I will introduce approaches to solve time domain decomposition (TDD)
formulations of large-scale optimal control problems. Optimal control
problems arise in many science and engineering applications, but their
numerical solution is expensive, both in terms of computing time and
memory requirements. TDD or direct multiple shooting formulations of
optimal control problems address these challenges by decomposing the
underlying differential equations into equations on shorter time
subintervals and couple these at the time interval boundaries. These
coupling conditions must be satisfied at the solution, but not during
the iteration of an optimization algorithm. This is exploited to
improve the numerical solution of such problems through superior
stability properties of sub-problems, introduction of parallelism, and
reduction of permanent memory requirements. However, TDD formulations
have a price: The auxiliary initial data at time interval boundaries
are additional optimization variables and the coupling conditions are
additional constraints. For problems governed by (discretized) PDEs
this leads to huge increases in optimization variables and
constraints. I will discuss methods to solve TDD formulations, present
convergence results, illustrate their performance on applications, and
sketch several open problems.
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