Material and Shape Derivative Method for Quasi-Linear Elliptic Systems with Applications in Inverse Electromagnetic Interface Problems

Second Order Derivatives , Newton Method , Application to Shape Optimization

{ We describe a Newton method applied to the evaluation of a critical point of a total energy associated to a shape optimization problem. The key point of these methods is the Hessian of the shape functional. We give an expression of the Hessian as well as the relation with the second-order Eulerian semi-derivative. An application to the electromagnetic shaping of liquid metals process is studi...

Shape-Measure Method for Solving Elliptic Optimal Shape Problems (Fixed Control Case)

Efficient evolution algorithms for embedded interfaces: from inverse parameter estimation to a level set method for ductile fracture

This thesis presents efficient and easy to implement algorithms for the simulation of embedded interface evolutions in applications of physically based modeling. It is organized in two parts. In the first part, we provide the mathematical and numerical background necessary for the development of our methods. We outline a state-of-the-art cut-cell algorithm that allows us to efficiently and conv...

Indicator Functions for Shape Reconstruction Related to the Linear Sampling Method

We provide exact shape reconstruction formulas in the spirit of the Linear Sampling method for a class of inverse problems in shape determination in the context of timeindependent partial differential equations. To this end, we prove a general theorem how, and under which assumptions, domain characterizations based on the range of the square root of an operator transform into domain characteriz...

Iterative Choices of Regularization Parameters in Linear Inverse Problems

We investigate possibilities of choosing reasonable regularization parameters for the output least squares formulation of linear inverse problems. Based on the Morozov and damped Morozov discrepancy principles, we propose two iterative methods, a quasi-Newton method and a two-parameter model function method, for finding some reasonable regularization parameters in an efficient manner. These dis...