12/01/1994 09:00 AM
Computer Science
This dissertation proposes, develops and evaluates multiresolution optimization methods for estimation of distributed parameters of mathematical models. The methods are based on the assumption that the distributed parameters are continuous almost everywhere in the defined field. The main idea in employing multiresolution optimization is to give priority to large-scale characteristics of the parameter distribution over smaller-scale ones in the estimation process. This allows the overall structure of the distribution to be found more quickly, which results in rapid approach of the estimation to the true distribution. This in turn allows more reliable search for the details. The multiresolution optimization method consists of a local search method and a multiresolution scheme that controls the resolution of the search. This dissertation employs the conjugate gradient method as a local search method, and the discrete Fourier transform and the Haar wavelet transform as a multiresolution scheme. Since the coefficients of these transforms are inherently sorted in frequency or scale, multiresolution estimation can be realized by estimating the transform coefficients with some controlled weights and inversely transforming the coefficients back to the parameter distribution. Two methods of controlling the resolution are devised. One is the \"step scheme\" in which the threshold frequency for a low-pass filter steps up every time the search converges. The other method is the \"weight scheme\" in which fixed weights are assigned so that the coefficients of low frequencies get larger weights than the coefficients of higher frequencies. This dissertation gives a proof that estimating the transform coefficients in either scheme is equivalent to directly estimating the desired parameter distribution by using the gradient that is filtered by the transform. In other words, we can realize a multiresolution optimization by simply filtering the gradient in a multiresolution manner to control the resolution of the search direction. The developed methods are evaluated in simulation of the so-called electrical impedance tomography, which is one of many potential applications. It is shown that the multiresolution optimization yields better estimates more rapidly than the conventional single-resolution method. Notes: Ph.D. Thesis