Abstract In this talk we consider the problem of reconstructing solutions x^† of ill-posed problems Ax=y where A є L(X,Y) is a linear operator with non-closed range R(A) between Hilbert spaces X and Y . We assume:

A1. Instead of exact data y є ℜ(A) , some noisy data y^δ є Y with || y-y^δ || ≤ δ are given.

A2. The unknown solution x^† possesses a certain solution smoothness which we describe by
x^† є M_φ,E ={ x є X ; x = [φ(A^*A) ]^1/2 v , || v || ≤ E }
with some monotonically increasing function φ satisfying lim_{λ → 0} φ(λ) = 0.

We answer the following questions:

Q1. Which best possible accuracy can be obtained for identifying the solution x^† of the operator equation Ax=y from noisy data y^δ under the two conditions A1 and A2?

Q2. How to regularize such that the best possible accuracy can be guaranteed?

We apply our general results to a Cauchy problem for the Helmholtz equation and show that, depending on different smoothness situations A2, the best possible accuracy for identifying x^† from noisy data may be of Hölder type, of logarithmic type or of some other type. In addition, we study regularization methods that provide the best possible accuracy without using smoothness information A2.

References

1. Regińska, T. and Regiński, K., Approximate solution of a Cauchy problem for the Helmholtz equation, Inverse Problems, 22 (2006), 975--989. doi:10.1088/0266-5611/22/3/015
2. Tautenhahn, U., Optimality for linear ill-posed problems under general source conditions, Num. Funct. Anal. and Optimiz., 19 (1998), 377--398. doi:10.1080/01630569808816834