RBFs are useful in scientific computing. In this thesis, we are interested in the positions of collocation points and RBF centers which causes the matrix for RBF interpolation singular and ill-conditioned. We explore the best bases by minimizing error function in supremum norm and root mean squares. We also use radial basis function to interpolate shifted data and find the best basis in certain sense. In the second part, we solve ill-posed problems by radial basis collocation method with different radial basis functions and various number of bases. If the solution is not unique, then the numerical solutions are different for different bases. To construct all the solutions, we can choose one approximation solution and add the linear combinations of the difference functions for various bases. If the solution does not exist, we show the numerical solution always fail to satisfy the origin equation.

The first processing stage in computational vision, also called early vision, consists in decoding 2D images in terms of properties of 3D surfaces. Early vision includes problems such as the recovery of motion and optical flow, shape from shading, surface interpolation, and edge detection. These are inverse problems, which are often ill-posed or ill-conditioned. We review here the relevant mathematical results on ill-posed and ill-conditioned problems and introduce the formal aspects of regularization theory in the linear and non-linear case. More general stochastic regularization methods are also introduced. Specific topics in early vision and their regularization are then analyzed rigorously, characterizing existence, uniqueness, and stability of solutions.

This is a thesis about how mollification can be used as a regularization method to reduce noise in ill-posed problems in order to make them well-posed. Ill-posed problems are problems where noise get magnified during the solution process. An example of this is how measurement errors increases with differentiation. To correct this we use mollification. Mollification is a regularization method that uses integration or weighted average to even out a noisy function. The different types of error that occurs when mollifying are the truncation error and the propagated data error. We are going to calculate these errors and see what affects them. An other thing worth investigating is the ability to differentiate a mollified function even if the function itself can not be differentiated. An application to mollification is a blood vessel problem in biometrics where the goal is to calculate the elasticity of the blood vessel’s wall. To do this measurements from the blood and the blood vessel are required, as well as equations for the calculations. The model used for the calculations is ill-posed with regard to specific variables which is why we want to apply mollification. Here we are also going to take a look at how the noise level affects the final result as well as the mollification radius.

No description available.

Doctor of Philosophy / Department of Mathematics / Alexander G. Ramm / Several methods for a stable solution to the equation $F(u)=f$ have been developed. Here $F:H\to H$ is an operator in a Hilbert space $H$, and we assume that noisy data $f_\delta$, $\|f_\delta-f\|\le \delta$, are given in place of the exact data $f$. When $F$ is a linear bounded operator, two versions of the Dynamical Systems Method (DSM) with stopping rules of Discrepancy Principle type are proposed and justified mathematically. When $F$ is a non-linear monotone operator, various versions of the DSM are studied. A Discrepancy Principle for solving the equation is formulated and justified. Several versions of the DSM for solving the equation are formulated. These methods consist of a Newton-type method, a gradient-type method, and a simple iteration method. A priori and a posteriori choices of stopping rules for these methods are proposed and justified. Convergence of the solutions, obtained by these methods, to the minimal norm solution to the equation $F(u)=f$ is proved. Iterative schemes with a posteriori choices of stopping rule corresponding to the proposed DSM are formulated. Convergence of these iterative schemes to a solution to the equation $F(u)=f$ is proved. This dissertation consists of six chapters which are based on joint papers by the author and his advisor Prof. Alexander G. Ramm. These papers are published in different journals. The first two chapters deal with equations with linear and bounded operators and the last four chapters deal with non-linear equations with monotone operators.

We present bounds on various quantities of interest regarding the central trajectory of a semi-definite program (SDP), where the bounds are functions of Renegar's condition number C(d) and other naturally-occurring quantities such as the dimensions n and m. The condition number C(d) is defined in terms of the data instance d = (A, b, C) for SDP; it is the inverse of a relative measure of the distance of the data instance to the set of ill-posed data instances, that is, data instances for which arbitrary perturbations would make the corresponding SDP either feasible or infeasible. We provide upper and lower bounds on the solutions along the central trajectory, and upper bounds on changes in solutions and objective function values along the central trajectory when the data instance is perturbed and/or when the path parameter defining the central trajectory is changed. Based on these bounds, we prove that the solutions along the central trajectory grow at most linearly and at a rate proportional to the inverse of the distance to ill-posedness, and grow at least linearly and at a rate proportional to the inverse of C(d)2 , as the trajectory approaches an optimal solution to the SDP. Furthermore, the change in solutions and in objective function values along the central trajectory is at most linear in the size of the changes in the data. All such bounds involve polynomial functions of C(d), the size of the data, the distance to ill-posedness of the data, and the dimensions n and m of the SDP.

A concept of a well-posed problem was initially introduced by J. Hadamard in 1923, who expressed the idea that every mathematical model should have a unique solution, stable with respect to noise in the input data. If at least one of those properties is violated, the problem is ill-posed (and unstable). There are numerous examples of ill- posed problems in computational mathematics and applications. Classical numerical algorithms, when used for an ill-posed model, turn out to be divergent. Hence one has to develop special regularization techniques, which take advantage of an a priori information (normally available), in order to solve an ill-posed problem in a stable fashion. In this thesis, theoretical and numerical investigation of Tikhonov's (variational) regularization is presented. The regularization parameter is computed by the discrepancy principle of Morozov, and a first-kind integral equation is used for numerical simulations.

The regularization of linear ill-posed problems is based on their conditional well-posedness when restricting the problem to certain classes of solutions. Given such class one may consider several related real-valued functions, which measure the wellposedness of the problem on such class. Among those functions the modulus of continuity is best studied. For solution classes which enjoy the additional feature of being star-shaped at zero, the authors develop a series of results with focus on continuity properties of the modulus of continuity. In particular it is highlighted that the problem is conditionally well-posed if and only if the modulus of continuity is right-continuous at zero. Those results are then applied to smoothness classes in Hilbert space. This study concludes with a new perspective on a concavity problem for the modulus of continuity, recently addressed by two of the authors in "Some note on the modulus of continuity for ill-posed problems in Hilbert space", 2011.

This work is devoted to the convergence analysis of a modified Runge-Kutta-type iterative regularization method for solving nonlinear ill-posed problems under a priori and a posteriori stopping rules. The convergence rate results of the proposed method can be obtained under Hölder-type source-wise condition if the Fréchet derivative is properly scaled and locally Lipschitz continuous. Numerical results are achieved by using the Levenberg-Marquardt and Radau methods.

Neste trabalho apresenta-se uma discussão geral sobre problemas inversos, problemas mal-postos c técnicas de regularização, visando sua aplicabilidade em problemas térmicos. Métodos numéricos especiais são discutidos para a solução de problemas que apresentam instabilidade em relação aos dados. Tais métodos baseiam-se na utilização de restrições ou informações adicionais sobre a solução procurada. O problema de determinação da condição inicial da equação do calor é resolvido numericamente através destas técnicas, particularmente a regularização de Tikhonov e o príncipio da máxima entropia conectados ao príncipio da discrepância de Morozov são utilizados. / In this work we present a general discussion on invcrse problems, ill-posed problems and regularization techniqucs, applying these techniques to thermal problcms. Special numerical methods are discusscd in order to solve problerns for which the solution is unstable under data perturbations. Such methods are based on the utilization of restrictions or additional information on thc solution. The problern of determining the initial condition of thc heat equation is numerically solved beyond thesc techniques, particularly thc T ikhonov regularization and thc maximum entropy principie connected to thc Morozov's discrepancy principie are used.