SNESFAS#

An implementation of the Full Approximation Scheme nonlinear multigrid solver, FAS, or nonlinear multigrid [BKST15] for solving nonlinear systems of equations with SNES. The nonlinear problem is solved by correction using coarse versions of the nonlinear problem. This problem is perturbed so that a projected solution of the fine problem elicits no correction from the coarse problem.

Options Database Keys and Prefixes#

-snes_fas_levels l - The number of levels
-snes_fas_cycles (1|2) - The number of cycles – 1 for V, 2 for W
-snes_fas_type (additive|multiplicative|full|kaskade) - Additive or multiplicative cycle
-snes_fas_galerkin (true|false) - Form coarse problems by projection back upon the fine problem
-snes_fas_smoothup u - The number of iterations of the post-smoother
-snes_fas_smoothdown d - The number of iterations of the pre-smoother
-snes_fas_monitor - Monitor progress of all of the levels
-snes_fas_full_downsweep (true|false) - Call the downsmooth on the initial downsweep of full FAS
-snes_fas_use_coarse_correction_linesearch (true|false) - Use a line search for the multiplicative coarse correction update on every level except the coarsest
-fas_coarse_correction_snes_linesearch_type (none|bt|secant|cp|nleqerr|bisection|shell) - Type of the enabled coarse correction line search, on all levels
-fas_coarse_correction_i_snes_linesearch_type (none|bt|secant|cp|nleqerr|bisection|shell) - Override the coarse correction line search type on level i only
-fas_levels_snes_ - Prefix for SNES options for all smoothers
-fas_levels_cycle_snes_ - Prefix for SNES options for all cycles
-fas_levels_i_snes_ - Prefix SNES options for the smoothers on level i
-fas_levels_i_cycle_snes_ - Prefix for SNES options for the cycle on level i
-fas_coarse_snes_ - Prefix for SNES options for the coarsest smoother

Notes#

The organization of the SNESFAS solver is slightly different from the organization of PCMG As each level has smoother SNES instances(down and potentially up) and a cycle SNES instance. The cycle SNES instance may be used for monitoring convergence on a particular level.

The coarse correction \(X += I(x^c - Rx)\) is applied directly (a unit step, equivalent to the original FAS correction) unless -snes_fas_use_coarse_correction_linesearch or SNESFASSetUseCoarseCorrectionLineSearch() is set, in which case a dedicated SNESLineSearch is created on every level, except the coarsest, (one per recursive SNESFAS instance) and applied instead. MG-Opt [Nas00] generalizes the correction this way, which is useful when monotonic decrease in the residual norm or energy functional is desired (e.g., non-convex problems where the full coarse correction may overshoot without an explicit convergence control strategy). As with PCMG’s -mg_levels_pc_type vs. -mg_levels_2_pc_type pattern, -fas_coarse_correction_snes_linesearch_type sets the line search type for all levels while -fas_coarse_correction_i_snes_linesearch_type overrides only level i; the line search can be retrieved via SNESFASGetCoarseCorrectionLineSearch().