BV Regularization Method For The Function Reconstruction From Noisy Local Averages

Function reconstruction from noisy local averages has important applications inenvironment science, statistics, image processing, computational mechanics and someother fields. Due to the ill-posedness, it is not easy to construct efficient numericalmethods for such a problem, and an approach is based on regularization. H1 semi-norm regularization method has been developed, but it can not deal with the discon-tinuous function effectively.To overcome the above difficulty, a BV regularization method is proposed andstudied in this thesis. To this end, the regularization term is taken as the TV-seminormof a function, leading to a regularization method with desirable performance in recon-structing discontinuous functions. Existence and uniqueness of the solution of thismethod have been established and error analysis has also been developed. After finiteelement discretization, the method can be reformulated as an optimization problem:min{ Au ? b 22 +α||Bu||1,2}. There are several methods for attacking this opti-mization problem, for example, Split Bregman, BOS, PDHG, etc. The Split Bregmanmethod is used in this thesis. Numerical experiments are given to show the accuracyand behavior of the method. The numerical comparison between this method andH1 semi-norm regularization method has been done, and the relevant conclusions aresummarized.