Numerical solution of optimal control of atherosclerosis using direct and indirect methods with shooting/collocation approach

TL;DR

This paper introduces a numerical approach combining direct and indirect methods, including collocation and shooting techniques, to solve complex optimal control problems modeling atherosclerosis progression.

Contribution

It develops a fixed point-collocation method for nonlinear PDE-constrained optimal control problems, transforming free boundary issues into fixed boundary problems and verifying solutions with indirect methods.

Findings

Efficient solution of complex PDE-based optimal control problems.

Validation of the numerical method through numerical experiments.

Demonstrated effectiveness of combined direct and indirect approaches.

Abstract

We present a direct numerical method for the solution of an optimal control problem controlling the growth of LDL, HDL and plaque. The optimal control problem is constrained with a system of coupled nonlinear free and mixed boundary partial differential equations consisting of three parabolics one elliptic and one ordinary differential equations. In the first step, the original problem is transformed from a free boundary problem into a fixed one and from the mixed boundary condition to a Neumann one. Then, employing a fixed point-collocation method, we solve the optimal control problem. In each step of the fixed point iteration, the problem is changed to a linear one and then, the equations are solved using the collocation method bringing about an NLP which is solved using sequential quadratic programming. Then, the obtained solution is verified using indirect methods originating from the first-order optimality conditions. Numerical results are considered to illustrate the efficiency of methods.