New optimized and accelerated pam methods for solving large non-symmetric linear systems: Theory and practice

The Projected Aggregation Methods generate the new point xk+1 as the projection ofxk onto an “aggregate” hyperplane usually arising from linear combinations of the hyperplanes planes defined by the blocks. In [13] an acceleration scheme was introduced for algorithms in which an optimized search direction arises from the solution of small quadratic subproblems. In this paper we extend that theory to classical methods like Cimmino's and to the generalized convex combination as defined in [5]. We prove that the resulting new highly parallel, algorithms improve the original convergence rate and present numerical results which show their outstanding computational efficiency.