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| In 2012, the Societe de Calcul Mathematique and the Federation Francaise des Jeux Mathematiques published a mathematical open competition riddling what the best way is to command fire fighting units over a vast spatial map over a long time period to fight wildland fire with minimal economic effort and maximal impact. While the original challenge narrative obviously roots in cellular games, one can translate the task into a coupled system of convection-diffusion partial differential equations modeling the fire spreading, time-dependent control functions modeling the fire fighters, and ordinary differential reaction equations modeling the burning. Within such a setting, the captain of the fire fighters faces an optimal control problem: His units have to minimize a cost functional.
The present talk first applies standard finite element techniques on adaptive Cartesian meshes to transform the reaction-diffusion problem into a discretised one-step time stepping formulation, and it applies a textbook adjoint formulation to translate the impact of unit movements into defect variables yielding an optimal control. This gives a second parabolic differential equation running backward in time as well as an additional evolution equation per finite element. With a parallel, memory-efficient realisation of simple relaxation solvers of all the equations on a space-time adaptive Cartesian grid, we are able to solve all equations simultaneously which in turn allows us to study some interesting techniques: local time stepping resulting from adaptivity in space-time grids, successive adoption of the control states in both space and time, and the multiscale interaction of control and the forward problem, e.g. Particular interesting is the opportunity to tailor hierarchically the unit movement in time to the fire spreading. |
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