Linköping Studies in Science and Technology

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

Iterative Methods for Solving the Cauchy Problem for the Helmholtz Equation

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.An invalid ISRN (LIU-TEK-LIC-2012:15) is stated on page 2. The ISRN belongs to the Licentiate thesis, published in 2012.</p

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

Let  be a bounded domain in Rn with a Lipschitz boundary Г divided into two parts Г0 and Г1 which do not intersect one another and have a common Lipschitz boundary. We consider the following Cauchy problem for the Helmholtz equation: where k, the wave number, is a positive real constant, аv denotes the outward normal derivative, and f and g are specified Cauchy data on Г0. This problem is ill–posed in the sense that small errors in the Cauchy data f and g may blow up and cause a large error in the solution. Alternating iterative algorithms for solving this problem are developed and studied. These algorithms are based on the alternating iterative schemes suggested by V.A. Kozlov and V. Maz’ya for solving ill–posed problems. Since these original alternating iterative algorithms diverge for large values of the constant k2 in the Helmholtz equation, we develop a modification of the alterating iterative algorithms that converges for all k2. We also perform numerical experiments that confirm that the proposed modification works

A Data Assimilation Approach to Coefficient Identification

The thermal conductivity properties of a material can be determined experimentally by using temperature measurements taken at specified locations inside the material. We examine a situation where the properties of a (previously known) material changed locally. Mathematically we aim to find the coefficient k(x) in the stationary heat equation (kTx)x = 0;under the assumption that the function k(x) can be parametrized using only a few degrees of freedom. The coefficient identification problem is solved using a least squares approach; where the (non-linear) control functional is weighted according to the distribution of the measurement locations. Though we only discuss the 1D case the ideas extend naturally to 2D or 3D. Experimentsdemonstrate that the proposed method works well.      

Adaptive optimization algorithm for the computational design of nanophotonic structures

We consider the problem of the construction of the nanophotonic structures of arbitrary geometry with prescribed desired properties. We illustrate the efficiency of our adaptive optimization algorithm on the construction of nanophotonic structure in two dimensions

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

N.B.: When citing this work, cite the original article. This is an electronic version of an article published in

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

We present a modification of the alternating iterative method, which was introduced by V.A. Kozlov and V. Maz’ya in for solving the Cauchy problem for the Helmholtz equation in a Lipschitz domain. The method is implemented numerically using the finite difference method

Numerical Solution of the Cauchy Problem for the Helmholtz Equation

The Cauchy problem for the Helmholtz equation appears in applications related to acoustic or electromagnetic wave phenomena. The problem is ill–posed in the sense that the solution does not depend on the data in a stable way. In this paper we give a detailed study of the problem. Specifically we investigate how the ill–posedness depends on the shape of the computational domain and also on the wave number. Furthermore, we give an overview over standard techniques for dealing with ill–posed problems and apply them to the problem

Relationships among anxiety, perceived stress, and burnout in young athletes

KOPSAVILKUMS Pētījuma mērķis bija noskaidrot, vai pastāv sakarība starp sportistu izdegšanas trim dimensijām – fizisko nogurumu, sporta vērtības samazināšanos un pazeminātu sniegumu - ar stresa, trauksmes līmeni, tā saistību ar traumatisma līmeni. Pētījumā piedalījās dažādu sporta veidu 104 abu dzimumu sportisti, Latvijas izlases dalībnieki, vecumā no astoņpadsmit līdz divdesmit gadiem. Pētījumā tika izmantota DASS-21 aptauja bez depresijas skalas un Sportistu izdegšanas aptauja. Aptauja tika papildināta ar četriem traumu smaguma pakāpi precizējošiem jautājumiem. Tika pierādīta cieši nozīmīga sakarība starp stresu un trauksmi, vidēji nozīmīga starp sportistu izdegšanas trim dimensijām un stresu, trauksmi. Netika apstiprināta fizisko traumu sakarība ar sportistu pašsajūtas aptaujas elementiem, stresu vai trauksmi. Korelācijas atklāja arī vidēji ciešu emocionālo traumu sakarību ar dzimumu, fizisko nogurumu, stresu un trauksmi. Atslēgas vārdi : stress, trauksmes līmenis, jauno sportistu pašsajūta, aptaujas, sakarība, traumas.SUMMARY The aim of the study was to find out whether there is a relationship between the three dimensions of athletes' well-being - physical fatigue, reduced sports value and reduced performance with stress, anxiety, how it relates to the level of trauma. The study involved 104 athletes of different sports, participants of both sexes, from 18 to twenty years of age. The study used the DASS-21 survey, with the depression subdivision removed, and the Athlete Well-Being Survey. Four questions were added to clarify the severity of the injuries. There was a strong relationship between stress and anxiety, a moderately significant relationship between the three dimensions of athletes' well-being and stress and anxiety. There was no correlation between physical injuries and the elements of the athletes' well-being survey, stress, anxiety. Showed moderate emotional correlation with gender, physical fatigue, stress, and anxiety. Keywords: stress, anxiety level, well-being of young athletes, surveys, correlation, injurie