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Sparse basis pursuit for compliance minimization in the vanishing volume ratio limit
Author(s) -
Evgrafov Anton,
Sigmund Ole
Publication year - 2020
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.202000008
Subject(s) - basis pursuit , limit (mathematics) , benchmark (surveying) , context (archaeology) , minification , basis (linear algebra) , mathematics , mathematical optimization , optimization problem , regular polygon , convex optimization , computer science , algorithm , compressed sensing , matching pursuit , mathematical analysis , paleontology , geometry , geodesy , biology , geography
We consider a non‐smooth convex variational problem appearing as a formal limit of compliance minimization in the vanishing volume ratio limit. The problem has a classical basis pursuit form, and several successful algorithms have been utilized to solve problems of this class in other application contexts. We discuss the well‐posedness and regularity of solutions to these problems, possible solution algorithms, and their discretizations as relevant in this mechanical engineering context. We then test the algorithms on a few benchmark problems with available analytical solutions. We find that whereas many algorithms are successful in estimating the optimal objective value to the problem to a high accuracy, the same cannot be said about finding the optimal solutions themselves. In particular, in some examples the algorithms struggle to properly identify the areas where the solutions should vanish entirely. We also discuss an example where the found optimal solutions are not sparse even though sparse(r) solutions exist.