Additive Schwarz Methods for Convex Optimization as Gradient Methods

Multilevel Additive Schwarz Methods

We consider the solution of the algebraic system of equations which result from the dis-cretization of elliptic equations. A class of multilevel algorithms are studied using the additive Schwarz framework. We establish that, in the general case, the condition number of the iterative operator grows at most linearly with the number of levels. The bound is independent of the mesh sizes and the num...

Universal gradient methods for convex optimization problems

In this paper, we present new methods for black-box convex minimization. They do not need to know in advance the actual level of smoothness of the objective function. Their only essential input parameter is the required accuracy of the solution. At the same time, for each particular problem class they automatically ensure the best possible rate of convergence. We confirm our theoretical results...

Asynchronous Weighted Additive Schwarz Methods

A class of asynchronous Schwarz methods for the parallel solution of nonsingular linear systems of the form Ax = f is investigated. This class includes, in particular, an asynchronous algebraic Schwarz method as well as asynchronous multisplitting. Theorems are obtained demonstrating convergence for the cases when A 1 is nonnegative and when A is an H-matrix. The results shown are for both the ...

Additive Schwarz Methods for Helmholtz Problems

Additive Schwarz Methods for Hyperbolic Equations 5153