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

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

Orthogonally Decoupled Variational Gaussian Processes

Gaussian processes (GPs) provide a powerful non-parametric framework for reasoning over functions. Despite appealing theory, its superlinear computational and memory complexities have presented a long-standing challenge. State-of-the-art sparse variational inference methods trade modeling accuracy against complexity. However, the complexities of these methods still scale superlinearly in the number of basis functions, implying that that sparse GP methods are able to learn from large datasets only when a small model is used. Recently, a decoupled approach was proposed that removes the unnecessary coupling between the complexities of modeling the mean and the covariance functions of a GP. It achieves a linear complexity in the number of mean parameters, so an expressive posterior mean function can be modeled. While promising, this approach suffers from optimization difficulties due to ill-conditioning and non-convexity. In this work, we propose an alternative decoupled parametrization. It adopts an orthogonal basis in the mean function to model the residues that cannot be learned by the standard coupled approach. Therefore, our method extends, rather than replaces, the coupled approach to achieve strictly better performance. This construction admits a straightforward natural gradient update rule, so the structure of the information manifold that is lost during decoupling can be leveraged to speed up learning. Empirically, our algorithm demonstrates significantly faster convergence in multiple experiments.Comment: Appearing NIPS 201

Strain Analysis by a Total Generalized Variation Regularized Optical Flow Model

In this paper we deal with the important problem of estimating the local strain tensor from a sequence of micro-structural images realized during deformation tests of engineering materials. Since the strain tensor is defined via the Jacobian of the displacement field, we propose to compute the displacement field by a variational model which takes care of properties of the Jacobian of the displacement field. In particular we are interested in areas of high strain. The data term of our variational model relies on the brightness invariance property of the image sequence. As prior we choose the second order total generalized variation of the displacement field. This prior splits the Jacobian of the displacement field into a smooth and a non-smooth part. The latter reflects the material cracks. An additional constraint is incorporated to handle physical properties of the non-smooth part for tensile tests. We prove that the resulting convex model has a minimizer and show how a primal-dual method can be applied to find a minimizer. The corresponding algorithm has the advantage that the strain tensor is directly computed within the iteration process. Our algorithm is further equipped with a coarse-to-fine strategy to cope with larger displacements. Numerical examples with simulated and experimental data demonstrate the very good performance of our algorithm. In comparison to state-of-the-art engineering software for strain analysis our method can resolve local phenomena much better