On a Conjugate Gradient/Newton/Penalty Method for the Solution of Obstacle Problems. Application to the Solution of an Eikonal System with Dirichlet Boundary Conditions

Abstract

As the dedication suggests, many scientists in several countries have contributed significantly during these last fifty years to make conjugate gradient the efficient and versatile method it is today. However, this article is not limited to conjugate gradient and from that point of view, it owes a lot to M. Hestenes for his major contributions, not only to conjugate gradient, but also to calculus of variations and augmented Lagrangian methods. Indeed, in this article after investigating the solution of obstacle problems for elliptic and parabolic linear operators by a methodology combining penalty, Newton’s method and conjugate gradient algorithms, we shall apply the above methodology to the search for non-negative solutions to an Eikonal system with Dirichlet boundary conditions, after transforming it into a (non-convex problem from Calculus of Variations; in order to solve this non-convex minimization problem we shall apply an operator-splitting m ethod, and as shown in, e.g., [1], [2] operator splitting methods have close links with augmented Lagrangian methods and therefore with M. Hestenes contributions to Applied and Computational Mathematics. Since, in this article, we make a systematic use of conjugate gradient algorithms, and of cyclic reduction methods for the fast solution of discrete linear elliptic problems, a special credit has to be given to G. H. Golub for his outstanding contributions to these topics (among many other contributions, which include, in fact, complementarity methods for the solution of discrete obstacle problems). We deeply regret he could not attend the Jyväskylä Conference on “50 Years of Conjugate Gradients”.