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Since the first international conference on domain decomposition methods in Paris in 1987 with legends attending like Pierre-Louis Lions, Gene H. Golub, Olof B. Widlund, Roland Glowinski, Tony Chan, Gerard Meurant and more, the field has reached a certain maturity for solving elliptic partial differential equations. In 2020 the 26th edition of this international conference on domain decomposition methods took place in Hong Kong. Research has not slowed down in this field and in particular, massively larger problems are treated with domain decomposition techniques these days, see for example the PinT approach for space-time problems.
Furthermore, heterogeneous domain decomposition methods for coupling different physics like fluid-structure interaction, or coupling different models like in discrete fracture networks, are still very much in development. Also, coarse spaces for domain decomposition methods are still not very well understood, with many new results like GenEO and SHEM, and their relation to reduced basis methods.
Our research school will address a further very active area of research for solving very large-scale problems, by bringing the community of domain decomposition and optimal control together.
Optimal control problems are solved by solving their optimality conditions, which are represented by coupled direct and adjoint equations (the Lagrange multiplier), and, for space-time optimal control problems, this requires storing the entire space-time trajectories both for the forward and backward problem, leading to extremely large systems that have to be solved.
Domain decomposition techniques are currently only being developed for optimal control problems, and a major difficulty is that one needs expertise in both research areas, domain decomposition and optimal control, to be effective for such development. The goal of our research school is then precisely to give an introduction to both fields by experts from the respective areas, to bring participants to a sufficient level to be able to understand then the more specialized lectures on the various recent domain decomposition methods, especially developed for optimal control and PDE constrained optimization problems.