A robust, discrete-gradient descent procedure for optimisation with time-dependent PDE and norm constraints

TL;DR

This paper introduces a robust, discrete-gradient descent method for solving norm-constrained optimisation problems involving time-dependent PDEs in fluid dynamics, ensuring better convergence and computational efficiency.

Contribution

It presents a numerically consistent, discrete formulation of the direct-adjoint looping method with global convergence guarantees, applicable to fluid dynamics problems.

Findings

Demonstrates robustness on three fluid dynamics problems

Provides a library SphereManOpt for implementation

Achieves improved convergence in PDE-constrained optimisation

Abstract

Many physical questions in fluid dynamics can be recast in terms of norm constrained optimisation problems; which in-turn, can be further recast as unconstrained problems on spherical manifolds. Due to the nonlinearities of the governing PDEs, and the computational cost of performing optimal control on such systems, improving the numerical convergence of the optimisation procedure is crucial. Borrowing tools from the optimisation on manifolds community we outline a numerically consistent, discrete formulation of the direct-adjoint looping method accompanied by gradient descent and line-search algorithms with global convergence guarantees. We numerically demonstrate the robustness of this formulation on three example problems of relevance in fluid dynamics and provide an accompanying library SphereManOpt