Mesh refinement and numerical sensitivity analysis for parameter calibration of partial differential equations
This paper deals with the problem of parameter calibration for a state equation in the form \[ a(q,u)(\phi) = f(\phi) \qquad\forall \phi\in V, \] where \(u\in V\) is a state variable (vector of all physical unknowns), \(q\in Q\) denotes the unknown parameters, \(V\), \(Q\) are appropriate Hilbert spaces and \(a(.,.)(.)\) defined on the Hilbert space \(Q\times V\times V\) is linear with respect to arguments in the second pair of parenthesis. The values of the parameters are estimated from a given set of measurements using a least squares approach which leads to the constraind minimization problem with a cost functional. The state equation (the constraint) is discretized by the Galerkin method based on a finite-dimensional space. Thus, two types of errors are involved: the measurement errors and the discretization errors. The aim of this paper is to analyze the dependency of the computed quantity of interest on both, the discretization and the measurement errors. In the first part of the paper, an a posteriori error estimator is derived which aims to control the error in the quantity of interest due to discretization. The second part is devoted to an adaptive mesh refinement algorithm based on the error estimator. The next section deals with the sensitivity analysis and finally, the whole approach is illustrated at hand of a flow problem in the last part of the paper. The new contributions of this paper are the combination of local mesh refinement with sensitivity analysis and a generalization of a posteriori error estimator established before. Concerning a posteriori error estimator, the authors generalize their previous work, since the interest functional depends both on the parameters and on the state variable in this paper.
scientific article; zbMATH DE number 2175561
scientific article