Title: Duality-based error estimates and their application to inverse problems

Speaker: Wolfgang Bangerth, Texas A&M

Date/Time: Tuesday, April 12, 2005, 1:00-2:00 pm (MT)

Location: Building 980, Room 95 (Sandia NM), Building 915, Room S101 (Sandia CA)

Brief Abstract: A posteriori error estimates for finite element discretizations have been investigated for some 25 years now, and there has been remarkable progress in estimating the discretization error. However, almost all of the approaches used in the field use various analytical tools that essentially measure the stability of solutions with regard to perturbations. While such stability estimates are available for many model equations, they are lacking for many practical cases of nonlinear equations, even if experimental evidence indicates that solutions of these equations indeed are stable.

One approach to avoid this problem is to numerically approximate the stability properties along with the computation of the solution. This leads to the formulation of "duality based error estimates" that are not only applicable to energy-norm estimates, but readily generalize to any output functional requested.

In this talk, we will explain the derivation of such estimates, and in particular apply it to the inverse problem of identifying parameters in PDEs from measurements of the state variable. This is a typical case of a problem for which stability-measures are notoriously hard to derive or may be even non-existent in general, and one that because of its numerical complexity greatly benefits from the adaptive meshes that can be derived from error estimators.