The effectiveness of band-Toeplitz preconditioners: A survey

Abstract

In many applications such as signal processing [26], differential equations [40], image restoration [20] and statistics [22] we have to solve n × n Hermitian Toeplitz linear systems A_n(f)x=b where the symbol f, the generating function, is an L¹ function and the entries of A_n(f) along the k- th diagonal coincide with the k-th Fourier coefficient a _k of f. When the generating function has (essential) zeros of even orders and is nonnegative, several very satisfactory band-Toeplitz preconditioners have been proposed [9, 18, 12, 32] leading to optimal and superoptimal preconditioned conjugate gradient (PCG) methods with a total cost of O(n log n) ops, even in the ill-conditioned case where other celebrated techniques fail [17]. More recently these preconditioners have been successfully extended to the block [27] and nondefinite [29] cases as well as to the case of zeros of any (also noninteger) orders [34]. The latter extension is very important because it concerns with a lot of practical situations not considered before. Therefore, the only relevant criticism to this approach continues to be the assumption that, at least, the sign of f, the position of the zeros and their order are known. In fact, in some applications, it is possible to assume the availability of this information but, in other fields, e.g. image restoration, this just results an ideal hypothesis. In the latter case, very recently, we have introduced [38] a practical and economical criterion in order to discover all the needed analytical properties of f by using as data input the coefficients a_k. In such way, we are convinced that the set of all the band-Toeplitz preconditioners based strategies can become much more attractive from an applicative point of view.