In this thesis we study the Cauchy problem for elliptic equations. It arises in many areas of application in science and engineering as a problem of reconstruction of solutions to elliptic equations in a domain from boundary measurements taken on a part of the boundary of this domain. The Cauchy problem for elliptic equations is known to be ill-posed.

We use an iterative regularization method based on alternatively solving a sequence of well-posed mixed boundary value problems for the same elliptic equation. This method, based on iterations between Dirichlet-Neumann and Neumann-Dirichlet mixed boundary value problems was first proposed by Kozlov and Maz’ya [13] for Laplace equation and Lame’ system but not Helmholtz-type equations. As a result different modifications of this original regularization method have been proposed in literature. We consider the Robin-Dirichlet iterative method proposed by Mpinganzima et.al [3] for the Cauchy problem for the Helmholtz equation in bounded domains.

We demonstrate that the Robin-Dirichlet iterative procedure is convergent for second order elliptic equations with variable coefficients provided the parameter in the Robin condition is appropriately chosen. We further investigate the convergence of the Robin-Dirichlet iterative procedure for the Cauchy problem for the Helmholtz equation in a an unbounded domain. We derive and analyse the necessary conditions needed for the convergence of the procedure.

In the numerical experiments, the precise behaviour of the procedure for different values of k² in the Helmholtz equation is investigated and the results show that the speed of convergence depends on the choice of the Robin parameters, μ₀ and μ₁. In the unbounded domain case, the numerical experiments demonstrate that the procedure is convergent provided that the domain is truncated appropriately and the Robin parameters, μ₀ and μ₁ are also chosen appropriately.

In this thesis, we study the Cauchy problem for an elliptic equation. We use Dirichlet-Robin iterations for solving the Cauchy problem. This allows us to include in our consideration elliptic equations with variable coefficient as well as Helmholtz type equations. The algorithm consists of solving mixed boundary value problems, which include the Dirichlet and Robin boundary conditions. Convergence is achieved by choice of parameters in the Robin conditions.

We have also reformulated the Cauchy problem for the Helmholtz equation as an operator equation. We investigate the conditions under which this operator equation is well-defined. Furthermore, we have also discussed possible extensions to the case where the Helmholtz operator is replaced by non-symmetric differential operators by using similar operator equations and model problems which are used for symmetric differential operators. We have observed that the Dirichlet - Robin iterations are equivalent to the classical Landweber iterations. Having formulated the problem in terms of an operator equation is an advantage since it lets us to implement more sophisticated iterative methods based on Krylov subspaces. In particular, we consider the Conjugate gradient method (CG) and the Generalized minimal residual method (GMRES). The numerical results shows that all the methods work well.

Cauchy problems for elliptic equations arise in applications in science and engineering. These problems often involve finding important information about an elliptical system from indirect or incomplete measurements. Cauchy problems for elliptic equations are known to be disadvantaged in the sense that a small pertubation in the input can result in a large error in the output. Regularization methods are usually required in order to be able to find stable solutions. In this thesis we study the Cauchy problem for elliptic equations in both bounded and unbounded domains using iterative regularization methods. In Paper I and II, we focus on an iterative regularization technique which involves solving a sequence of mixed boundary value well-posed problems for the same elliptic equation. The original version of the alternating iterative technique is based on iterations alternating between Dirichlet-Neumann and Neumann-Dirichlet boundary value problems. This iterative method is known to possibly work for Helmholtz equation. Instead we study a modified version based on alternating between Dirichlet-Robin and Robin-Dirichlet boundary value problems. First, we study the Cauchy problem for general elliptic equations of second order with variable coefficients in a limited domain. Then we extend to the case of unbounded domains for the Cauchy problem for Helmholtz equation. For the Cauchy problem, in the case of general elliptic equations, we show that the iterative method, based on Dirichlet-Robin, is convergent provided that parameters in the Robin condition are chosen appropriately. In the case of an unbounded domain, we derive necessary, and sufficient, conditions for convergence of the Robin-Dirichlet iterations based on an analysis of the spectrum of the Laplacian operator, with boundary conditions of Dirichlet and Robin types.

In the numerical tests, we investigate the precise behaviour of the Dirichlet-Robin iterations, for different values of the wave number in the Helmholtz equation, and the results show that the convergence rate depends on the choice of the Robin parameter in the Robin condition. In the case of unbounded domain, the numerical experiments show that an appropriate truncation of the domain and an appropriate choice of Robin parameter in the Robin condition lead to convergence of the Robin-Dirichlet iterations.

In the presence of noise, additional regularization techniques have to implemented for the alternating iterative procedure to converge. Therefore, in Paper III and IV we focus on iterative regularization methods for solving the Cauchy problem for the Helmholtz equation in a semi-infinite strip, assuming that the data contains measurement noise. In addition, we also reconstruct a radiation condition at infinity from the given Cauchy data. For the reconstruction of the radiation condition, we solve a well-posed problem for the Helmholtz equation in a semi-infinite strip. The remaining solution is obtained by solving an ill-posed problem. In Paper III, we consider the ordinary Helmholtz equation and use seperation of variables to analyze the problem. We show that the radiation condition is described by a non-linear well-posed problem that provides a stable oscillatory solution to the Cauchy problem. Furthermore, we show that the ill–posed problem can be regularized using the Landweber’s iterative method and the discrepancy principle. Numerical tests shows that the approach works well.

Paper IV is an extension of the theory from Paper III to the case of variable coefficients. Theoretical analysis of this Cauchy problem shows that, with suitable bounds on the coefficients, can iterative regularization methods be used to stabilize the ill-posed Cauchy problem.

