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A new optimal control algorithm for the Keller-Segel problem

F. Guillen-Gonzalez, F. Palmero-Ramos, M. A. Rodriguez-Bellido, G. Tierra

Abstract

In this work we introduce a new optimal control algorithm for the Keller-Segel chemo-attraction system, where both boundary and distributed controls are considered and both are associated with introducing/removing the amount of chemical substances in the system. The key idea of our approach is to design the optimal control algorithm after discretizing the state problem system, which is done using an upwind finite volume scheme in space and a semi-implicit finite difference in time. Then, the discrete optimal control is approximated identifying the gradient of the reduced discrete cost via the discrete adjoint scheme. Finally, to minimize the reduced cost functional, we use a gradient descent type method (Adam scheme). Moreover, several numerical results are presented to illustrate the efficiency of the proposed approach.

A new optimal control algorithm for the Keller-Segel problem

Abstract

In this work we introduce a new optimal control algorithm for the Keller-Segel chemo-attraction system, where both boundary and distributed controls are considered and both are associated with introducing/removing the amount of chemical substances in the system. The key idea of our approach is to design the optimal control algorithm after discretizing the state problem system, which is done using an upwind finite volume scheme in space and a semi-implicit finite difference in time. Then, the discrete optimal control is approximated identifying the gradient of the reduced discrete cost via the discrete adjoint scheme. Finally, to minimize the reduced cost functional, we use a gradient descent type method (Adam scheme). Moreover, several numerical results are presented to illustrate the efficiency of the proposed approach.
Paper Structure (26 sections, 5 theorems, 61 equations, 14 figures, 2 tables)

This paper contains 26 sections, 5 theorems, 61 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The optimal control problem
  3. General formulation of a chemotaxis optimal control problem
  4. Linearized problems
  5. Adjoint problem
  6. Study case: Quadratic tracking cost with bilinear distributed control and boundary control, of either bilinear or Robin type.
  7. Discretize-then-Optimize approach.
  8. Numerical scheme for the state variables
  9. Discrete sensitivity problem
  10. Differentiating with respect to distributed control
  11. Differentiating with respect to boundary control
  12. Implementation of the discrete adjoint system
  13. Computation of discrete gradient
  14. Adam descent gradient algorithm
  15. Numerical Simulations
  16. ...and 11 more sections

Key Result

Theorem 3.2

The decoupled and linear scheme eq:scheme_v-eq:scheme_u is uniquely solvable, positivity-preserving and mass-conservative

Figures (14)

  • Figure 1: Dynamics of the manufactured control $f$ (left) and the associated variables $u$ (center) and variable $v$ (right).
  • Figure 2: Approximation a manufactured solution. Top row: Dynamics of the obtained control $f$ (left) and the associated variables $u$ (center) and variable $v$ (right). Bottom row: Evolution of the cost functional $J(f)$ (left), evolution of $\|\nabla J(f)\|$ (right) and the influence of perturbing the obtained control.
  • Figure 3: Dynamics of variables $u$ (left) and $v$ (right) when no control is acting on the system.
  • Figure 4: Results for $\Omega_c=[-1,1]$ and $\Omega_o=[-1,1]$. Top row: Dynamics of control $f$ (left) and the associated variables $u$ (center) and variable $v$ (right). Bottom row: Evolution of the cost functional $J(f)$ (left), evolution of $\|\nabla J(f)\|$ (center) and the influence of perturbing the obtained control (right).
  • Figure 5: Results for $\Omega_c=[-0.5,0.5]$ and $\Omega_o=[-1,1]$. Top row: Dynamics of control $f$ (left) and the associated variables $u$ (center) and variable $v$ (right). Bottom row: Evolution of the cost functional $J(f)$ (left), evolution of $\|\nabla J(f)\|$ (center) and the influence of perturbing the obtained control (right).
  • ...and 9 more figures

Theorems & Definitions (10)

  • Remark 3.1
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • proof
  • Theorem 3.5
  • Corollary 3.6
  • Remark 3.7