Iterated regularization methods for solving inverse problems

Typical inverse problems are ill-posed which frequently leads to difficulties in calculatingnumerical solutions. A common approximation method to solve ill-posed inverse problemsis iterated Tikhonov-Lavrentiev regularization.We examine iterated Tikhonov-Lavrentiev regularization and show that, in the casethat regularity properties are not globally satisfied, certain projections of the error converge faster than the theoretical predictions of the global error. We also explore the sensitivity of iterated Tikhonov regularization to the choice of the regularization parameter. We show that by calculating higher order sensitivities we improve the accuracy. We present a simple to implement algorithm that calculates the iterated Tikhonov updates and the sensitivities to the regularization parameter. The cost of this new algorithm is one vector addition and one scalar multiplication per step more than the standard iterated Tikhonov calculation.In considering the inverse problem of inverting the Helmholz-differential filter (with filterradius δ), we propose iterating a modification to Tikhonov-Lavrentiev regularization (withregularization parameter α and J iteration steps). We show that this modification to themethod decreases the theoretical error bounds from O(α(δ^2 +1)) for Tikhonov regularizationto O((αδ^2)^(J+1) ). We apply this modified iterated Tikhonov regularization method to theLeray deconvolution model of fluid flow. We discretize the problem with finite elements inspace and Crank-Nicolson in time and show existence, uniqueness and convergence of thissolution.We examine the combination of iterated Tikhonov regularization, the L-curve method,a new stopping criterion, and a bootstrapping algorithm as a general solution method inbrain mapping. This method is a robust method for handling the difficulties associated withbrain mapping: uncertainty quantification, co-linearity of the data, and data noise. Weuse this method to estimate correlation coefficients between brain regions and a quantified performance as well as identify regions of interest for future analysis

An inverse problem for a one-dimensional time-fractional diffusion problem

Over the last two decades, anomalous diusion processes in which the mean squares variance grows slower or faster than that in a Gaussian process have found many applications. At a macroscopic level, these processes are adequately described by fractional dierential equations, which involves fractional derivatives in time or/and space. The fractional derivatives describe either history mechanism or long range interactions of particle motions at a microscopic level. The new physics can change dramatically the behavior of the forward problems. For example, the solution operator of the time fractional diusion diusion equation has only limited smoothing property, whereas the solution for the space fractional diusion equation may contain weakly singularity. Naturally one expects that the new physics will impact related inverse problems in terms of uniqueness, stability, and degree of ill-posedness. The last aspect is especially important from a practical point of view, i.e., stably reconstructing the quantities of interest. In this paper, we employ a formal analytic and numerical way, especially the two-parameter Mittag-Leer function and singular value decomposition, to examine the degree of ill-posedness of several \classical" inverse problems for fractional dierential equations involving a Djrbashian-Caputo fractional derivative in either time or space, which represent the fractional analogues of that for classical integral order dierential equations. We discuss four inverse problems, i.e., backward fractional diusion, sideways problem, inverse source problem and inverse potential problem for time fractional diusion, and inverse Sturm-Liouville problem, Cauchy problem, backward fractional diusion and sideways problem for space fractional diusion. It is found that contrary to the wide belief, the in uence of anomalous diusion on the degree of ill-posedness is not denitive: it can either signicantly improve or worsen the conditioning of related inverse problems, depending crucially on the specic type of given data and quantity of interest. Further, the study exhibits distinct new features of \fractional" inverse problems, and a partial list of surprising observations is given below. (a) Classical backward diusion is exponentially ill-posed, whereas time fractional backward diusion is only mildly ill-posed in the sense of norms on the domain and range spaces. However, this does not imply that the latter always allows a more eective reconstruction. (b) Theoretically, the time fractional sideways problem is severely ill-posed like its classical counterpart, but numerically can be nearly well-posed. (c) The classical Sturm-Liouville problem requires two pieces of spectral data to uniquely determine a general potential, but in the fractional case, one single Dirichlet spectrum may suce. (d) The space fractional sideways problem can be far more or far less ill-posed than the classical counterpart, depending on the location of the lateral Cauchy data. In many cases, the precise mechanism of these surprising observations is unclear, and awaits further analytical and numerical exploration, which requires new mathematical tools and ingenuities. Further, our ndings indicate fractional diusion inverse problems also provide an excellent case study in the dierences between theoretical ill-conditioning involving domain and range norms and the numerical analysis of a nite-dimensional reconstruction procedure. Throughout we will also describe known analytical and numerical results in the literature

Numerical Treatment of Non-Linear singular pertubation problems

Magister Scientiae - MScThis thesis deals with the design and implementation of some novel numerical methods for non-linear singular pertubations problems (NSPPs). It provide a survey of asymptotic and numerical methods for some NSPPs in the past decade. By considering two test problems, rigorous asymptotic analysis is carried out. Based on this analysis, suitable numerical methods are designed, analyzed and implemented in order to have some relevant results of physical importance. Since the asymptotic analysis provides only qualitative information, the focus is more on the numerical analysis of the problem which provides the quantitative information.South Afric

Extracting discontinuity using the probe and enclosure methods

This is a review article on the development of the probe and enclosure methods from past to present, focused on their central ideas together with various applications.Comment: 121 pages, minor modificatio