Local Convergence of the Lavrentiev Method for the Cauchy Problem via a Carleman Inequality

Abstract

The purpose is to perform a sharp analysis of the Lavrentiev method applied to the regularization of the ill-posed Cauchy problem, set in the Steklov-Poincare variational framework. Global approximation results have been stated earlier that demonstrate that the Lavrentiev procedure yields a convergent strategy. However, no convergence rates are available unless a source condition is assumed on the exact Cauchy solution. We pursue here bounds on the approximation (bias) and the noise propagation (variance) errors away from the incomplete boundary where instabilities are located. The investigation relies on a Carleman inequality that enables enhanced local convergence rates for both bias and variance errors without any particular smoothness assumption on the exact solution. These improved results allows a new insight on the behavior of the Lavrentiev solution, look similar to those established for the Quasi-Reversibility method in [Inverse Problems 25, 035005, 2009]. There is a case for saying that this sort of `super-convergence' is rather inherent to the nature of the Cauchy problem and any reasonable regularization procedure would enjoy the same locally super-convergent behavior.