Smolyax

Fast interpolation of high-dimensional and vector-valued functions - avoiding the curse-of-dimensionality by using sparse-grid interpolation nodes, - ensuring numerical stability by using a barycentric Smolyak interpolation formulation and - providing hardware-agnostic high performance by implementing key algorithms in JAX.

Features

• Node sequences for interpolation on bounded or unbounded domains (Leja nodes and Gauß-Hermite nodes, respectively)
• General anisotropic multi-index sets $\Lambda \subset \mathbb{N}^d0$ of the form $\Lambda := \{\boldsymbol{\nu} \in \mathbb{N}^d0 \ : \ \sum{j=1}^{d} kj \nu_j < t \}$ where $\boldsymbol{k}\in \mathbb{R}^{d}$ is monotonically increasing and controls the anisotropy while the threshold $t > 0$ controls the cardinality of the set.
• Heuristics to determine the threshold parameter $t$ to construct a set $\Lambda$ with a specified cardinality (which is typically not analytically available)
• Smolyak operator for interpolating high-dimensional and vector-valued functions $f : \mathbb{R}^{d1} \to \mathbb{R}^{d2}$ for $d1, d2 \in \mathbb{N}$ potentially large.
• Functionality to compute the integral of the interpolant (quadrature) as well as evaluate its derivative.

The implementation is designed for maximal efficiency. As a rough example consider interpolation a scalar function with $d_1 = 10^3$ inputs using $n = 10^4$ interpolation nodes. With smolyax you can expect to both generate the multi-index set as well as evaluate the interpolant from the corresponding polynomial space in well less under $0.1$ seconds on a contemporary laptop CPU.

Documentation

• For an introduction to relevant literature and the key implementation concepts, see the JOSS paper accompanying this repository.
• For code documentation see here.

Get started

Dependencies

smolyax requires python >= 3.9. Core dependencies are jax and numba, for more details see pyproject.toml.

Installation

To install smolyax and its dependencies, run: pip install "smolyax @ git+https://github.com/JoWestermann/smolyax.git"

To install smolyax with GPU support enabled, run: pip install "smolyax[cuda] @ git+https://github.com/JoWestermann/smolyax.git"

In order to run notebooks and/or tests, install via git clone git@github.com:JoWestermann/smolyax.git pip install -e "smolyax[notebook]" # for additionally installing notebook and plotting dependencies pip install -e "smolyax[dev]" # for additionally installing testing, code quality and documentation dependencies

Usage

To construct the interpolant to a function f, which has d_in inputs and d_out outputs, first choose the polynomial space in which to interpolate by setting up a node generator object, e.g. Leja nodes: node_gen = nodes.Leja(dim=d_in) and choosing a weight vector k controlling the anisotropy as well as a threshold t controlling the size of the multi-index set: k = [np.log((2+j)/np.log(2)) for j in range(d_in)] t = 5. Then, initialize the interpolant as f_ip = interpolation.SmolyakBarycentricInterpolator(node_gen=node_gen, k=k, d_out=d_out, t=t, f=f) and evaluate it at a point x by calling y = f_ip(x)

For more examples and visualizations see notebooks, in particular see the examples for interpolating a one-dimensional, two-dimensional or high-dimensional function.

Help

If you have questions or need help, please reach out through our Github Discussions!

Contribute

Need a feature?

We keep track of features that could be implemented without too much trouble and that we will work on prioritized on demand via our open issues.

Submit a feature!

If you want to submit a feature, please do so via a pull request. Ensure that all tests run through by running pytest from the project root directory, and ensure that performance has not degraded by first creating a benchmark on the main branch via pytest --benchmark-only --benchmark-save=baseline and compare performance on your feature branch against this baseline via pytest --benchmark-only --benchmark-compare=0001_baseline --benchmark-sort=name --benchmark-compare-fail=min:5%

Cite

If you used this library for your research, please cite it as:

@article{westermann2025smolyax, title={Smolyax: a high-performance implementation of the {S}molyak interpolation operator}, author={Westermann, Josephine and Chen, Joshua}, journal = {Journal of Open Source Software}, publisher = {The Open Journal}, doi = {10.21105/joss.08505}, url = {https://doi.org/10.21105/joss.08505}, volume = {10}, number = {112}, pages = {8505}, year = {2025} }