Parameter Estimation in Variational Inequalities
This project is concerned with the identification of a (positive) distributed parameter u in variational inequalities of the type
Introducing a non-negative slack variable (multiplier)the variational inequality is equivalent to
Identification problems for variational inequalities frequently occur in practical applications. For instance, the estimation of the height function u between two rotating surfaces in elastohydrodynamic lubrication problems belongs to the present problem class. In this situatuation the variational inequality stems from the Reynolds lubrication equation and the additional requirement that the pressure y is nonnegative.
A common technique to identify the parameter u from measurements z of y is based on a regularized least squares formulation
?Typical results on existence of multipliers available from the literature either guarantee multipliers in very general situations but are not amenable for numerical realization, or the corresponding first order conditions are numerically implementable but existence of multipliers fails in specific important cases.
Our aim is to derive multipliers that exist in general
the corresponding first order characterization is immediately amenable for numerical realization.
This is achived by the following strategy:
The optimal solution (u*,y*,l*) of the regularized least squares problem with corresponding multipliers (p*,m*) is characterized by the following first order necessary optimality conditions: