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

Abstract

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]

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

“…We solve (63)-(65) in two space dimensions using the direct solver UMFPACK [17] and in three space dimensions we use MINRES. For a more efficient solver with preconditioning, we refer to [10].…”

Preconditioners for state-constrained optimal control problems with Moreau-Yosida penalty function

“…A preconditioner for MINRES and the aforementioned problem could look like the following: with A 0 , A 1 , and being approximations to the (1,1)‐block, the (2,2)‐block, and the Schur complement, respectively. The use of MINRES for optimal control problems has been recently investigated in . Note that MINRES is also applicable in the case of a semidefinite (1,1)‐block, which is the case if we were to consider the minimization of J ( y , u ) as in , but with the ∥ y − y d ∥ 2 term given on some subdomain Ω 1 ⊂ Ω (as opposed to Ω itself).…”

Allen-Cahn and Cahn-Hilliard Variational Inequalities Solved with Optimization Techniques

“…Only for each individual component can we split the set of nodes into nodes which are active (for this component) and its complement. The resulting linear system is hence quite complex but can be solved efficiently with the help of MINRES, see [6]. ii) There is a straightforward variant of (PDAS-Vector) without mass constraints.…”