Linear estimation is a fundamental branch of signal processing that deals with estimating the values of parameters from a corrupted measured data. Throughout the years, several optimization criteria have been used to achieve this task. The most astonishing attempt among theses is the linear least-squares. Although this criterion enjoyed a wide popularity in many areas due to its attractive properties, it appeared to suffer from some shortcomings. Alternative optimization criteria, as a result, have been proposed. These new criteria allowed, in one way or another, the incorporation of further prior information to the desired problem. Among theses alternative criteria is the regularized least-squares (RLS). In this thesis, we propose two new algorithms to find the regularization parameter for linear least-squares problems. In the constrained perturbation regularization algorithm (COPRA) for random matrices and COPRA for linear discrete ill-posed problems, an artificial perturbation matrix with a bounded norm is forced into the model matrix. This perturbation is introduced to enhance the singular value structure of the matrix. As a result, the new modified model is expected to provide a better stabilize substantial solution when used to estimate the original signal through minimizing the worst-case residual error function. Unlike many other regularization algorithms that go in search of minimizing the estimated data error, the two new proposed algorithms are developed mainly to select the artifcial perturbation bound and the regularization parameter in a way that approximately minimizes the mean-squared error (MSE) between the original signal and its estimate under various conditions. The first proposed COPRA method is developed mainly to estimate the regularization parameter when the measurement matrix is complex Gaussian, with centered unit variance (standard), and independent and identically distributed (i.i.d.) entries. Furthermore, the second proposed COPRA method deals with discrete ill-posed problems when the singular values of the linear transformation matrix are decaying very fast to a significantly small value. For the both proposed algorithms, the regularization parameter is obtained as a solution of a non-linear characteristic equation. We provide a details study for the general properties of these functions and address the existence and uniqueness of the root. To demonstrate the performance of the derivations, the first proposed COPRA method is applied to estimate different signals with various characteristics, while the second proposed COPRA method is applied to a large set of different real-world discrete ill-posed problems. Simulation results demonstrate that the two proposed methods outperform a set of benchmark regularization algorithms in most cases. In addition, the algorithms are also shown to have the lowest run time.

This work is about solving an inverse problem for an elliptic equation. An inverse problem is often ill-posed, which means that a small measurement error in data can yield a vigorously perturbed solution. Regularization is a way to make an ill-posed problem well-posed and thus solvable. Two important tools to determine if a problem is well-posed or not are norms and convergence. With help from these concepts, the error of the reg- ularized function can be calculated. The error between this function and the exact function is depending on two error terms. By solving the problem with an elliptic equation, a linear operator is eval- uated. This operator maps a given function to another function, which both can be found in the solution of the problem with an elliptic equation. This opera- tor can be seen as a mapping from the given function’s Fourier-sine coefficients onto the other function’s Fourier-sine coefficients, since these functions are com- pletely determined by their Fourier-sine series. The regularization method in this thesis, uses a chosen number of Fourier-sine coefficients of the function, and the rest are set to zero. This regularization method is first illustrated for a simpler problem with Laplace’s equation, which can be solved analytically and thereby an explicit parameter choice rule can be given. The goal with this work is to show that the considered method is a reg- ularization of a linear operator, that is evaluated when the problem with an elliptic equation is solved. In the tests in Chapter 3 and 4, the ill-posedness of the inverse problem is illustrated and that the method does behave like a regularization is shown. Also in the tests, it can be seen how many Fourier-sine coefficients that should be considered in the regularization in different cases, to make a good approximation. / Det här arbetet handlar om att lösa ett inverst problem för en elliptisk ekvation. Ett inverst problem är ofta illaställt, vilket betyder att ett litet mätfel i data kan ge en kraftigt förändrad lösning. Regularisering är ett tillvägagångssätt för att göra ett illaställt problem välställt och således lösbart. Två viktiga verktyg för att bestämma om ett problem är välställt eller inte är normer och konvergens. Med hjälp av dessa begrepp kan felet av den regulariserade lösningen beräknas. Felet mellan den lösningen och den exakta är beroende av två feltermer. Genom att lösa problemet med den elliptiska ekvationen, så är en linjär operator evaluerad. Denna operator avbildar en given funktion på en annan funktion, vilka båda kan hittas i lösningen till problemet med en elliptisk ekva- tion. Denna operator kan ses som en avbildning från den givna funktions Fouri- ersinuskoefficienter på den andra funktionens Fouriersinuskoefficienter, eftersom dessa funktioner är fullständigt bestämda av sina Fouriersinusserier. Regularise- ringsmetoden i denna rapport använder ett valt antal Fouriersinuskoefficienter av funktionen, och resten sätts till noll. Denna regulariseringsmetod illustreras först för ett enklare problem med Laplaces ekvation, som kan lösas analytiskt och därmed kan en explicit parametervalsregel anges. Målet med detta arbete är att visa att denna metod är en regularisering av den linjära operator som evalueras när problemet med en elliptisk ekvation löses. I testerna i kapitel 3 och 4, illustreras illaställdheten av det inversa problemet och det visas att metoden beter sig som en regularisering. I testerna kan det också ses hur många Fouriersinuskoefficienter som borde betraktas i regulariseringen i olika fall, för att göra en bra approximation.

