smolpack¶

Multidimensional Quadrature Using Sparse Grids for Python

Overview¶

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 package allows you to easily and efficiently integrate high-dimensional functions without the exponential cost of classical tensor-product rules.

Numerical integration in multiple 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 constructs sparse grids that achieve comparable polynomial exactness with dramatically fewer nodes. smolpack approximates integrals of the form:

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

Requirements¶

• NumPy

Example Usage¶

import numpy as np
import smolpack


# Integrate exp(x1 + x2 + x3) over [0,1]^3
# Exact value: (e - 1)^3 ≈ 5.073214
def exp_sum(dim, x):
    return np.exp(np.sum(x))


result = smolpack.int_smolyak(exp_sum, dim=3, qq=5)
print(f"Result = {result:.6f}")

# Convergence with increasing level
exact = (np.e - 1.0) ** 3
for qq in range(4, 9):
    result = smolpack.int_smolyak(exp_sum, dim=3, qq=qq)
    print(
        f"qq={qq}  k={qq - 3}  result={result:.10f}  error={abs(result - exact):.2e}"
    )

# Compare both solvers
r1 = smolpack.int_smolyak(exp_sum, dim=3, qq=7)
r2 = smolpack.cc_int_smolyak(exp_sum, dim=3, qq=7)
print(f"Delayed CC:  {r1:.10f}")
print(f"Standard CC: {r2:.10f}")

# Query function-evaluation count
n = smolpack.get_count()
print(f"Evaluations: {n}")

Main Features¶

1. Sparse-grid cubature over \([0,1]^d\) for dimensions \(1 \le d < 40\).
2. Two solver variants: delayed and standard Clenshaw-Curtis Smolyak rules.
3. High-performance C core compiled via f2py, with a clean NumPy-based Python API.
4. Simple integrand signature — any callable f(dim, x) -> float works.
5. Built-in function-evaluation counter via get_count().
6. print_stats flag for runtime diagnostics.

See Also¶

• NumPy: Array library used for the integrand interface
• SciPy: General scientific computing — complements smolpack for pre/post-processing
• SMOLPACK C library: The original C implementation by Knut Petras, distributed by John Burkardt

Acknowledgements¶

The author thanks Knut Petras for the original SMOLPACK C library, which provides the numerical core of this package and is distributed via John Burkardt's page.

References¶

• Smolyak, S., "Quadrature and Interpolation Formulas for Tensor Products of Certain Classes of Functions," Doklady Akademii Nauk SSSR, 4, 240–243, 1963.
• Petras, K., "Fast Calculation of Coefficients in the Smolyak Algorithm," Numerical Algorithms, 26(2), 93–109, 2001.
• Petras, K., "Smolyak Cubature of Given Polynomial Degree with Few Nodes for Increasing Dimension," Numerische Mathematik, 93(4), 729–753, 2003.