In this thesis, we study Cauchy problems for the elliptic and degenerate elliptic equations. These problems are ill-posed. We split the boundary of the domain into two parts. On one of them, say Γ₀, we have available Cauchy data and on remaining part Γ₁ we introduce unknown Robin data. To construct the operator equation which replaces our Cauchy problem we use two boundary value problems (BVP). The first one is the mixed BVP with Robin condition on Γ₁ and with Dirichlet condition on Γ₀ and the second BVP with Dirichlet Data on Γ₁ and with Robin data on Γ₀. The well–posedness of these problems is achieved by an appropriate choice of parameters in Robin boundary conditions. The first Dirichlet–Robin BVP is used to construct the operator equation replacing the Cauchy problem and the second Robin–Dirichlet problem for adjoint operator. Using these problems we can apply various regularization methods for stable reconstruction of the solution. In Paper I, the Cauchy problem for the elliptic equation with variable coefficients, which includes Helmholtz type equations, is analyzed. A proof showing that the Dirichlet–Robin alternating algorithm is convergent is given, provided that the parameters in the Robin conditions are chosen appropriately. Numerical experiments that shows the behaviour of the algorithm are given. In particular, we show how the speed of convergence depends on the choice of Robin parameters. 

In Paper II, the Cauchy problem for the Helmholtz equation, for moderate wave numbers k², is considered. The Cauchy problem is reformulated as an operator equation and iterative method based on Krylov subspaces are implemented. The aim is to achieve faster convergence in comparison to the Alternating algorithm from the previous paper. Methods such as the Landweber iteration, the Conjugate gradient method and the generalized minimal residual method are considered. We also discuss how the algorithms can be adapted to also cover the case of non–symmetric differential operators. 

In Paper III, we look at a steady state heat conduction problem in a thin plate. The plate connects two cylindrical containers and fix their relative positions. A two dimensional mathematical model of heat conduction in the plate is derived. Since the plate has sharp edges on the sides we obtained a degenerate elliptic equation. We seek to find the temperature on the interior cylinder by using data on the exterior cylinder. We reformulate the Cauchy problem as an operator equation, with a compact operator. The operator equation is solved using the Landweber method and the convergence is investigated. 

In Paper IV, the Cauchy problem for a more general degenerate elliptic equation is considered. We stabilize the computations using Tikhonov regularization. The normal equation, in the Tikhonov algorithm, is solved using the Conjugate gradient method. The regularization parameter is picked using either the L–curve or the Discrepancy principle. 

In all papers, numerical examples are given where we solve the various boundary value problems using a finite difference scheme. The results show that the suggested methods work quite well. 

I denna avhandling studerar vi Cauchy-problem för elliptiska och degenererade elliptiska ekvationer. Dessa problem är illa ställda. Vi delar upp randen till området i två delar. På en av dem, säg Γ0, har vi Cauchy–data tillgängligt, och på den återstående delen, Γ1 introducerar vi okända Robin-villkor.

För att konstruera operatorekvationen som ersätter vårt Cauchy-problem använder vi två randvärdesproblem (BVP). Det första problemet är ett BVP med Robin–villkor på Γ1 och Dirichlet–villkor på Γ0. Det andra problemet är ett BVP med Dirichlet–data på Γ1 och med Robin–data på Γ0. Dessa problem är välställda om parametrar i Robinvillkoren väljs lämpligt. Det första Dirichlet–Robin problemet används för att konstruera operatorekvationen som ersätter Cauchy problemet, och det andra Robin–Dirichlet-problemet används för att definiera den adjungerande operatorn. Vi kan sedan tillämpa olika regulariseringsmetoder och återskapa lösningen till problemet på ett stabilt sätt.

I Artikel I analyseras Cauchy-problemet för den elliptiska ekvationen med variabla koefficienter, vilket inkluderar ekvationer av Helmholtz-typ. Ett bevis som visar att den Dirichlet–Robin alternerande algoritmen är konvergent ges, förutsatt att parametrarna i Robin–villkoren väljs på lämpligt sätt. Numeriska experiment som illustrerar algoritmens beteende ges. I synnerhet visar vi hur konvergen-shastigheten beror på valet av Robin-parametrar.

I Artikel II behandlas Cauchy–problemet för Helmholtz ekvation, för medelstora vågtal k2. Cauchy–problemet omformuleras som en operatorekvation och iterativa metoder, baserade på Krylov rum, implementeras. Syftet är att uppnå snabbare konvergens jämfört med den ursprungliga alternerande algoritmen som studerades i den föregående artikeln. Vi diskuterar också hur algoritmerna kan anpassas till fallet med icke-symmetriska differentialoperatorer.

I Artikel III tittar vi på ett stationärt värmeledningsproblem i en tunn platta. Plattan sammanbinder två cylindriska behållare och fixerar deras relativa position. En tvådimensionell matematisk modell av värmeledning i plattan härleds. Eftersom plattan har vassa kanter på sidorna får vi en degenererad elliptisk ekvation. Vi försöker hitta temperaturen på den inre cylindern genom att använda data på den yttre cylindern. Vi omformulerar Cauchy–problemet som en operatorekvation, med en kompakt operator. Operatorekvationen löses med Landwebers metod och konvergensen undersöks.

I Artikel IV behandlas Cauchy problemet för en mer allmän degenererad elliptisk ekvation. Vi stabiliserar beräkningarna med hjälp av Tikhonov–regularisering, där normal ekvationen löses med Konjugerade gradientmetoden. Reguleringsparametern väljs med antingen L–kurva eller Diskrepansprincipen.

I alla artiklar ges numeriska exempel där vi löser de olika randvärdesproblemen med hjälp av finita differenser. Resultaten visar att de föreslagna metoderna fungerar ganska bra.