Iterative exponential filtering for large discrete ill-posed problems

Summary. 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 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 is a regularization parameter. The iterative method is derived by expanding 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 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. Received January 25, 1997 / Revised version received February 9, 1998 / Published online July 28, 1999