Asymptotic numerical method (ANM)

Dimensions

This is a method for small dimensions, less than several thousands.

To access this algorithm, you have to use the package AsymptoticNumericalMethod.jl

The method [Rubbert],[Charpentier] seeks a Taylor approximation of the solutions of

\[\mathbf F(X,p)=0\in\mathbb R^n\]

where $X\in\mathbb R^n, p\in\mathbb R$. The solution is found by solving

\[F(x(s),p(s))= 0\]

\[\langle x(s)-x(0),\frac{\partial u}{\partial s}(0)\rangle + (p(s)-p(0))\frac{\partial p}{\partial s}(0) = s\]

where

\[x(s)=\sum\limits_{k=0}^K x_ks^k,\quad p(s)=\sum\limits_{k=0}^K p_ks^k\]

for some user passed $K>0$. It gives

\[F(x(s),p(s)) = \sum\limits_{k=0}^K F_ks^k+h.o.t.\]

from which we deduce the equations $F_k=0$. We then find:

\[F_{1, x_1=Id, p_1=1}(x_k,p_k) = -F_{k,x_k=0,p_k=0}.\]

The validity range of the solution can be estimated by

\[r_K = \left(\frac{\epsilon}{\lvert\lvert F_K\lvert\lvert}\right)^{1/K}\]

This allows to iterate and find a sequence of series which spans the parameter range.

Implementation

The method is based on the package TaylorSeries.jl which makes it easy to manipulate Taylor series based on Automatic Differentiation.

Method

See AsymptoticNumericalMethod.ANM for more information.

Example

We provide an example of use. We define a BifurcationProblem as usual and pass the continuation algorithm ANM.

using AsymptoticNumericalMethod, Plots, Parameters
using BifurcationKit
const BK = BifurcationKit

function F(x, p)
	@unpack α = p
	f = similar(x)

	f[1] = (-2x[1]+x[2]) + α * exp(x[1])
	f[2] = ( x[1]-2x[2]) + α * exp(x[2])

	return f
end

sol0 = zeros(2)
par = (α = 0.0, )
prob = BifurcationProblem(F, sol0, par, (@lens _.α); record_from_solution = (x,p) -> norminf(x))
┌─ Bifurcation Problem with uType Vector{Float64}
├─ Inplace:  false
├─ Symmetric: false
└─ Parameter: α
optanm = ContinuationPar(dsmin = 0.01, dsmax = 0.15, detect_bifurcation = 3, ds= 0.01, newton_options = NewtonPar(tol = 1e-9), n_inversion = 6, max_bisection_steps = 15, max_steps = 15, )

branm = continuation(prob, ANM(20, 1e-8), optanm, normC = norminf, verbosity = 2)
 ┌─ Curve type: EquilibriumCont
 ├─ 15 series of degree 20, tol = 1.0e-8
 ├─ Number of points: 301
 ├─ Type of vectors: Vector{Float64}
 ├─ Parameter α starts at 6.666666666666667e-5, ends at 2.539417525123997e-12
 ├─ Algo: ANM
 └─ Special points:

- #  1,       bp at α ≈ +0.36763519 ∈ (+0.36787944, +0.36787944), |δp|=2e-11, [converged], δ = ( 1,  0), step =  27
- #  2,       bp at α ≈ +0.14900840 ∈ (+0.14936118, +0.14936122), |δp|=4e-08, [converged], δ = ( 1,  0), step =  67

You can plot the result as usual:

plot(branm)
Example block output

You can also show the radius of convergence of each series:

plot(branm, plotseries = true)
Example block output

Finally, for each series, we ca evaluate the residual norm:

plot()
for ii in eachindex(branm.polU)
	s = LinRange(-0*branm.radius[ii], branm.radius[ii], 20)
	plot!([branm.polp[ii].(s)], [norminf(F(branm.polU[ii](_s), BK.setparam(prob,branm.polp[ii](_s)))) for _s in s], legend = false, linewidth=5)#, marker=:d)
end
title!("")
Example block output

References

  • Charpentier

    Charpentier, Isabelle, Bruno Cochelin, and Komlanvi Lampoh. “Diamanlab - An Interactive Taylor-Based Continuation Tool in MATLAB,” n.d., 12.

  • Rubbert

    Rubbert, Lennart, Isabelle Charpentier, Simon Henein, and Pierre Renaud. “Higher-Order Continuation Method for the Rigid-Body Kinematic Design of Compliant Mechanisms”, n.d., 18.