Computability of solutions of operator equations
We study operator equations within the Turing machine based framework for computability in analysis. Is there an algorithm that maps pairs (T, u) (where T is given in form of a program) to solutions of Tx u Here we consider the case when T is a bounded linear mapping between Hilbert spaces. We are in particular interested in computing the generalized inverse T+u, which is the standard concept of solution in the theory of inverse problems. Typically, T+ is discontinuous (i. e. the equation Tx u is ill posed) and hence no computable mapping. However, we will use effective versions of theorems from the theory of regularization to show that the mapping (T, T u, |T+u T+u is computable. We then go on to study the computability of average case solutions with respect to Gaussian measures which have been considered in information based complexity. Here, T+ is considered as an element of an L2 space. We define suitable representations for such spaces and use the results from the first part of the paper to show that (T, T |T+ T+ is computable. (c) 2007 WILEY VCH Verlag GmbH Co. KGaA, Weinheim)
Math. Log. Q.