Background

The application that motivated the development of Newton-Krylov_OOC is spinning up biogeochemical tracers in an ocean general circulation model (OGCM). The target model configuration is multi-year runs with O(106-7) grid points. Such runs take hours of time on a supercomputer and are typically done through a batch submission system.

Literature on the topic demonstrates that Newton-Krylov (NK) based solvers can be used effectively to solve this problem [Kha08, LP08]. These solvers utilize Newton’s method to solve for an initial model state such that end state from running the model forward in time is the same as the initial state. Newton’s method is a general iterative method for solving non-linear equations. Successive iterations are computed by adding an increment to the previous iterate. The increment is the solution of a large system of linear equations. It turns out that it is not practical to directly solve the system of linear equations that determine the increment, so an iterative solver is used. Krylov-based iterative solvers are well suited to solve the system of equations that arise. Each iteration of the Krylov solver requires a forward model run, which in our target application is a multi-year run. Overall, using an NK solver requires at least 10s of forward model runs.

An excellent book on NK solvers, [Kel03], advises that it is preferable to use an existing Newton–Krylov implementation instead of implementing your own. However, existing implementations of NK solvers have some features that make their usage impractical for the target model configuration.

  1. In-Core Memory Requirement: Existing NK solvers utilize a driver program that stays resident in memory while function evaluations occur. On some computing systems, it is not possible to have the driver program reside in memory for the duration of multiple model runs.

  2. Solving One Problem at a Time:

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The application of a preliminary version of Newton-Krylov_OOC to spinning up passive tracers being run online in an OGCM is documented in [Lin17].

References

Kel03

C. T. Kelley. Solving Nonlinear Equations with Newton's Method. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2003. doi:10.1137/1.9780898718898.

Kha08

Samar Khatiwala. Fast spin up of ocean biogeochemical models using matrix-free Newton-Krylov. Ocean Model., 23(3–4):121–129, 2008. doi:10.1016/j.ocemod.2008.05.002.

LP08

Xingwen Li and François W. Primeau. A fast Newton-Krylov solver for seasonally varying global ocean biogeochemistry models. Ocean Model., 23(1–2):13–20, 2008. doi:10.1016/j.ocemod.2008.03.001.

Lin17

K. Lindsay. A Newton-Krylov solver for fast spin-up of online ocean tracers. Ocean Model., 109:33–43, 2017. doi:10.1016/j.ocemod.2016.12.001.