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 time-dependent 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, three-dimensional 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 steady-state problem or from solving a time step within an implicit, time-dependent 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 Newton-Krylov 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 matrix-vector 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 Newton-Krylov approach for solving nonlinear systems. While FAS methods may require less storage than Newton-Krylov 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 Newton-Krylov-multigrid methods. As the success of FAS methods can be problem-dependent, 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 grid-related data as part of the coarse grid operator formulation.

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For more information, contact: Carol Woodward, Project Lead, 925-424-6013,