Quasi-Monte Carlo Community Software
Quasi-Monte Carlo (QMC) methods are used to approximate multivariate integrals. They have four main components: an integrand, a discrete distribution, summary output data, and stopping criterion. Information about the integrand is obtained as a sequence of values of the function sampled at the data-sites of the discrete distribution. The stopping criterion tells the algorithm when the user-specified error tolerance has been satisfied. We are developing a framework that allows collaborators in the QMC community to develop plug-and-play modules in an effort to produce more efficient and portable QMC software. Each of the above four components is an abstract class. Abstract classes specify the common properties and methods of all subclasses. The ways in which the four kinds of classes interact with each other are also specified. Subclasses then flesh out different integrands, sampling schemes, and stopping criteria. Besides providing developers a way to link their new ideas with those implemented by the rest of the QMC community, we also aim to provide practitioners with state-of-the-art QMC software for their applications.
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Installation
pip install qmcpy
The QMCPy Framework
The central package including the 5 main components as listed below. Each component is implemented as abstract classes with concrete implementations. For example, the lattice and Sobol' sequences are implemented as concrete implementations of the DiscreteDistribution
abstract class. An overview of implemented componenets and some of the underlying mathematics is available in the QMCPy README. A complete list of concrete implementations and thorough documentation can be found in the QMCPy Read the Docs .
- Stopping Criterion: determines the number of samples necessary to meet an error tolerance.
- Integrand: the function/process whose expected value will be approximated.
- True Measure: the distribution to be integrated over.
- Discrete Distribution: a generator of nodes/sequences, that can be either IID (for Monte Carlo) or low-discrepancy (for quasi-Monte Carlo), that mimic a standard distribution.
- Accumulate Data: stores and updates data used in the integration process.
Quickstart
Note: If the following mathematics is not rendering try using Google Chrome and installing the Mathjax Plugin for GitHub.
Will will find the expected value of the Keister integrand [18]
$$g(\boldsymbol{x})=\pi^{d/2}\cos(||\boldsymbol{x}||)$$
integrated over a $\mathcal{N}(\boldsymbol{0},\boldsymbol{I}/2)$ Gaussian in $d$ dimensions.
We may choose a Sobol' discrete distribution with a corresponding Sobol' cubature stopping criterion to preform quasi-Monte Carlo numerical integration.
import qmcpy as qp
from numpy import pi, cos, sqrt, linalg
d = 2
s = qp.Sobol(d)
g = qp.Gaussian(s, covariance=1./2)
k = qp.CustomFun(g, lambda x: pi**(d/2)*cos(linalg.norm(x,axis=1)))
cs = qp.CubQMCSobolG(k, abs_tol=1e-3)
solution,data = cs.integrate()
print(data)
A more detailed quickstart can be found in our GitHub repo at QMCSoftware/demos/quickstart.ipynb
or in this Google Colab quickstart notebook.
We also highly recommend you take a look at Fred Hickernell's tutorial at the Monte Carlo Quasi-Monte Carlo 2020 Conference and the corresponding MCQMC2020 Google Colab notebook.
Developers
- Sou-Cheng T. Choi
- Fred J. Hickernell
- Michael McCourt
- Jagadeeswaran Rathinavel
- Aleksei Sorokin
Collaborators
- Mike Giles
- Marius Hofert
- Pierre L’Ecuyer
- Christiane Lemieux
- Dirk Nuyens
- Art Owen
- Pieterjan Robbe
Contributors
- Jungtaek Kim
Citation
If you find QMCPy helpful in your work, please support us by citing the following work:
Choi, S.-C. T., Hickernell, F. J., McCourt, M., Rathinavel, J. & Sorokin, A.
QMCPy: A quasi-Monte Carlo Python Library. Working. 2020.
https://qmcsoftware.github.io/QMCSoftware/
BibTex citation available here
References
[1] F. Y. Kuo and D. Nuyens. "Application of quasi-Monte Carlo methods to elliptic PDEs with random diffusion coefficients - a survey of analysis and implementation," Foundations of Computational Mathematics, 16(6):1631-1696, 2016. (springer link, arxiv link)
[2] Fred J. Hickernell, Lan Jiang, Yuewei Liu, and Art B. Owen, "Guaranteed conservative fixed width confidence intervals via Monte Carlo sampling," Monte Carlo and Quasi-Monte Carlo Methods 2012 (J. Dick, F.Y. Kuo, G. W. Peters, and I. H. Sloan, eds.), pp. 105-128, Springer-Verlag, Berlin, 2014. DOI: 10.1007/978-3-642-41095-6_5
[3] Sou-Cheng T. Choi, Yuhan Ding, Fred J. Hickernell, Lan Jiang, Lluis Antoni Jimenez Rugama, Da Li, Jagadeeswaran Rathinavel, Xin Tong, Kan Zhang, Yizhi Zhang, and Xuan Zhou, GAIL: Guaranteed Automatic Integration Library (Version 2.3.1) [MATLAB Software], 2020. Available from http://gailgithub.github.io/GAIL_Dev/.
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