Convergence of a Greedy Algorithm on Nonlinear Convex Problems and Application to Uncertainty Quantification on Obstacle Problems

In this article, a method is proposed to study uncertainty propagation on high-dimensional obstacle problems. A greedy algorithm, based on variable decomposition, is used to approximate the solution of regularized problems obtained by penalization of the initial problem. The convergence of this algorithm is a consequence of a more general theorem. Indeed, the algorithm converges for the minimization of a strongly convex functional whose derivative is Lipschitz on bounded sets. We describe how this algorithm was numerically implemented and present the results which were obtained with a one-dimensional membrane problem.