MultiParamContinuation.jl

This Julia package aims at performing multi parameter continuation of possibly large dimensional equations $F(u, par) = 0$ where $F:\mathbb R^n\times \mathbb R^{par}\to\mathbb R^m$ by taking advantage of iterative methods, dense / sparse formulation and specific hardwares (e.g. GPU).

It incorporates a continuation algorithm ^{[Henderson][Dankowicz]} based on a Newton method to correct a predictor step and a Matrix-Free/Dense/Sparse eigensolver can be used to compute stability and bifurcation points.

Limitations

As this is early work, the following limitations need to be addressed.

• It is partially optimized for speed (allocations, static arrays, etc)
• It is not suitable as is for large scale problems although it is very simple to address this. Note that the interface for jacobian free computation has yet to been pushed.
• It only computes 2d manifolds for now, i.e. $n=m+2$
• It allows loose detection of continuous events (no bisection)
• One needs to improve interface for using BVH search tree in large dimensions.

Installation

To install it, please run

] add MultiParamContinuation

To install the bleeding edge version, please run

] add MultiParamContinuation#master

Citing this work

If you use this package for your work, we ask that you cite the following paper!! Open source development strongly depends on this. It is referenced on HAL-Inria as follows:

@misc{veltz:hal-02902346,
  TITLE = {{BifurcationKit.jl}},
  AUTHOR = {Veltz, Romain},
  URL = {https://hal.archives-ouvertes.fr/hal-02902346},
  INSTITUTION = {{Inria Sophia-Antipolis}},
  YEAR = {2020},
  MONTH = Jul,
  KEYWORDS = {pseudo-arclength-continuation ; periodic-orbits ; floquet ; gpu ; bifurcation-diagram ; deflation ; newton-krylov},
  PDF = {https://hal.archives-ouvertes.fr/hal-02902346/file/354c9fb0d148262405609eed2cb7927818706f1f.tar.gz},
  HAL_ID = {hal-02902346},
  HAL_VERSION = {v1},
}

Other softwares

There are many good softwares already available.

• For continuation in small dimension, there is only COCO which is very reliable and for now more general than MultiParamContinuation.jl as it can compute k-d manifolds. Of course, we have the original C++ implementation by M. Henderson Multifario.

• For large scale problems, there is only Trilinos-LOCA

In Julia, there is no other alternative to the current package.

Reproducibility

References