Scalable optimal control for inequality-constrained discretizations of scalar conservation laws

TL;DR

This paper introduces a scalable optimal control approach for inequality-constrained discretizations of scalar conservation laws, combining high accuracy and conservation with efficient iterative solvers.

Contribution

It proposes a new optimal control formulation using flux potentials, and develops scalable matrix-free trust-region Newton solvers with multigrid methods.

Findings

Solvers achieve machine precision constraint satisfaction.

Scaling of inequality constraints improves convergence.

Demonstrated effectiveness on linear problems and Cahn-Hilliard equation.

Abstract

Optimization-based (OB) alternatives to traditional flux limiters couch preservation of properties such as local bounds and maximum principles into optimization problems, which impose these properties through inequality constraints. In this paper, we propose a new potential-target OB approach that enforces these properties using an optimal control formulation, in which the control is the source term expressed through flux potentials. The resulting OB formulation combines superb accuracy with excellent local conservation properties, but complicates the development of scalable iterative solvers, which is greatly influenced by the choice of semi-norms for the objective function. We use this fact to design scalable iterative solvers based on matrix-free trust-region Newton methods with projections onto convex sets. These solvers leverage inexpensive multigrid V-cycles while satisfying all constraints to machine precision. Numerical experiments reveal that the convergence behavior of the solvers can be greatly improved by a simple scaling of the inequality constraints. We demonstrate excellent performance in applications to linear test problems, such as $L^2$ projection and solid body rotation, and to the Cahn-Hilliard equation.