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Suppose we have a PDE that is solvable in terms of certain boundary conditions. Suppose further that rather than the boundary conditions being known, we know what solution we would like for the PDE, and we wish to find the "optimal" boundary conditions that will lead to our known solution.

As an example, suppose we wanted to get a certain temperature distribution in a room, and we wanted to set a radiator temperature over time optimally to achieve that distribution.

Are there general methods to solve this type of problem, and if so, what are they?

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