Nonlinear Krylov Acceleration

Nonlinear Krylov Acceleration (NKA) [1] is a method for accelerating the convergence of fixed-point (Picard) iterations. Many Newton-like and inexact Newton methods are fixed point iterations. The NKA project provides the canonical implementation of the method for several programming languages. The black-box accelerator is simple to integrate into existing code. Placed in the iteration loop, it observes the sequence of solution updates and replaces them with improved updates using information it has gleaned from previous solution iterates.

It was only recently recognized (2011 [2]) that NKA is essentially equivalent to Anderson Acceleration [3] for a specific choice of mixing parameter. NKA was independently devised by Miller in 1990 in a different application context using a very different approach, and though it leads to the same algebraic method, NKA's organization is somewhat different, and arguably superior. The NKA approach also provides clear rationale for the proper choice of Anderson's arbitrary mixing parameter.

The NKA method was first described in the final section of [1]. A much more detailed description of it and a comparison with Anderson Acceleration can be found in [4]. A description can also be found in the slides doc/nlk.pdf.

Using NKA

Several different versions of NKA are provided here:

• The directory src-F95 contains the original Fortran 95 implementation. (The original implementation was in Fortran 77.)
• The directory src-F08 contains a newer object-oriented version implemented in modern Fortran. That version requires a compiler that supports the 2008 standard plus perhaps some minor 2018 features.
• The directory src-F08-vector contains a version of src-F08 that operates with abstract vector objects rather than contiguous rank-1 arrays as in the other versions. This provides much greater flexibility, but at the expense of requiring an application specific implementation of the base vector class.
• The directory src-C contains a C version, which is a straightforward translation of the F95 version.

The source for these versions consists of one or two source files that can be easily incorporated into your own software project. They all feature essentially the same interface, which is documented in the comments at the top of the source file.

Each of these versions also contain an example program that illustrates how to use NKA by solving a nonlinear elliptic equation on a regular 2D grid. There is a simple CMake-based build system. A simple cmake . in the sub- directory, followed by make will build the nka_example program. If cmake has problems finding your Fortran compiler, try setting the FC environment variable to the path to it. For a test, output from nka_example should be compared to that in reference_output. The "F08" and "F08-vector" example programs are a bit more elaborate, allowing several problem and method parameters to be set on the command line. Use the --help option to get usage information. You can get a better idea of how NKA behaves by experimenting with these programs.

References

1. N.N. Carlson and K. Miller. Design and Application of a Gradient- Weighted Moving Finite Element Code I: in One Dimension. SIAM Journal on Scientific Computing, 19(3):728-765, 1998. NKA is described in the final section.

2. H.F. Walker and P. Ni. Anderson Acceleration for Fixed-Point Iterations. SIAM Journal on Numerical Analysis, 49(4), 2011.

3. D. Anderson. Iterative Procedures for Nonlinear Integral Equations. Journal of the ACM, 12(4), 1965.

4. M.T. Calef, E.D. Fichtl, J.S. Warsa, M. Berndt, and N.N. Carlson. Nonlinear Krylov acceleration Applied to a Discrete Ordinates Formulation of the k-Eigenvalue Problem. Journal of Computational Physics, 238:188–209, 2013.