## 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).