This paper deals with the characterization of multiplication operators, especially with its behavior in the ill-posed case. We want to classify the different types and degrees of ill-posedness. We give some connections between this classification and regularization methods.

This thesis presents how Tikhonov’s regularization can be used to solve an inverse problem of Helmholtz equation inside of a rectangle. The rectangle will be met with both Neumann and Dirichlet boundary conditions. A linear operator containing a Fourier series will be derived from the Helmholtz equation. Using this linear operator, an expression for the inverse operator can be formulated to solve the inverse problem. However, the inverse problem will be found to be ill-posed according to Hadamard’s definition. The regularization used to overcome this ill-posedness (in this thesis) is Tikhonov’s regularization. To compare the efficiency of this inverse operator with Tikhonov’s regularization, another inverse operator will be derived from Helmholtz equation in the partial frequency domain. The inverse operator from the frequency domain will also be regularized with Tikhonov’s regularization. Plots and error measurements will be given to understand how accurate the Tikhonov’s regularization is for both inverse operators. The main focus in this thesis is the inverse operator containing the Fourier series. A series of examples will also be given to strengthen the definitions, theorems and proofs that are made in this work.

In 1991, A. Máté precisely calculated the first-order asymptotic behavior of the sequence of Christoffel functions associated with Szego measures on the unit circle. Our principal goal is the abstraction of his result in two directions: We compute the translated asymptotics, limn λn(µ, x + a/n), and obtain, as a corollary, a universality limit for the fairly broad class of Szego weights. Finally, we prove Máté’s result for measures supported on smooth curves in the plane. Our proof of the latter derives, in part, from a precise estimate of certain weighted means of the Faber polynomials associated with the support of the measure. Finally, we investigate a variety of applications, including two novel applications to ill-posed problems in Hilbert space and the mean ergodic theorem.

No description available.

We study a Cauchy problem arising in uid mechanics, involving the socalled stationary generalized Stokes system, where one should recover the ow from boundary measurements. The problem is ill-posed in the sense that the solution does not depend continuously on data. Two iterative procedures for solving this problem are proposed and investigated. These methods are regularizing and in each iteration one solves a series of well-posed problems obtained by changing the boundary conditions. The advantage with this approach, is that these methods place few restrictions on the domain and on the coefficients of the problem. Also the structure of the equation is preserved. Well-posedness of the problems used in these procedures is demonstrated, i.e., that the problems have a unique solution that depends continuously on data. Since we have numerical applications in mind, we demonstrate well-posedness for the case when boundary data is square integrable. We give convergence proofs for both of these methods.

The aim of this report is to study how machine learning can be used to reconstruct 2 dimensional computed tomography images from limited angle data. This could be used in a variety of applications where either the space or timeavailable for the CT scan limits the acquired data.In this study, three different types of models are considered. The first model uses filtered back projection (FBP) with a single learned filter, while the second uses a combination of multiple FBP:s with learned filters. The last model instead uses an FNO (Fourieer Neural Operator) layer to both inpaint and filter the limited angle data followed by a backprojection layer. The quality of the reconstructions are assessed both visually and statistically, using PSNR and SSIM measures.The results of this study show that while an FBP-based model using one or more trainable filter(s) can achieve better reconstructions than ones using an analytical Ram-Lak filter, their reconstructions still fail for small angle spans. Better results in the limited angle case can be achieved using the FNO-basedmodel.

No description available.

No description available.