Regularization Methods in Multiple Hilbert Scales

RUBEN D. SPIES

Antalya

Conferencia; The 6th Inernational Conference on "inverse Problems: Modeling and Simulation".; 2012

Izmir University

Quite often an inverse problem can be formulated as the need for determining x in an equation of the form Tx = y, where T is a linear bounded operator between two in¯nite dimensional Hilbert spaces X and Y, the range of T, R(T), is non-closed and y is the data, which is known, perhaps with a certain degree of error. It is well known that under these hypotheses, the problem Tx = y is ill-posed in the sense of Hadamard ([2]) and solving it requires regularization. The Tikhonov-Phillips method, which is undoubtedly the best known and most commonly used way of regularizing an ill-posed problem, was originally proposed by Tikhonov and Phillips in 1962 and 1963 (see [5], [6], [7]) and consist of finding the minimizer of the functional Ja;L(x) := ||Tx-y ||^2  + a||Lx||^2  (1),  where L is an operator defined on a certain subspace D(L) of X, into a Hilbert space Z. The use of (1) to regularize problem Tx = y automatically implies the a-priori knowledge or assumption that the exact solution belongs to D(L). This approach gives rise to the theory of generalized inverses and regularization with seminorms (see for instance [1], Chapter 8). Hilbert scales become appropriate when there is no certainty that the exact solution is an element of D(L). The idea of using Hilbert scales for regularizing inverse ill-posed problems was first introduced by Natterer in 1984 ([4]) for the special case of the classical Tikhonov-Phillips method. In his work Natterer regularized the problem Tx = y by minimizing the functional  ||Tx - y^delta||+ ||x||_s over the space Xs, where ||y^delta- k||<=delta and (Xt)t denotes the Hilbert scale induced by theoperator L over X. In certain cases a value of s0 > 0 for which the exact solution x0 belongs toXs0 , could be known. In such cases it is possible to proceed with regularization of the problem Tx = y by means of the traditional methods, by replacing the Hilbert space X by Xs0 and T by its restriction to D(Ls0 ). In other cases, however, it is possible that such a value of s0 be not exactly known, although it could be reasonable to assume the existence of some u > 0 for which xy 2 Xu, (although the exact value of u may be unknown). It is precisely in this case in which Hilbert scales provide a solid mathematical framework for the development of convergent regularization methods which allow us to take advantage, in a optimal and adaptive" way, of the source condition x0 in Xu in order to obtain the best possible convergence speed, even though u is unknown. The first result about convergence on Hilbert scales is due to F. Natterer ([4]). In this work we summarize a number of recent results on regularization methods in multiple Hilbert scales ([3]). We shall introduce the concept of a multiple or vectorial Hilbert scale on aproduct space, define appropriate regularization methods on these scales, both for the case of a single observation and for the case of multiple observations. In the latter case, it is shown how vector-valued regularization functions in these multiple Hilbert scales can be used. In all cases convergence is proved and orders and optimal orders of convergence are shown.  [1] H. W. Engl, M. Hanke, and A. Neubauer, Regularization of inverse problems, Mathematics and its Applications, vol. 375, Kluwer Academic Publishers Group, Dordrecht, 1996.  [2] J. Hadamard, Sur les problemes aux derivees partielles et leur signification physique, Princeton University Bulletin 13 (1902), 49-52.  [3] G. L. Mazzieri and R. D. Spies, Regularization methods for ill-posed problems in multiple Hilbert scales, Inverse Problems, IOP, (2012), to appear.  [4] F. Natterer, Error bounds for Tikhonov regularization in Hilbert scales, Applicable Anal. 18 (1984), no. 1-2, 29-37.  [5] D. L. Phillips, A technique for the numerical solution of certain integral equations of the first kind, J. Assoc. Comput. Mach. 9 (1962), 84-97.  [6] A. N. Tikhonov, Regularization of incorrectly posed problems, Soviet Math. Dokl. 4 (1963), 1624-1627.  [7] , Solution of incorrectly formulated problems and the regularization method, SovietMath. Dokl. 4 (1963), 1035-1038.