Mid Sweden University

In this paper, the Lame coefficients in the linear elasticity problem are estimated by using the measurements of displacement. Some a posteriori error estimators for the approximation error of the parameters are derived, and then adaptive finite element schemes are developed for the discretization of the parameter estimation problem, based on the error estimators. The Gauss-Newton method is employed to solve the discretized nonlinear least-squares problem. Some numerical results are presented.

In this paper, adaptive finite element method is developed for the estimation of distributed parameter in elliptic equation. Both upper and lower error bound are derived and used to improve the accuracy by appropriate mesh refinement. An efficient preconditioned project gradient algorithm is employed to solve the nonlinear least-squares problem arising in the context of parameter identification problem. The efficiency of our error estimators is demonstrated by some numerical experiments

A discrete ordinate method is developed for solving the radiative transfer equation, and the corresponding parameter estimation problem is given a least-squares formulation. Two Levenberg-Marquardt methods, a feasible-path approach and an SQP type method, are analyzed and compared. A sensitivity analysis is given, and it is shown how it can be used for designing measurements with minimal impact of measurement noise. Numerical experiments are performed to exemplify the usefulness of the theory.

In this paper, the elastic constants of a material are recovered from measured displacements where the model is the equilibrium equations for the orthotropic case. The finite element method is used for the discretization of the state equation and the Gauss–Newton method is used to solve the nonlinear least squares problem attained from the parameter estimation problem. A posteriori error estimators are derived and used to improve the accuracy by an appropriate mesh refinement. A numerical experiment is presented to show the applicability of the approach.

Physical and chemical phenomena are often described by a system of partial di®erential equations. These equations usually involve unknown parameters, which cannot be measured directly but which can be adjusted to make the model predictions match the observed data. The process of ¯tting these para- meters to laboratory or plant data is called parameter estimation. In order to recover these parameters, the well-known output least squares formulation is of- ten utilized. To solve the optimization problem governed by partial di®erential equations, the in¯nite-dimensional problem must be approximated by introduc- ing discretizations such as a ¯nite elements or di®erences. It is clear that the e±ciency of the numerical methods dealt with here will be in°uenced by the discretization scheme. The goal of this thesis is to develop e±cient numerical methods for the parameter estimation problems governed by partial di®erential equations, based on adaptive ¯nite element methods. This work was initiated by an investigation into an a posteriori error esti- mator of residual type for parameter estimation problems with a ¯nite number of unknown parameters. It appears that an adaptive ¯nite element algorithm guided by the derived a posteriori error estimator produces a sequence of eco- nomical, locally re¯ned meshes. The methods are then applied to the identi¯ca- tion of elastic constants in paper from measured displacements. Further, some a posteriori error estimators of gradient recovery type are derived for the error in parameters due to the discretization. The main advantages of using error estimators of this type are the simplicity of their implementation and their cost e®ectiveness. Often, the unknown parameters are functions, which also need to be dis- cretized. Adaptive ¯nite element method is developed for the estimation of distributed parameters in elliptic equations with multi-mesh techniques. Finally, a goal-oriented adaptive method, dual weighted residual methods (DWR methods) are employed determining the elastic constants in paper from measured displacements.