Approximate source conditions in Tikhonov regularization‐new analytical results and some numerical studies

Abstract

We present some new ideas and results for finding convergence rates in Tikhonov regularization for ill‐posed linear inverse problems with compact and non‐compact forward operators based on the consideration of approximate source conditions and corresponding distance functions. The new results and studies complement and extend in numerous points the recent papers [5, 7, 8, 10] that also exploit the distance functions originally introduced in [2] which measure the violation of a moderate source condition that works as a benchmark. In this context, we distinguish as in [8] logarithmic, power and exponential decay rates for the distance functions and their consequences. Under specific range inclusions the decay rate of distance functions is verified explicitly, whereas in [10] this result is also used but formulated only in an implicit manner. Applications to non‐compact multiplication operators are briefly reviewed from [8]. An important new result is that we can show for compact operators a one‐to‐one correspondence between the maximal power type decay rates for the distance functions and maximal exponents of Holder rates in Tikhonov regularization linked by the specific singular value expansion of the solution element. Some numerical studies on simple integration illustrate the compact operator case and the specific situation of discretized problems. Finally, some ideas of generalization are mentioned concerning the fact that the benchmark of the distance function can be shifted.

Key words:

• linear ill‐posed problems
• Tikhonov regularization
• approximate source conditions
• distance functions
• convergence rates
• compact operator
• singular value asymptotics
• multiplication operators
• range inclusions