Optimization Methods: Inverse Problems, Algorithms and Applications

Abstract

This thesis deals with inverse problems and it is focused on practical applications arising in geomechanics. By inverse problems we understand an identification of a finite set of material parameters. The thesis provides proofs of existence of at least one solution to the continuous setting of inverse problems, which are governed either by scalar elliptic or linear elasticity state problems. For both state problems we assume a priori known material distributions. We prove the convergence of parameters, which are identified in state problems discretized by FEM, to the parameters which solve the continuous state problems. Algebraic sensitivity analysis of the first and the second order for a large class of nonlinear least squares cost functionals is provided. Gradient based optimization methods with various line search techniques are used for the numerical solution of the introduced problems.

Proposed numerical methods are first tested on model state problems and subsequently applied to a parameter identification of a model problem of groundwater flow in a confined aquifer in a few settings where the effects of an over and under determination of inverse problems are studied. A two level approach is used for a more complicated parameter distribution.

Results of the three following geomechanical engineering applications are shown: a large scale identification problem which was a part of an evaluation of the in situ experiment concerning the rock damage in repositories of spent nuclear fuel; a parameter fitting for hydro-mechanical model of unsaturated fluid flow in a borehole sealing structure; and an identification of material properties for a constituent of a geological composite using laboratory data from the uniaxial load test applied to the composite.