ALFA.jl

Automated Local Fourier Analysis
Author NilsKintscher
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Started In
February 2020

ALFA.jl : Automated Local Fourier Analysis

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This is a Julia implementation of the framework described in

[1] Kahl, K., Kintscher, N. Automated local Fourier analysis (aLFA). Bit Numer Math (2020). https://doi.org/10.1007/s10543-019-00797-w

The main purpose of this framework is to enable the reliable and easy-to-use analysis of complex methods on repetitive structures, e.g., multigrid methods with complex overlapping block smoothers.

Throughout this framework we refer to definitions, theorems, lemmata and algorithms in [1].

Important remarks:

  • If you only want to use this framework to analyze an operator or a method/composition of operators, you may try to proceed in the same way as in the examples: The examples in [1] can be found in the documentation of this framework.
  • If you are interested in the core algorithms and want to understand this framework completely, I recommend to read [1] completely before digging into the source code of this repository.
  • Some unit tests of this framework are done with randomly generated crystal operators which can lead to very ill-conditioned problems. Thus, these tests may fail in case the datatype used for the lattice basis and structure elements is T::Float64. As this cannot be avoided when using floating-point arithmetic, the tests don't throw an error if at least $95%$ pass. From my own experience I can say that this is not a problem in actual applications. Nevertheless, there exist rational versions ( T::Rational{BigInt}) of these algorithms which are very reliable, but also slower.

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