Brief Overview of Nonlinear Solvers Research
Scope of the Project
With the advent of powerful, massively parallel supercomputers and highly
effective solution methods, the solution to fully implicit formulations of many
application models has become possible, whereas in the recent past, solving
these formulations had been considered intractable. Solutions to fully implicit
formulations provide more accuracy and, due to the ability to take larger time
steps in resolving timedependent behavior, can be faster to solve than
explicit formulations.
NSDE project members conduct research in the solution of implicit nonlinear
differential equations discretized on large, threedimensional meshes and
arising in the LLNL mission areas of radiation transport and groundwater
flow. Both of these applications require the solution of large systems of
coupled, nonlinear equations. Current project research leverages advances in
each application area to develop better solvers for both. This work examines
parallel solution strategies for nonlinear discrete systems arising from
solving a steadystate problem or from solving a time step within an implicit,
timedependent problem. Investigators also pursue solution strategies for the
linear systems arising in each nonlinear iteration.
Algorithmic Research Areas
A primary focus has been in the application of NewtonKrylov methods for
solving the nonlinearities in the applications of interest. These methods can
result in quadratic convergence of the nonlinear iteration, provided the linear
systems have been solved sufficiently accurately. One difficulty, however,
arises in iterating the linear solver to a low enough tolerance, a requirement
that often necessitates an efficient and scalable preconditioner for the Krylov
method. Project members have been investigating various multigrid methods
applied as preconditioners in the solution of the linear Jacobian systems at
each nonlinear iteration. In particular, they are studying the choice of
approximation of the Jacobian as a preconditioning matrix, and ways of
selecting the best approximation. They have also been investigating the use of
stale values of tabulated data in the Krylov method matrixvector multiply.
Additionally, for both applications under consideration, NSDE project members
are investigating the viability of nonlinear multigrid schemes such as the Full
Approximation Storage (FAS) method, for problems of interest at the
laboratory. These schemes provide an alternative to the NewtonKrylov approach
for solving nonlinear systems. While FAS methods may require less storage than
NewtonKrylov schemes, they are not as fully understood. Project members are
examining these methods for the specified applications, and comparing the
resulting performance to that of the NewtonKrylovmultigrid methods. As the
success of FAS methods can be problemdependent, project researchers are trying
to determine the best method for each of the application areas noted
earlier. Specific research topics include: determining the radius of
convergence of the FAS method; developing techniques for choosing the coarse
grid operators, and coarsening gridrelated data as part of the coarse grid
operator formulation.
