Numerical methods for non-smooth problems from Calculus of Variations: Applications

The numerical solutions of non-smooth problems from the Calculus of Variations are addressed, these problems having implications in image denoising, plasticity theory, interface reconstruction for free surface flows, generalized eigenvalue problems and computation of best constants in Sobolev inequalities. Numerical algorithms are developed to treat efficiently the corresponding non-smooth operators.;In a second part, the numerical solution of non-smooth eigenvalue problems from the Calculus of Variations are addressed. A scalar problem is considered first. It consists of minimizing the L1 norm of the gradient of a function vanishing on the boundary, over the ball composed of functions whose L2 norm is one. In a second step the vectorial cases are addressed, in which the objective functions vary between the gradient and the rate deformation tensor of the function under the same constraints. Piecewise linear or Bercovier-Pironneau ( P1 -iso- P2/P1 ) finite elements are used for the space discretizations. The previous problems are solved using an augmented Lagrangian method together with an Uzawa-Douglas-Rachford scheme. The smallest eigenvalues are shown to be constant for all bounded domains in two dimensions and the finite element approximations converge to the exact solutions. Numerical solutions are exhibited for the computation of the smallest eigenvalues and eigenfunctions on various domains.;In a first part, numerical methods for non-smooth optimization problems based on L1 norms are presented for smoothing of signals with noise or functions with sharp gradients. The problems addressed here consist of the minimization of the distance between a given signal typically with jumps or noise, and a smooth approximation. A smoothing term is introduced to add regularity. The L1 distance between the noisy signal and its approximation with a L2 smoothing term is considered first. In a second step, the L 2 distance together with a L1 smoothing term is considered. Finally the L1 distance with a L1 smoothing term is considered. The previous problems are treated with an over-relaxation algorithm, an augmented Lagrangian method or a combination of both. Piecewise finite elements are used for the space discretization.