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Calculus of variations problems are generally cast in in the following simple form: find $u(t)$ that satisfies some boundary conditions and minimises

$$ F[u] = \int_{t=0}^{t=t_f} f(u(t),u'(t),t) dt. $$

I have run into a far more general problem. I have a functional $F[u]$ that is substantially more complicated. Given $u$ (or in this case the set of functions $u^i(\mathbf{x})$ where $\mathbf{x}\in\mathbb{R}^n$) I solve a pde which depends on these functions, and the solution is analysed to give the value of the functional $F$. Given a set of functions $u^i(\mathbf{x})$, I can numerically evaluate $F[u^i]$ very efficiently. However, I want to explore the function space to find the set of $u^i$ that maximises $F$. To complicate matters, I have further constraints on the $u^i$ I can choose: for instance, I require a particular path integral of $u^i$ to equal a certain value.

All the methods I have found in the calculus of variations seem to assume, in one way or another, that the functional $F$ is of the simple form outlined above. Do more general numerical methods exist? I'm looking for a method or algorithm that requires nothing more than the ability to numerically evaluate the functional for any given function, and from that will search the function space and converge on an optimal solution. The ability to place (ideally arbitrary) constrains on $u^i$ would be an added bonus.

Perhaps I ask for too much. I'd appreciate any help or advice.

Thanks

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    $\begingroup$ Can you tell us more about your functional and the PDE that the functions satisfy? It's hard to say much otherwise. However, I'd point you to the field of "PDE-constrained optimization". Here's a presentation on the topic by @WolfgangBangerth (who happens to be active on this site), and he may be more helpful than I can be. Unfortunately, the field is extremely difficult, so the answers may be unsatisfactory. homes.esat.kuleuven.be/~optec/events/workshops/PDEworkshop/… $\endgroup$ – Tyler Olsen Oct 11 '17 at 14:59
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    $\begingroup$ Ha, how funny that you'd find such an old presentation of mine :-) And yes, more detail in the question would definitely help! $\endgroup$ – Wolfgang Bangerth Oct 12 '17 at 1:42
  • $\begingroup$ Hi all - thanks for your help. The presentation was exactly what I was looking for - I was unaware of PDE-constrained optimisation, and the breadth of the world I was walking into. I'm going to try solving my problem with dolfin-adjoint, and see how we go. Thanks again. $\endgroup$ – tom Oct 13 '17 at 6:07

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