Rosemary Renaut In earlier work, presented at the 1994 Iterative Methods meeting, a multisplitting (MS) method of block relaxation type was utilised for the solution of the least squares problem, and nonlinear unconstrained problems. This talk will focus on recent developments of the general approach and represents joint work both with Andreas Frommer, University of Wupertal for the linear problems and with Hans Mittelmann, Arizona State University for the nonlinear problems. The least squares problem can be solved using MS at the domain level. This amounts to a column partitioning of the underlying system matrix. Equivalently, it can be shown that the MS solution actually finds the solution of the normal equations using MS on the system matrix defined by the normal equations. Hence, this corresponds to the solution of a system of equations with SPD system matrix. According to the standard theory for MS methods this iteration will converge if and only if the underlying split is P-regular. The theory for convergence of the least squares problem shows, however, that convergence is in this case guaranteed. Hence the splitting must be P-regular. This leads us to suppose that the MS of any SPD linear system is necessarily P-regular, and provides a convergent iteration. In this talk I will indicate the convergence theory that leads to this conclusion and investigate how we might seek an alternative proof using the usual techniques from linear algebra. Block-Jacobi methods are practical for parallel implementations but the rate of convergence is limited. To attempt to remedy this inherent problem MS can be reformulated as a two-phase process. Two-stage methods, in which a splitting is applied to the already split method have already shown to be advantageous in some cases, see Szyld et al. Here a different strategy, which originates in the Parallel Variable Distribution idea of Mangasarian et al, is suggested. At each iterative step there is some flexibility in the manner in which the updated global solution is obtained. Standard MS uses a convex combination of the recent local solutions with a fixed choice of weights. Instead the minimal solution can be found by carrying out a local optimization with respect to these weights. Typically the number of subproblems indicated by the MS is substantially less than the original size of the problem. Therefore this optimization can be carried out locally to one processor with minimal extra cost. For the linear problems this minimization can actually be formulated as a least squares problem with a system matrix that only needs to be factorized at the very first iteration. Results demonstrating the increased efficiency will be presented. These ideas have also been applied for the solution of large-scale unconstrained convex minimization problems. Again, in order to not have the convergence order limited to linear we have started to explore an alternative approach. This combines an exact augmented Lagrangian with a truncated Newton iteration. Preliminary results will be presented. Rosemary A. Renaut, Department of Mathematics Arizona State University, email renaut@math.la.asu.edu