## Submodules

Estimate the 1D ground truth integral.

Parameters:
Returns:

The integral estimate (output of scipy.integrate.quad).

Estimate the 2D ground truth integral.

Parameters:
Returns:

The integral estimate (output of scipy.integrate.dblquad).

2D toy integrand that is a Gaussian on a circle.

$f(x) = (2\pi \sigma^2)^{-\frac{1}{2}} r^2 e^{-\frac{(r - \mu)^2}{2 \sigma^2}}$

where $$\sigma^2$$ is the variance attribute, $$\mu$$ is the mean attribute and $$r = \|x\|$$ is the length of the input $$x$$.

Parameters:
• mean (float) – The mean of the circular Gaussian in units of radius (must be >= 0, defaults to 0).

• variance (float) – The variance of the Gaussian (must be > 0, defaults to 1.).

Return type:
Returns:

The wrapped test function, and the integrals bounds (the latter defaults to [-3, 3]^2).

1D toy integrand coined by Philipp Hennig.

One of the earlier mentions e.g., in this talk (external link).

$f(x) = e^{-x^2 -\sin^2(3x)}$
Return type:
Returns:

The wrapped test function, and the integrals bounds (the latter default to [-3, 3]).

2D Sombrero function.

$f(x) = \frac{\operatorname{sin}(\pi r \omega)}{\pi r \omega}$

where $$\omega$$ is the freq parameter and $$r=\|x\|$$ is the length of the input vector $$x$$.

Parameters:

freq (float) – The frequency of the sombrero (must be > 0, defaults to 1).

Return type:
Returns:

The wrapped test function, and the integrals bounds (the latter defaults to [-3, 3]^2).