Primal-dual active set methods for Allen-Cahn variational inequalities

This thesis aims to introduce and analyse a primal-dual active set strategy for solving Allen-Cahn variational inequalities. We consider the standard Allen-Cahn equation with non-local constraints and a vector-valued Allen-Cahn equation with and without non-local constraints. Existence and uniqueness results are derived in a formulation involving Lagrange multipliers for local and non-local constraints. Local Convergence is shown by interpreting the primal-dual active set approach as a semi-smooth Newton method. Properties of the method are discussed and several numerical simulations in two and three space dimensions demonstrate its efficiency. In the second part of the thesis various applications of the Allen-Cahn equation are discussed. The non-local Allen-Cahn equation can be coupled with an elasticity equation to solve problems in structural topology optimisation. The model can be extended to handle multiple structures by using the vector-valued Allen-Cahn variational inequality with non-local constraints. Since many applications of the Allen-Cahn equation involve evolution of interfaces in materials an important extension of the standard Allen-Cahn model is to allow materials to exhibit anisotropic behaviour. We introduce an anisotropic version of the Allen-Cahn variational inequality and we show that it is possible to apply the primal-dual active set strategy efficiently to this model. Finally, the Allen-Cahn model is applied to problems in image processing, such as segmentation, denoising and inpainting. The primal-dual active set method proves exible and reliable for all the applications considered in this thesis

Preconditioning for Allen-Cahn variational inequalities with non-local constraints

The solution of Allen-Cahn variational inequalities with mass constraints is of interest in many applications. This problem can be solved both in its scalar and vector-valued form as a PDE-constrained optimization problem by means of a primal-dual active set method. At the heart of this method lies the solution of linear systems in saddle point form. In this paper we propose the use of Krylov-subspace solvers and suitable preconditioners for the saddle point systems. Numerical results illustrate the competitiveness of this approach

Allen-Cahn and Cahn-Hilliard variational inequalities solved with Optimization Techniques

Parabolic variational inequalities of Allen-Cahn and Cahn- Hilliard type are solved using methods involving constrained optimization. Time discrete variants are formulated with the help of Lagrange multipliers for local and non-local equality and inequality constraints. Fully discrete problems resulting from finite element discretizations in space are solved with the help of a primal-dual active set approach. We show several numerical computations also involving systems of parabolic variational inequalities

Phase-field approaches to structural topology optimization

The mean compliance minimization in structural topology optimization is solved with the help of a phase field approach. Two steepest descent approaches based on L2- and H-1 gradient flow dynamics are discussed. The resulting flows are given by Allen-Cahn and Cahn-Hilliard type dynamics coupled to a linear elasticity system. We finally compare numerical results obtained from the two different approaches

Non-local Allen-Cahn systems: Analysis and a primal dual active set method

We show existence and uniqueness of a solution for the non-local vector-valued Allen-Cahn variational inequality in a formulation involving Lagrange multipliers for local and non-local constraints. Furthermore, we propose and analyze a primal-dual active set method for local and non-local vector-valued Allen-Cahn variational inequalities. Convergence of the primal-dual active set algorithm is shown by interpreting the approach as a semi-smooth Newton method and numerical simulations are presented demonstrating its efficiency

Non-local Allen-Cahn systems: analysis and a primal-dual active set method

We show existence and uniqueness of a solution for the non-local vector-valued Allen-Cahn variational inequality in a formulation involving Lagrange multipliers for local and non-local constraints. Furthermore, we propose and analyze a primal-dual active set method for local and non-local vector-valued Allen-Cahn variational inequalities. The local convergence behaviour of the primal-dual active set algorithm is studied by interpreting the approach as a semi-smooth Newton method and numerical simulations are presented demonstrating its efficiency

Resurgence of Field Fever in a Temperate Country: An Epidemic of Leptospirosis among Seasonal Strawberry Harvesters in Germany in 2007

Background: Although leptospirosis is a reemerging zoonosis of global importance, outbreaks related to agricultural exposures are primarily situated in tropical countries. In July 2007, a suspected leptospirosis outbreak was recognized among strawberry harvesters from Eastern Europe who were working in Germany. An investigation was initiated to identify the outbreak source and the risk factors for infection. Methods: We conducted a retrospective cohort study with use of a questionnaire administered to harvesters by health authorities in Romania, Slovakia, and Poland. Collected serum samples were tested by microscopic agglutination test and immunoglobulin M enzyme‐linked immunosorbent assay. A case patient was defined as a person who worked in the strawberry field during the period 5 June–8 September 2007 and had leptospirosis‐compatible symptoms and either an antibody titer 1:800 and a positive immunoglobulin M enzyme‐linked immunosorbent assay result (for a confirmed case) or no serological confirmation (for a suspected case). Local rodents were examined for leptospirosis. Results: Among 153 strawberry harvesters, we detected 13 confirmed case patients who had test results positive for antibodies against Leptospira species serogroup Grippotyphosa and 11 suspected case patients (attack rate, 16%). Risk of disease increased with each day that an individual worked in the rain with hand wounds (odds ratio, 1.1; 95% confidence interval, 1.04–1.14) and accidental rodent contact (odds ratio, 4.8; 95% confidence interval, 1.5–15.9). Leptospires of the serogroup Grippotyphosa were isolated from the kidneys of 7 (64%) of 11 voles. Conclusions: This is, to our knowledge, the largest leptospirosis epidemic to occur in Germany since the 1960s. Contact between hand lesions and contaminated water or soil and infected voles was the most likely outbreak source. The unusually warm winter of 2006–2007 supported vole population growth and contributed to this resurgence of leptospirosis in Germany. Because of ongoing climate change, heightened awareness of leptospirosis in temperate regions is warranted