A SENSITIVITY ANALYSIS FOR LEAST-SQUARES ILL-POSED PROBLEMS USING THE HAAR BASIS

This paper applies the wavelet theory to provide some information about the stability and resolution limit of inverse problems. A sensitivity analysis far one-dimensional least-squares ill-posed problems is done by using the Haar basis. Three mappings from the parameter space L(2)(R) into the observation space L(2)(R) are considered. The sensitivity in the direction of a Haar function of width h is O(h(gamma)) (which approximates the lower bound of sensitivity of the mapping), where gamma > 0, implying that the sensitivity in the Haar basis directions has a rapid decay when h --> 0. This gives a quantitative relation between the sensitivity of the mapping and the parameter resolution scale. In this way, the Haar basis decomposes the parameters to be estimated into parts with different sensitivity order, providing an easy method to identify the suitability of multiresolution algorithms for inverse problems and to choose a convenient discretization size h.