Package: qmcpy
Quasi-Monte Carlo Community SoftwareQuasi-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.Homepage~Article~GitHub~Read the Docs~PyPI~Blogs~DockerHub~Contributing~IssuesInstallationpip install qmcpyThe QMCPy FrameworkThe 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 theDiscreteDistributionabstract class. A complete list of concrete implementations and thorough documentation can be found in theQMCPy 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.QuickstartNote: If the following mathematics is not rendering try using Google Chrome and installing theMathjax Plugin for GitHub.We will approximate the expected value of the $d$ dimensional Keister integrand [18]$$g(X)=\pi^{d/2}\cos(\lVert X \rVert)$$where $X \sim \mathcal{N}(\boldsymbol{0},\boldsymbol{I}/2)$.We may choose a Sobol' discrete distribution with a corresponding Sobol' cubature stopping criterion to preform quasi-Monte Carlo integration.importqmcpyasqpfromnumpyimportpi,cos,sqrt,linalgd=2dnb2=qp.DigitalNetB2(d)gauss_sobol=qp.Gaussian(dnb2,mean=0,covariance=1/2)k=qp.CustomFun(true_measure=gauss_sobol,g=lambdax:pi**(d/2)*cos(linalg.norm(x,axis=1)))qmc_sobol_algorithm=qp.CubQMCSobolG(k,abs_tol=1e-3)solution,data=qmc_sobol_algorithm.integrate()print(data)Running the above code outputsLDTransformData (AccumulateData Object)
    solution        1.808
    error_bound     4.68e-04
    n_total         2^(13)
    time_integrate  0.008
CubQMCSobolG (StoppingCriterion Object)
    abs_tol         0.001
    rel_tol         0
    n_init          2^(10)
    n_max           2^(35)
CustomFun (Integrand Object)
Gaussian (TrueMeasure Object)
    mean            0
    covariance      2^(-1)
    decomp_type     PCA
Sobol (DiscreteDistribution Object)
    d               2^(1)
    dvec            [0 1]
    randomize       LMS_DS
    graycode        0
    entropy         127071403717453177593768120720330942628
    spawn_key       ()A more detailed quickstart can be found in our GitHub repo atQMCSoftware/demos/quickstart.ipynbor inthis Google Colab quickstart notebook.We also highly recommend you take a look atFred Hickernell's tutorial at the Monte Carlo Quasi-Monte Carlo 2020 Conferenceandthe corresponding MCQMC2020 Google Colab notebook.CommunityPlease refer tothis documentfor the key roles in the QMCPy community.CitationIf 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 availablehereVideo TutorialPlease refer tothis videofor a quick introduction to QMCPy.For a more detail introduction refer tothis video.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. 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