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Filtered Gradient Algorithms for Inverse Design Problems of One-Dimensional Burgers Equation

  • Laurent Gosse
  • Enrique Zuazua
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 16)

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.

Notes

Acknowledgements

This work was partially supported by the Advanced Grant 694126-DYCON (Dynamic Control) of the European Research Council Executive Agency, ICON of the French ANR (2016-ACHN-0014-01), FA9550-15-1-0027 of AFOSR, A9550-14-1-0214 of the EOARD-AFOSR, and the MTM2014-52347 Grant of the MINECO (Spain). Both the authors thank Prof. Andreas Griewank who suggested to study convex combinations of inverse designs during a meeting in September 2015.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.IACCNRRomaItaly
  2. 2.DeustoTech, Fundación DeustoAvda UniversidadesBilbao, Basque CountrySpain
  3. 3.Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain
  4. 4.Facultad IngenieríaUniversidad de Deusto, Avda. UniversidadesBasque CountrySpain

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