## Multisplitting For Linear, Least Squares and Nonlinear Problems.

```                       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

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