Numerical approximation of young measuresin non-convex variational problems |
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Authors: | Carsten Carstensen Tomáš Roubíček |
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Institution: | Mathematisches Seminar, Christian-Albrecht-Universit?t zu Kiel, Ludewig-Meyn-Strasse 4, D-24098 Kiel, Germany, DE Mathematical Institute, Charles University, Sokolovská 83, CZ-186 75 Praha 8, Czech Republic, CZ
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Abstract: | Summary. In non-convex optimisation problems, in particular in non-convex variational problems, there usually does not exist any classical
solution but only generalised solutions which involve Young measures. In this paper, first a suitable relaxation and approximation
theory is developed together with optimality conditions, and then an adaptive scheme is proposed for the efficient numerical
treatment. The Young measures solving the approximate problems are usually composed only from a few atoms. This is the main
argument our effective active-set type algorithm is based on. The support of those atoms is estimated from the Weierstrass
maximum principle which involves a Hamiltonian whose good guess is obtained by a multilevel technique. Numerical experiments
are performed in a one-dimensional variational problem and support efficiency of the algorithm.
Received November 26, 1997 / Published online September 24, 1999 |
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Keywords: | Mathematics Subject Classification (1991):65K10 65N50 49M40 |
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