Iterative exponential filtering for large discrete ill-posed problems

Abstract

We describe a new iterative method for the solution of large, very ill-conditioned linear systems of equations that arise when discretizing linear ill-posed problems. The right-hand side vector represents the given data and is assumed to be contaminated by measurement errors. Our method applies a filter function of the form \(\varphi_\beta(t):=1-\exp(-\beta t^2)\) with the purpose of reducing the influence of the errors in the right-hand side vector on the computed approximate solution of the linear system. Here \(\beta\) is a regularization parameter. The iterative method is derived by expanding \(\varphi_\beta(t)\) in terms of Chebyshev polynomials. The method requires only little computer memory and is well suited for the solution of large-scale problems. We also show how a value of \(\beta\) and an associated approximate solution that satisfies the Morozov discrepancy principle can be computed efficiently. An application to image restoration illustrates the performance of the method.