Optimal control for mass conservative level set methods

This paper presents two different versions of an optimal control method for enforcing mass conservation in level set algorithms. The proposed PDE-constrained optimization procedure corrects a numerical solution to the level set transport equation so as to satisfy a conservation law for the corresponding Heaviside function. In the original version of this method, conservation errors are corrected by adding the gradient of a scalar control variable to the convective flux in the state equation. In the present paper, we investigate the use of vector controls. The alternative formulation offers additional flexibility and requires less regularity than the original method. The nonlinear system of first-order optimality conditions is solved using a standard fixed-point iteration. The new methodology is evaluated numerically and compared to the scalar control approach.