14:00h – 14:40h

A randon domain decomposition scheme for parabolic PDE constrained optimization problems

Probability theory has had a significant impact on various areas of mathematics, particularly in algorithms and
combinatorics. The use of randomness in algorithms dates back to the Monte Carlo methods [6, 7]. In subsequent
years, random algorithms gained strength in optimization, notably with the emergence of techniques such as
simulated annealing [5] and genetic algorithms [4], which were used in complex optimization problems. In
recent years, stochastic gradient descent has attracted attention in machine learning and artificial intelligence; its
development has been crucial in training large-scale models, specially machine learning algorithms, see [1, 2].
In this talk, we discuss a random domain decomposition scheme for parabolic optimal control problems,
inspired by mini-batch algorithms used in machine learning. This scheme is based on the papers [3, 8] The method
discretizes the parabolic equation using the explicit Euler scheme and replaces the full elliptic operator with a
randomly selected elliptic operator acting on a part of the domain at each step. This approach aims to reduce
computational time. We will demonstrate the convergence of the scheme, highlighting the trade-off between speed
and accuracy. Special attention is paid to optimal control problems with bang-bang minimizers.
Acknowledgements
The author has been funded by means of an Alexander von Humboldt scholarship.
References
[1] Léon Bottou. Large-scale machine learning with stochastic gradient descent. In Proceedings of COMPSTAT’2010, pages 177–186.
Physica-Verlag/Springer, Heidelberg, 2010.
[2] Léon Bottou, Frank E. Curtis, and Jorge Nocedal. Optimization methods for large-scale machine learning. SIAM Rev., 60(2):223–311,
2018. doi:10.1137/16M1080173.
[3] Monika Eisenmann and Tony Stillfjord. A randomized operator splitting scheme inspired by stochastic optimization methods. arXiv
preprint arXiv:2210.05375, 2022.
[4] John H. Holland. Adaptation in natural and artificial systems. University of Michigan Press, Ann Arbor, MI, 1975. An introductory
analysis with applications to biology, control, and artificial intelligence.
[5] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi. Optimization by simulated annealing. Science, 220(4598):671–680, 1983.
URL: https://www.science.org/doi/abs/10.1126/science.220.4598.671, arXiv:https://www.science.org/doi/pdf/
10.1126/science.220.4598.671, doi:10.1126/science.220.4598.671.
[6] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. Teller. Equation of state calculations by fast computing machines.
The Journal of Chemical Physics, 21(6):1087–1092, 1953.
[7] Nicholas Metropolis and S. Ulam. The Monte Carlo method. J. Amer. Statist. Assoc., 44:335–341, 1949. URL: http://links.jstor.
org/sici?sici=0162-1459(194909)44:247<335:TMCM>2.0.CO;2-3&origin=MSN.
[8] D. W. M. Veldman and E. Zuazua. A framework for randomized time-splitting in linear-quadratic optimal control. Numer. Math.,
151(2):495–549, 2022. doi:10.1007/s00211-022-01290-3.

14:50h – 15:30h

Nonlocal Diffusion Equations with Nonlocal reaction.