiska fält i ett område från inexakta mätningar på en del av områdets randär viktigt för att t.ex. detektera källan för akustiskt brus. Sådana problem beskrivs av Helmholtz ekvation.I avhandlingen studerar vi sådana problem där Cauchydataär givna endast på en del av områdets rand och man vill finna lösningen i hela området. Vi vill alltså lösa Cauchyproblem för Helmholtz ekvation. Sådana problemär illa ställda, vilket innebär att små fel i Cauchydata kan medföra stora fel i lösningen. Särskilda regulariseringsmetoder måste därför användas för att ge stabilitet med avseende på mätfel.Avhandlingens syfteär att utveckla iterativa lösningsmetoder för dessa Cauchyproblem. Metoderna går ut ...

An alternating iterative procedure for the Cauchy problem for the Helmholtz equation

“…In the new algorithm, we alternate not only the Dirichlet-Neumann boundary conditions on Γ 0 and Γ 1 , but we also alternate boundary conditions on γ. More details about this modification can be found in [27].…”

“…Now consider the second term in the right-hand side of (27). Since we have assumed that χ j ∈ R, a µ (u n (χ j ), u n (χ j )) tends to zero as n → ∞.…”