We propose and analyze a primal-dual active set method for local and nonlocal Allen-Cahn variational inequalities. An existence result for the non-local variational inequality is shown in a formulation involving Lagrange multipliers for local and non-local constraints. Superlinear local convergence is shown by interpreting the approach as a semi-smooth Newton method. Properties of the method are discussed and several numerical simulations demonstrate its efficiency.

Abstract. The mean compliance minimization in structural topology optimization is solved with the help of a phase field approach. Two steepest descent approaches based on L 2 -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. Mathematics Subject Classification (2000). 74P15, 74P05, 74S03, 35K99.

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.

Abstract. Parabolic variational inequalities of Allen-Cahn and CahnHilliard 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.Mathematics Subject Classification (2010). 35K55, 35S85, 65K10, 90C33, 90C53, 49N90, 65M60.

Abstract. 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.AMS subject classifications. Primary 65F10, 65N22, 65F50 Secondary 76D07 Key words. PDE-constrained optimization, Allen-Cahn model, Newton method, Saddle point systems, Preconditioning, Krylov subspace solver.1. Introduction. The solution of Allen-Cahn variational inequalities with nonlocal constraints can be formulated as an optimal control problem, which can be solved using a primal-dual active set method [4,6,7]. This method has proven very efficient in a variety of applications [27,29,35,36]. As we will show in the course of this paper the solution to a linear system of the form Kx = b with K a real symmetric matrix is at the heart of this method. The sparse linear systems are usually of very large dimension and in combination with 3-dimensional experiments the application of direct solvers such as UMFPack [11] becomes infeasible. As a result iterative methods have to be employed (see e.g. [24,43] for introductions to this field). For symmetric and indefinite systems the Minimal Residual method (minres) [39] is a common solver as it minimizes the 2-norm of the residual r k = b − Kx k over the Krylov subspace span r 0 , Kr 0 , . . . , K k−1 r 0 . The convergence behaviour of the iterative scheme depends on the conditioning of the problem and the clustering of the eigenvalues and can usually be enhanced with preconditioning techniques.P −1 Kx = P −1 b In this paper, we provide an efficient preconditioner P for the solution of Allen-Cahn variational inequalities combining methods for indefinite problems [31,39,48,3] and algebraic multigrid developed for elliptic systems [15,43,42]. The arising linear systems lead to matrices K which have the following saddle point block-structure which arise in a variety of applications [3]