Parameter Choice Rules for Discrete Ill-Posed Problems Based on Extrapolation Methods

Linear discrete inverse problems are common in many applicative fields. Regularization consists of substituting to the original ill-conditioned problem an approximated formulation depending on a parameter, which has to be chosen so that the new problem is well-conditioned and its solution is close enough to the ideal solution. When the parameter is discrete, like in the truncated singular value decomposition (TSVD) and in the generalized TSVD (TGSVD), one has to choose a vector out of a sequence. In this paper we explore the possibility to employ a sequence of extrapolated solutions to estimate the best parameter, as well as substituting to the regularized solution an extrapolated one. We investigate the use of three classical vector extrapolation methods, MPE (minimal polynomial extrapolation), RRE (reduced rank extrapolation), and VEA (vector epsilon algorithm). For the VEA method we also develop a new computational scheme which reduces memory storage and computing time. Numerical experiments compare the performance of the newly introduced approaches with other well-known methods.