Mean-field optimal control for biological pattern formation

Martin Burger; Lisa Maria Kreusser; Claudia Totzeck

doi:10.1051/cocv/2021034

Mean-field optimal control for biological pattern formation

Martin Burger, Lisa Maria Kreusser, Claudia Totzeck

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

2 Citations (SciVal)

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Abstract

We propose a mean-field optimal control problem for the parameter identification of a given pattern. The cost functional is based on the Wasserstein distance between the probability measures of the modeled and the desired patterns. The first-order optimality conditions corresponding to the optimal control problem are derived using a Lagrangian approach on the mean-field level. Based on these conditions we propose a gradient descent method to identify relevant parameters such as angle of rotation and force scaling which may be spatially inhomogeneous. We discretize the first-order optimality conditions in order to employ the algorithm on the particle level. Moreover, we prove a rate for the convergence of the controls as the number of particles used for the discretization tends to infinity. Numerical results for the spatially homogeneous case demonstrate the feasibility of the approach.

Original language	English
Article number	40
Number of pages	24
Journal	ESAIM - Control, Optimisation and Calculus of Variations
Volume	27
Early online date	30 Apr 2021
DOIs	https://doi.org/10.1051/cocv/2021034
Publication status	Published - 31 Dec 2021

Bibliographical note

Funding Information:
Acknowledgements. MB has been partially supported by the German Science Foundation (DFG) through CRC TR 154 ”Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks”. MB and LMK acknowledge support from the European Union Horizon 2020 research and innovation programmes under the Marie Sk lodowska-Curie grant agreement No. 777826 (NoMADS). LMK acknowledges support from the European Union Horizon 2020 research and innovation programmes under the Marie Sk lodowska-Curie grant agreement No. 691070 (CHiPS), the EPSRC grant EP/L016516/1, the German National Academic Foundation (Studienstiftung des Deutschen Volkes), the Cantab Capital Institute for the Mathematics of Information and Magdalene College, Cambridge (Nevile Research Fellowship). CT was partly supported by the European Social Fund and by the Ministry Of Science, Research and the Arts Baden-Württemberg. Moreover, CT acknowledges support by the state of Baden-Württemberg through bwHPC, in particular the bwForCluster MLS&WISO Production.

Keywords

Dynamical systems
Interacting particle systems
Mean-field limits
Optimal control with ODE/PDE constraints
Pattern formation

ASJC Scopus subject areas

Control and Systems Engineering
Control and Optimization
Computational Mathematics

Access to Document

10.1051/cocv/2021034

bkt_optimalcontrol_accepted.PDF
© EDP Sciences, SMAI 2021 Burger, M, Kreusser, LM & Totzeck, C 2021, 'Mean-field optimal control for biological pattern formation', ESAIM - Control, Optimisation and Calculus of Variations, vol. 27, 40. https://doi.org/10.1051/cocv/2021034
Accepted author manuscript, 731 KB

Cite this

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title = "Mean-field optimal control for biological pattern formation",

abstract = "We propose a mean-field optimal control problem for the parameter identification of a given pattern. The cost functional is based on the Wasserstein distance between the probability measures of the modeled and the desired patterns. The first-order optimality conditions corresponding to the optimal control problem are derived using a Lagrangian approach on the mean-field level. Based on these conditions we propose a gradient descent method to identify relevant parameters such as angle of rotation and force scaling which may be spatially inhomogeneous. We discretize the first-order optimality conditions in order to employ the algorithm on the particle level. Moreover, we prove a rate for the convergence of the controls as the number of particles used for the discretization tends to infinity. Numerical results for the spatially homogeneous case demonstrate the feasibility of the approach. ",

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author = "Martin Burger and Kreusser, {Lisa Maria} and Claudia Totzeck",

note = "Funding Information: Acknowledgements. MB has been partially supported by the German Science Foundation (DFG) through CRC TR 154 ”Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks”. MB and LMK acknowledge support from the European Union Horizon 2020 research and innovation programmes under the Marie Sk lodowska-Curie grant agreement No. 777826 (NoMADS). LMK acknowledges support from the European Union Horizon 2020 research and innovation programmes under the Marie Sk lodowska-Curie grant agreement No. 691070 (CHiPS), the EPSRC grant EP/L016516/1, the German National Academic Foundation (Studienstiftung des Deutschen Volkes), the Cantab Capital Institute for the Mathematics of Information and Magdalene College, Cambridge (Nevile Research Fellowship). CT was partly supported by the European Social Fund and by the Ministry Of Science, Research and the Arts Baden-W{\"u}rttemberg. Moreover, CT acknowledges support by the state of Baden-W{\"u}rttemberg through bwHPC, in particular the bwForCluster MLS&WISO Production. ",

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N2 - We propose a mean-field optimal control problem for the parameter identification of a given pattern. The cost functional is based on the Wasserstein distance between the probability measures of the modeled and the desired patterns. The first-order optimality conditions corresponding to the optimal control problem are derived using a Lagrangian approach on the mean-field level. Based on these conditions we propose a gradient descent method to identify relevant parameters such as angle of rotation and force scaling which may be spatially inhomogeneous. We discretize the first-order optimality conditions in order to employ the algorithm on the particle level. Moreover, we prove a rate for the convergence of the controls as the number of particles used for the discretization tends to infinity. Numerical results for the spatially homogeneous case demonstrate the feasibility of the approach.

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Mean-field optimal control for biological pattern formation

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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