A Moment Approach to Approximating Functional Solutions

Tillmann Weisser
Los Alamos National Laboratory

During the last decade different non-linear problems in fields such as optimal control, uncertainty quantification, and recently partial differential equations have been shown to possess equivalent formulations as linear optimization problems over the space of measures. If these linear problems have some additional algebraic properties the so-called Moment-SOS-hierarchy provides a tool to approximate the moments of the optimizing measures – and eventually solve the original non-linear problems – via semi-definite programming.
A well-founded objection to this approach is that, while the solution to the original problem usually is a function, the hierarchy solutions typically are the moments of the associated Young measure. It is therefore an active field of research to recover the graph of a function based on a finite number of moments of the corresponding Young measure only. Beyond the mentioned context, this question is interesting itself and pops up, e.g., in machine learning.
We present a new way of solving scalar hyperbolic PDEs employing the Moment-SOS-hierarchy and use this as an example to show our approach to recover the graph of a functional solution from knowledge of a finite sequence of approximated moments.

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