IMAL   13325
INSTITUTO DE MATEMATICA APLICADA DEL LITORAL "DRA. ELEONOR HARBOURE"
Unidad Ejecutora - UE
artículos
Título:
Arbitrary Divergence Speed of the Least Squares Method in Infinite Dimensional Inverse Ill-Posed Problems
Autor/es:
RUBEN D. SPIES AND KARINA G. TEMPERINI
Revista:
INVERSE PROBLEMS
Editorial:
Institute of Physics Publishing
Referencias:
Lugar: London; Año: 2006 vol. 22 p. 611 - 626
ISSN:
0266-5611
Resumen:
A standard engineering procedure for approximating the solutions of an infinite dimensional inverse problem of the form Ax = y, where A is a given compact linear operator on a Hilbert space X and y is the given data, is to find a sequence {XN} of finite-dimensional approximating subspaces of X whose union is dense in X and to construct the sequence {xN} of least-squares solutions of the problem in XN. In 1980, Seidman showed that if the problem is ill-posed, then, without any additional assumptions on the exact solution or on the sequence of approximating subspaces XN, it cannot be guaranteed that the sequence {xN} will converge to the exact solution. In this paper, this result is extended in the following sense: it is shown that if X is separable, then for any y in X, y not equal 0 and for any arbitrarily given function s : N − > R+ there exists an injective, compact linear operator A and an increasing sequence of finite-dimensional subspaces XN of X such that the norm of xN − A−1y  is greater or equal than s(N) for all N in N, where xN is the least-squares solution of Ax = y in XN.