Filtered gradient algorithms for inverse design problems of one-dimensional Burgers equation

Gosse L., Zuazua E Filtered gradient algorithms for inverse design problems of one-dimensional Burgers equation
Innovative Algorithms and Analysis,Gosse, Laurent, Natalini, Roberto (Eds.), Springer INdAM Series 16, 2017, 197-228. DOI: 10.1007/978-3-319-49262-9_7

Abstract: Inverse design for hyperbolic conservation laws is exemplified through the 1D Burgers equation which is motivated by aircraft’s sonic-boom minimization issues. In particular, we prove that, as soon as the target function (usually a N-wave) isn’t continuous, there is a whole convex set of possible initial data, the backward entropy solution being possibly its centroid. Further, an iterative strategy based on a gradient algorithm involving “reversible solutions” solving the linear adjoint problem is set up. In order to be able to recover initial profiles different from the backward entropy solution, a filtering step of the backward adjoint solution is inserted, mostly relying on scale-limited (wavelet) subspaces. Numerical illustrations, along with profiles similar to F-functions, are presented.

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