Iterative exponential filtering for large discrete ill-posed problems

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 lter function of the form ' (t) := 1 ?exp(?t 2) with the purpose of reducing the innuence 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 ' (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 and an associated approximate solution that satisses the Morozov discrepancy principle can be computed eeciently. An application to image restoration illustrates the performance of the method.