Numerical computation of rank-one convex envelopes. (English) Zbl 0941.65062

Dolzmann, Georg

SIAM J. Numer. Anal. 36, No. 5, 1621-1635 (1999).

A real-valued function \(f\) defined on the space \(M^{m\times n}\) of all real \(m\times n\) matrices is said to be rank-one convex if the functions \(\varphi:\mathbb{R}\to \mathbb{R}\) defined by \[ \varphi(t)= f(F+ t\cdot R) \] are convex \(\forall F\), \(R\in M^{m\times n}\) and rank \(R\leq 1\). The computation of a rank-one convex envelope \(f^{rc}\) for a given not rank-one convex function \(f\) is a difficult problem.
The author describes a convergent numerical algorithm and gives an error estimation in \(L^\infty\). Some numerical tests are presented.

Reviewer: H.Benker (Merseburg)

MSC:

65K10	Numerical optimization and variational techniques
49M25	Discrete approximations in optimal control
49J40	Variational inequalities

Keywords:

rank-one convexity; generalized convex envelopes; nonconvex variational problems; convergence; algorithm; error estimation; numerical tests