Approximating maps into manifolds

Approximate functions of type $$\mathbb{R}^m \to M^n$$ where $M^n$ is an $n$-dimensional Riemannian manifold.

To learn more about approximating maps into Riemannian manifolds, check out our preprint at arxiv.org/abs/2403.16785.

If you use this software in your work, please cite

@misc{jacobsson2024,
      title={Approximating maps into manifolds with lower curvature bounds}, 
      author={Simon Jacobsson and Raf Vandebril and Joeri van der Veken and Nick Vannieuwenhoven},
      year={2024},
      eprint={2403.16785},
      archivePrefix={arXiv},
      primaryClass={math.NA}
}

Example 1

using ManiFactor
using Manifolds: Sphere, get_point, StereographicAtlas

n = 2
M = Sphere(n)

m = 2
f(x) = get_point(M, StereographicAtlas(), :south, [x[1]^2 - x[2]^2, -x[1] * x[2]])
fhat = approximate(m, M, f)

x0 = rand(m)
f(x0) - fhat(x0)

Approximate $$f \colon [-1, 1]^2 \to S^2, (x, y) \mapsto \mathrm{stereographic~projection}(x^2 - y^2, 2 x y)$$ using a varying number of sample points. This figure illustrates the approximation accuracy by showing the image on $S^2$ of a grid in $[-1, 1]^2$:

Example 2

Approximate $$f \colon [-1, 1]^2 \to H^2, (x, y) \mapsto \mathrm{stereographic~projection}(x^2 - y^2, 2 x y)$$ using a varying number of sample points. This figure illustrates the approximation accuracy by showing the image on $H^2$ of a grid in $[-1, 1]^2$:

Example 3

Approximate $$ f \colon [1, 2] \to \mathrm{Gr}(100, 3), t \mapsto \mathrm{span}{b, A(t) b, A(t)^2} $$ where $b$ is a random $100$-vector and $A(t)$ is the evaluation of the kernel $$ K(x, x', t) = \frac{2}{\pi} \sum_{l = 1}^\infty \sin(l x) \sin(l x') \exp(-t l^2 / 4) $$ on a $100 \times 100$ grid on $[0, \pi] \times [0, \pi]$. The kernel appears when solving the heat equation on a finite interval with endpoints held at a fixed temperature. This figure illustrates the approximation accuracy compared to what is predicted by the theory:

$N$ is the number of sample points in each direction, so that the total number of sample points is $N^2$.

Example 4

Approximate $$f \colon [-1, 1]^3 \to \mathrm{Segre}(30, 30), x \mapsto \frac{1}{2} \exp{x_1} \exp{(W_1 x_2)} e_1 (\exp{(W_2 x_3)} e_1)^\mathrm{T}$$ where $W_1$ and $W_2$ are randomly chosen antisymmetric $30 \times 30$ matrices and $e_1 = (1, 0, \dots, 0)$. This figure illustrates the approximation accuracy compared to what is predicted by the theory:

$N$ is the number of sample points in each direction, so that the total number of sample points is $N^3$.