Camel 3 Function

using Pkg
Pkg.activate("../../.")
using Globtim
using DynamicPolynomials, DataFrames
using ProgressLogging

  Activating project at `~/globtim`

We define the center and range of the square domain of approximation.

# Constants and Parameters
const n, a, b = 2, 5, 1
const scale_factor = a / b
f = camel # Objective function
d = 6 # Initial Degree 
SMPL = 200   
println("Number of samples: ", SMPL^2)
TR = test_input(f, 
                dim = n,
                center = [0.0, 0.0],
                GN = SMPL, 
                sample_range = scale_factor
                )

Number of samples: 40000

test_input(2, [0.0, 0.0], 200, (0.1, 0.5), 0.002, (0.0, 0.0), 5.0, 1.0, 6, Globtim.camel)

The Constructor function computes the coefficients of the polynomial approximant in the basis specified, with GN^2 samples distributed following the measure with respect to which the polynomial basis is orthogonal. The RationalPrecision specifies that the conversion of the polynomial’s coefficients from the tensorized orthogonal basis to the standard monomial basis is carried out in high precision.

pol_cheb = Constructor(TR, d, basis=:chebyshev, precision=RationalPrecision)
pol_lege = Constructor(TR, d, basis=:legendre, precision=RationalPrecision, normalized=true);

current L2-norm: 5.403750270905881e-12
current L2-norm: 5.0293810818049854e-12

Note that with the Legendre polynomials, it is necessary to run with the normalized option to ensure that the polynomials are orthonormal with respect to the uniform measure on the domain of approximation.

@polyvar(x[1:n]) # Define polynomial ring 
real_pts_cheb = solve_polynomial_system(
    x, n, d, pol_cheb.coeffs;
    basis=pol_cheb.basis,
    precision=pol_cheb.precision,
    normalized=pol_cheb.normalized,
)

real_pts_lege = solve_polynomial_system(
    x, n, d, pol_lege.coeffs;
    basis=pol_lege.basis,
    precision=pol_lege.precision,
    normalized=pol_lege.normalized)

df_cheb = process_crit_pts(real_pts_cheb, f, TR)
df_lege = process_crit_pts(real_pts_lege, f, TR);

using Optim
df_cheb, df_min_cheb = analyze_critical_points(f, df_cheb, TR, tol_dist=0.00125);
df_lege, df_min_lege = analyze_critical_points(f, df_lege, TR, tol_dist=0.00125);

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using CairoMakie
CairoMakie.activate!

activate! (generic function with 1 method)

fig_1 = cairo_plot_polyapprox_levelset(pol_cheb, TR, df_cheb, df_min_cheb, chebyshev_levels=true)

fig_2 = cairo_plot_polyapprox_levelset(pol_lege, TR, df_lege, df_min_lege, chebyshev_levels=true)