PDE control and numerics: Some recent progress and challenges
Most applications not only involve modelling and forward resolution of PDE models but also parameter identification, control and observation problems. It turns out that a successful development of the needed computational tools cannot be achieved by simply superposing the state of the art on PDE, Control and Inverse Problem Theory, with classical Numerical Analysis.
The number of interesting and often difficult issues that arise in the interplay of numerics and PDE inversion and control are huge.
In this lecture we shall address some of them, with special focus on the following two ones:
* We first address the problem of inverse design, aiming to identify the initial source leading to a desired final configuration, in the context of hyperbolic or viscous conservation laws. This problem is ill-posed because of the instabilities of backward parabolic problems and, in the inviscid hyperbolic context, because of the lack of uniqueness. We discuss numerical strategies for an efficient inversion, in particular, in the presence of multiple solutions. This leads, in particular, to interesting open problems on the numerical approximation of all possible weak solutions of conservation laws and not only the entropic ones.
* We then analyse the impact of numerical meshes on the propagation properties of numerical solutions. When dealing with transport and wave equations, Fourier analysis allows showing that high frequency wave packets have the tendency of evolving very slowly, ruining the efficiency of inversion procedures. We show how, for a given inversion problem, taking its geometric features into account, an ad hoc computational mesh can be built so to ensure the efficient observation, inversion and control of all numerical waves.
We shall conclude describing some challenging open problems and possible directions for future research.