Regularization by projection in variable Hilbert scales

Abstract

We introduce and analyze projection schemes for solving ill-posed linear operator equations in Hilbert space. Control parameters are shown to yield convergence. Emphasis is on their approximation theoretic content. For smoothness given in terms of general source conditions we establish convergence rates, shown to be optimal in some cases. The study is accomplished with a discussion on condition numbers.

Keywords:

• Projection method
• Regularization
• General source condition
• Condition number

AMS Subject Classification::

• 65J20
• Secondary 47B06

Acknowledgement

The authors want to thank an unknown referee for initiating a discussion on qualification of projection methods. This led to the considerations in section 4.2, and may be viewed as a first attempt to tackle this problem.

Notes

¹The symbol f(t) ≈ g(t) means, that there are constants 0 < c < C < ∞ for which c ≤ f(t)/g(t) ≤ C as t → 0.