Optimality for ill-posed problems under general source conditions

In this paper we consider linear ill-posed problems Ax = y where instead of y noisy data y(delta) are available with parallel to y -y(delta)parallel to less than or equal to delta and A : X --> Y is a linear operator between Hilbert spaces X and Y. Assuming the general source condition x epsilon M-phi,M-E = (x epsilon X : x - (x) over bar = phi(A*A)(1/2)v, parallel to v parallel to less than or equal to E) With appropriate functions cp we study following questions: (i) which (best possible) accuracy can be obtained for identifying a: from y(delta) epsilon Y under the assumptions parallel to y -y(delta)parallel to less than or equal to delta and x epsilon M-phi,M-E? (ii) are there special regularization methods which guarantee this best possible accuracy, i.e., which are optimal on the set M-phi,M-E? Concerning question (i) we prove that under certain conditions there holds inf sup parallel to Ry(delta) - x parallel to = E root p(-1)(delta(2)/E-2 with p(lambda) = lambda phi(-1)(lambda) where the 'inf' is taken over all methods R:Y --> X and the 'sup' is taken over all x epsilon M-phi,M-E Y-delta epsilon Y and parallel to y-y(delta)parallel to less than or equal to delta. Concerning question (ii) we prove the optimality of a general class of regularization methods and specify our general optimality results to Tikhonov type methods and to spectral methods. Heat equation problems backward in time which are characterized by different functions phi(lambda) serve as model examples.