Proximal Methods for the Resolution of Variational Problems

PARENTE L. A.; ARAGONE L. S.; LOTITO P. A

Mambucaba, Brasil

Workshop; Brazopt 2009 - VIII Brazilian Workshop on Continuous Optimization; 2009

IMPA

We consider a variational problem which consists in the minimization of the functional J(u) = ess supx2f(x; u;Du) where u 2 W1;1() with v = g on @ and  Rn is bounded. This work deals with the numerical approximation of the solution. Given the convexity of the data functions, through a discretization procedure, we obtain a nite dimensional nonsmooth optimization problem, and we nd a minimum by applying a proximal-like method. In general, solutions are not unique, but a canonical extension is given by an absolute minimizer, which is de ned as the function that solves the problem on every subdomain of : It is known that an absolute minimizer u is a viscosity solution of an Euler-Lagrange equation, the so-called Aronsson equation Af [u] = 0, which in the case of optimal Lipschitz extensions, i.e. with f(x; u;Du) = kD(u)k2, becames the 1-Laplace equation. We show numerical examples and we discuss some ideas to obtain optimality conditions for absolute minimizers.J(u) = ess supx2f(x; u;Du) where u 2 W1;1() with v = g on @ and  Rn is bounded. This work deals with the numerical approximation of the solution. Given the convexity of the data functions, through a discretization procedure, we obtain a nite dimensional nonsmooth optimization problem, and we nd a minimum by applying a proximal-like method. In general, solutions are not unique, but a canonical extension is given by an absolute minimizer, which is de ned as the function that solves the problem on every subdomain of : It is known that an absolute minimizer u is a viscosity solution of an Euler-Lagrange equation, the so-called Aronsson equation Af [u] = 0, which in the case of optimal Lipschitz extensions, i.e. with f(x; u;Du) = kD(u)k2, becames the 1-Laplace equation. We show numerical examples and we discuss some ideas to obtain optimality conditions for absolute minimizers.