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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Keywords: Optimal control with ODE/PDE constraints, interacting particle systems, mean-field limits, dynamical systems, pattern formation
@article{COCV_2021__27_1_A42_0,
author = {Burger, Martin and Kreusser, Lisa Maria and Totzeck, Claudia},
title = {Mean-field optimal control for biological pattern formation},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021034},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021034/}
}
TY - JOUR AU - Burger, Martin AU - Kreusser, Lisa Maria AU - Totzeck, Claudia TI - Mean-field optimal control for biological pattern formation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021034/ DO - 10.1051/cocv/2021034 LA - en ID - COCV_2021__27_1_A42_0 ER -
%0 Journal Article %A Burger, Martin %A Kreusser, Lisa Maria %A Totzeck, Claudia %T Mean-field optimal control for biological pattern formation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021034/ %R 10.1051/cocv/2021034 %G en %F COCV_2021__27_1_A42_0
Burger, Martin; Kreusser, Lisa Maria; Totzeck, Claudia. Mean-field optimal control for biological pattern formation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 40. doi: 10.1051/cocv/2021034
[1] , , and , Mean field control hierarchy. Appl. Math. Optim. 76 (2017) 93–135.
[2] , and , Gradient Flows: In Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics. ETH Zürich. Birkhäuser Basel (2005).
[3] , , and , Mean-field pontryagin maximum principle. Opt. Theo. Appl. 175 (2017) 1–38.
[4] , , and , On a boltzmann-type price formation model. Proc. Roy. Soc. A Math. Phys. Eng. Sci. 469 (2013) 20130126.
[5] , , , and , Pattern formation of a nonlocal, anisotropic interaction model. Math. Models Methods Appl. Sci. 28 (2018) 409–451.
[6] , , and , Mean field games with nonlinear mobilities in pedestrian dynamics. Disc. Cont. Dyn. Syst. B 19 (2014) 1311–1333.
[7] , , and , Mean-field optimal control and optimality conditions in the space of probability measures. SIAM: J. Control Optim. 59 (2021) 977–1006.
[8] , , , and , Instantaneous control of interacting particle systems in the mean-field limit. J. Computat. Phys. 405 (2020) 109181.
[9] , , , and , An analytical framework for a consensus-based global optimization method. Math. Mod. Meth. Appl. Sci. 28 (2018).
[10] , , and , Equilibria of an anisotropic nonlocal interaction equation: analysis and numerics. Preprint (2019). | arXiv
[11] , , and , Stability analysis of line patterns of an anisotropic interaction model. SIAM J. Appl. Dyn. Syst. 18 (2019) 1798–1845.
[12] , and , Meanfield games and model predictive control. Commun. Math. Sci. 15 (2017) 1403–1422.
[13] , , and , Self-propelled particles with soft-core interactions: patterns, stability, and collapse. Phys. Rev. Lett. 96 (2006) 104302.
[14] , , , and , An anisotropic interaction model for simulating fingerprints. J. Math. Biol. 78 (2019) 2171–2206.
[15] , , and , Boltzmann and Fokker–Planck equations modelling opinion formation in the presence of strong leaders. Proc. Roy. Soc. A Math. Phys. Eng. Sci. 465 (2009) 3687–3708.
[16] and , Pot: Python optimal transportlibrary (2017). https://pythonot.github.io.
[17] , and , Distributed optimal control for multi-agent trajectory optimization. Automatica 50 (2014) 149–154.
[18] and , Mean-field optimal control. ESAIM: Cont. Optim. Calcul. Variat. 20 (2014) 1123–1152.
[19] and , Mathematical modelling of cancer cell invasion of tissue: local and non-local models and the effect of adhesion. J. Theoret. Biol. 250 (2008) 684–704.
[20] , Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, chapter On the Dynamics of Large Particle Systems in the Mean Field Limit. Springer International Publishing, Cham (2016), 1–144.
[21] , and , Instantaneous control for traffic flow. Math. Methods Appl. Sci. 30 (2007) 153–169.
[22] , , and , Optimization with PDE Constraints. Springer (2009).
[23] and , On anisotropic diffusion equations for label propagation. Preprint (2020). | arXiv
[24] and , Merkel cells and the individuality of friction ridge skin. J. Theoret. Biol. 317 (2013) 229–237.
[25] , and Control to flocking of the kinetic cucker–smale model. SIAM J. Math. Anal. 47 (2015) 4685–4719.
[26] , , and , A consensus-based model for global optimization and its mean-field limit. Math. Mod. Meth. Appl. Sci. 27 (2017).
[27] , , and , Mathematical modelling of turning delays in swarm robotics. IMA J. Appl. Math. 80 (2015) 1454–1474.
[28] , An anisotropic interaction model with collision avoidance. Kin. Rel. Mod. 13 (2019) 1219–1242.
[29] and , Consensus-based global optimization with personal best. Math. Bio. Eng. 17 (2020) 6026–6044.
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