On the average case solvability of III-posed problems

Abstract

We wish to solve an ill-posed problem whose solution operator S is a measurable unbounded linear transformation of a Banach space into a Hilbert space. Let ε > 0. It is known that the e-complexity of this problem is infinite in the worst case setting. Suppose we turn to an average case or probabilistic setting, the domain of S being equipped with a zero-mean Gaussian measure μ. It is known that the problem has finite ε-complexity iff S ϵ L 2( μ), and optimal information and algorithms are essentially the same as if S were bounded. We show that any such unbounded operator S belongs to L 2( μ). Hence, the ε-complexity of any such illposed problem is finite in the average case and probabilistic settings.