Iterative exponential filtering for large discrete ill-posed problems

We describe a new iterative method for the solution of large, very ill-cond itioned linear systems of equations that arise when discretizing linear ill -posed problems. The right-hand side vector represents the given data and i s assumed to be contaminated by measurement errors. Our method applies a fi lter function of the form phi(beta)(t) := 1 - exp(-beta t(2)) with the purp ose of reducing the influence of the errors in the right-hand side vector o n the computed approximate solution of the linear system. Here beta is a re gularization parameter. The iterative method is derived by expanding phi(be ta)(t) in terms of Chebyshev polynomials. The method requires only little c omputer 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 th at satisfies the Morozov discrepancy principle can be computed efficiently. An application to image restoration illustrates the performance of the met hod.