Superfast Tikhonov Regularization of Toeplitz Systems

TL;DR

This paper introduces a superfast algorithm for Tikhonov regularization of Toeplitz systems, reducing computational complexity to O(n (log n)^2) and ensuring stable solutions through numerical validation.

Contribution

It presents a novel extension-and-transformation technique that converts Tikhonov-regularized Toeplitz systems into tangential interpolation problems for efficient solving.

Findings

Algorithm achieves O(n (log n)^2) complexity

Numerical simulations confirm stability and efficiency

Method outperforms previous algorithms in speed

Abstract

Toeplitz-structured linear systems arise often in practical engineering problems. Correspondingly, a number of algorithms have been developed that exploit Toeplitz structure to gain computational efficiency when solving these systems. The earliest "fast" algorithms for Toeplitz systems required O(n^2) operations, while more recent "superfast" algorithms reduce the cost to O(n (log n)^2) or below. In this work, we present a superfast algorithm for Tikhonov regularization of Toeplitz systems. Using an "extension-and-transformation" technique, our algorithm translates a Tikhonov-regularized Toeplitz system into a type of specialized polynomial problem known as tangential interpolation. Under this formulation, we can compute the solution in only O(n (log n)^2) operations. We use numerical simulations to demonstrate our algorithm's complexity and verify that it returns stable solutions.