Arbitrary divergence speed of the least-squares method in infinite-dimensional inverse ill-posed problems

R D Spies, K G Temperini
2006 Inverse Problems  
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 {X N } of finite-dimensional approximating subspaces of X whose union is dense in X and to construct the sequence {x N } of least squares solutions of the problem in X N . In [1] , Seidman showed that if the problem is ill-posed, then, without any additional
more » ... ut any additional assumptions on the exact solution or on the sequence of approximating subspaces X N , it cannot be guaranteed that the sequence {x N } will converge to the exact solution. In this article this result is extended in the following sense: it is shown that if X is separable, then for any y ∈ X, y = 0 and for any arbitrarily given function s : IN → IR + there exists an injective, compact linear operator A and an increasing sequence of finitedimensional subspaces X N ⊂ X such that x N − A −1 y ≥ s(N ) for all N ∈ IN, where x N is the least squares solution of Ax = y in X N .
doi:10.1088/0266-5611/22/2/014 fatcat:f7mefbpx2nhatp7tgjkgqudloq