Abstract

Stability and Convergence Results for a Class of Sequential Predictor-Corrector Regularization Methods for Ill-Posed Volterra Equations

Wolfgang Ring
University of Graz
Special Research Center on Optimzation and Control, Institute of

A class of sequential predictor-corrector regularization
method for the stable solution of ill-posed Volterra problems is analyzed. In
the discrete setting, the regularization technique is based on constant or
affine continuation of the solution into a `future' time interval. Using this
continuation, the solution at the current time-step is determined such that the
given data are fitted on the future interval best possible in a least squares
sense. An infinite dimensional analogue for the discrete regularization
technique is derived. The infinite dimensional formulation has the form of a
Volterra integro-differential equation of the second kind. Under a certain
stability assumption well-posedness and convergence results for the infinite
dimensional formulation are obtained. Stability estimates and convergence in the
cases of exact and noisy data are proved. A convergence rate result is proved
for finitely smoothing kernels if the exact solution is C^1,α-smooth.
It is also proved that the stability condition cannot be satisfied for Volterra
problems with ν-smoothing kernels,
where ν ≥ 5. Numerical examples are
presented showing the stable and the unstable situation and verifying
numerically the convergence rate result.