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

Fast Calculation of Flow Ensembles

Computing Ensembles occurs frequently in the simulation of complex flows to increase forecasting skill, quantify uncertainty and estimate flow sensitivity. The main issue with ensemble calculation is its high demand of computer resources vs. the limited computer resources existing. Generally computing a large ensemble is prohibitive due to the high computational cost of numerical simulation of nonlinear dynamical systems. Moreover, to compute ensembles of moderate/small size, resolution is very often sacrificed to reduce computation time. In this thesis, we study an efficient ensemble simulation algorithm that can reduce the computing cost significantly making computing a large ensemble or an ensemble of high resolution possible. The motivation for the new algorithm is that for linearly implicit methods, the linear solve is a large contributor to overall complexity and it is far cheaper in both storage and solution time to solve linear systems with the same coefficient matrix than with different coefficient matrices. We present this algorithm with different ensemble time stepping methods. These methods are carefully derived and both theoretically and numerically investigated. Computing an ensemble simultaneously allows each realization to access ensemble data and the use of means and fluctuations in numerical regularizations for each realization. We put forth two ensemble eddy viscosity regularizations that remove severe timestep condition for high Reynolds number flows. The study of the ensemble eddy viscosity regularizations also suggests reconsidering an old but not as well developed definition of the mixing length. This mixing length vanishes at solid walls without van Driest damping, increases stability and improves flow predictions in our preliminary tests. The goal of conventional turbulence models is to produce a model that accurately predicts time averaged or ensemble averaged flow statistics. In this thesis, we develop a new family of ensemble based turbulence models and study their convergence by analyzing the evolution of model variance. For these new turbulence models from the calculated ensemble (at low cost), the kinetic energy in fluctuations can be directly calculated without additional modeling, reducing the computing cost while increasing the physical fidelity of the models