smolpack¶

Sparse-grid Smolyak cubature over \([0,1]^d\) for Python.

What is smolpack?¶

smolpack is a Python library for efficient numerical integration (cubature) over the unit hypercube \([0,1]^d\) using Smolyak's algorithm with Clenshaw-Curtis quadrature rules. The numerical core is written in C and compiled via f2py, giving near-native performance while providing a clean, NumPy-based Python API.

Numerical integration (quadrature in 1-D, cubature in higher dimensions) is a fundamental problem in computational science. Standard tensor-product rules scale exponentially with dimension — a 1-D rule with \(n\) nodes becomes \(n^d\) nodes in \(d\) dimensions, quickly making the computation intractable. Smolyak's algorithm circumvents this curse of dimensionality by constructing sparse grids that achieve comparable polynomial exactness with dramatically fewer nodes.

smolpack approximates integrals of the form:

\[ I[f] = \int_{[0,1]^d} f(\mathbf{x})\,d\mathbf{x}, \]

using the Smolyak combination technique with Clenshaw-Curtis basic rules.

Available solvers¶

Solver	Underlying 1-D rule	Characteristics
`int_smolyak`	Delayed Clenshaw-Curtis	Fewer function evaluations for a given accuracy level
`cc_int_smolyak`	Standard Clenshaw-Curtis	Classical nested rule (1, 3, 5, 9, 17, 33, 65, … nodes)

Choosing a solver

For most applications, start with int_smolyak — the delayed variant achieves the same polynomial exactness with fewer integrand evaluations. Use cc_int_smolyak when compatibility with the classical Clenshaw-Curtis node hierarchy is required.

Quick example¶

import numpy as np
import smolpack

# Integrate exp(x1 + x2 + x3) over [0,1]^3
# Exact value: (e - 1)^3 ≈ 5.073214

def my_func(dim, x):
    return np.exp(np.sum(x))

result = smolpack.int_smolyak(my_func, dim=3, qq=5)
print(f"Integral ≈ {result:.6f}")

License¶

smolpack is distributed under the LGPL-2.1 license.