Title: Two reduced non-smooth Newton's methods for discretised state constrained optimal control problem governed by advection-diffusion equation

Authors: Kheira Mostefa Hanchour; Boubakeur Benahmed

Addresses: Higher School of Economics of Oran, Algeria ' National Polytechnic School of Oran – Maurice Audin, Algeria

Abstract: We are interested in solving numerically an optimal control problem governed by advection-diffusion equation with state inequality constraints. This problem can be implemented to modelise the minimisation of air pollution. First the infinite dimensional problem is discretised by application of the upwind symmetric interior penalty galerkin (SIPG) method. Then, we write the optimality conditions for the discretised problem. We reformulate these conditions into an equivalent system of nonlinear and non-smooth equations by the use of the Fischer-Burmeister function. Then, two effective algorithms that combine the non-smooth Newton's method and an active (index) set technique or pseudo inverse method are applied to solve this system. At the end, numerical examples are presented to illustrate the performance of our methods.

Keywords: PDE-constrained optimisation; advection-diffusion equation; Fisher-Burmeister function; non-smooth Newton's method; active index set method; Pseudo inverse of a matrix; saddle point system; air pollution minimisation.

DOI: 10.1504/IJMMNO.2023.129922

International Journal of Mathematical Modelling and Numerical Optimisation, 2023 Vol.13 No.2, pp.147 - 172

Received: 02 Jun 2022
Accepted: 20 Aug 2022
Published online: 03 Apr 2023 *

Full-text access for editors Full-text access for subscribers Purchase this article Comment on this article