PECOS Developers Manual

Version 1.0

Author:

Michael S. Eldred, William J. Bohnhoff, Richard V. Field, Jr.

Main Page Table of Contents

• Introduction
• Overview of PECOS
  • Data Transformations
  • Probability Transformations
  • Multivariate Basis Functions
  • Univariate Basis Functions
  • Integration Drivers
• Services
• Code Examples
• Additional Resources

Introduction

The PECOS (Parallel Environment for Creation Of Stochastics) library initiated as a capability to generate samples of random fields (RFs) and stochastic processes (SPs) from a set of user-defined power spectral densities (PSDs). The RF/SP may be either Gaussian or non-Gaussian and either stationary or nonstationary, and the resulting sample is intended for run-time query by parallel finite element simulation codes.

PECOS has grown to include univariate and multivariate polynomial approximations, using either orthogonal or interpolation polynomials, along with numerical integration drivers for computing expansion coefficients.

The Developers Manual documents the class structures used by the PECOS library. It derives directly from annotation of the source code.

Overview of PECOS

The PECOS library collects a variety of utilities for transformation of probability distributions, transformations between frequency and time domain data, and modeling using basis functions.

The use of class hierarchies provides a mechanism for extensibility in PECOS components. In each of the various class hierarchies, adding a new capability typically involves deriving a new class and providing a few virtual function redefinitions. These redefinitions define the coding portions specific to the new derived class, with the common portions already defined at the base class. Thus, with a relatively small amount of new code, the existing facilities can be extended, reused, and leveraged for new purposes.

The software components are presented in the following sections using a top-down order.

Data Transformations

Class hierarchy: DataTransformation.

Data transformations provide a mechanism for moving data sets between the frequency domain and the time domain. These transformations typically involve modeling through the use of scalar random variables (transformed to standardized probability distributions) and a set of spatial and/or temporal basis functions. Thus the BasisFunction and ProbabilityTransformation capabilities are components of the DataTransformation capability. Specific data transformations include:

• ForwardTransformation: from time domain to frequency domain based on FFT. A placeholder for now.

• InverseTransformation: from frequency domain (PSD) to time domain.
  • FourierInverseTransformation: from frequency domain to time domain based on inverse fast Fourier transform. This approach generates a complete time series all at once.
  • KarhunenLoeveInverseTransformation: from frequency domain (PSD) to time domain based on Karhunen-Loeve. This approach generates a complete time series all at once.
  • SamplingInverseTransformation: from frequency domain (PSD) to time domain based on random sampling. This approach supports incremental generation of time series. This approach will be adapted from the Salinas implementation by Richard Koteras.

Probability Transformations

Class hierarchy: ProbabilityTransformation.

Probability transformations provide a mechanism for modifying an input probability space to have desired modeling characteristics. In particular, by transforming from a given type of probability distribution to a standardized probability distribution, algorithms designed for these standard probability distributions can be applied. Specific transformations include:

• NatafTransformation: a nonlinear variable transformation from original, correlated probability distributions to uncorrelated standardized probability distributions. The transformation is based on CDF equivalence and application of the Cholesky factor of a modified correlation matrix. Nataf is currently operational and in use by DAKOTA.

• RosenblattTransformation: another nonlinear variable transformation from original, correlated probability distributions to uncorrelated standardized probability distributions. Whereas Nataf operates on CDF marginal distributions, Rosenblatt operates on joint distributions and is the preferred approach when this level of data is known. Rosenblatt is not yet supported but may be added in time.

Integration Drivers

Class hierarchy: IntegrationDriver.

This hierarchy provides a set of drivers for multivariate numerical integration for computing coefficients of a multivariate BasisApproximation. Specific drivers include:

• CubatureDriver: Stroud cubature rules and extensions
• SparseGridDriver: Smolyak sparse grids; isotropic, anisotropic, and generalized
• TensorProductDriver: tensor-product quadrature; isotropic and anisotropic
• LocalRefinableDriver: structured grids for local h-refinement

LHSDriver is currently separate from this hierarchy.

Multivariate Basis Approximations

Class hierarchy: BasisApproximation.

This hierarchy provides sets of multivariate basis functions for spatial and/or time domains in support of DataTransformation capabilities. Specific basis function sets may include:

• FourierBasis: Fourier basis functions which map frequencies to the time domain.
• KLBasis: discrete Karhunen-Loeve eigenfunctions from singular value decomposition of covariance matrices.
• PolynomialApproximation:
  • OrthogPolyApproximation: a multivariate orthogonal basis defined from one-dimensional orthogonal polynomials (see OrthogonalPolynomial below) and a multi-index.
  • InterpPolyApproximation: a multivariate interpolation basis defined from one-dimensional interpolation polynomials (local or global, value-based or gradient-enhanced, nodal or hierarchical; see InterpolationPolynomial below) and a multi-index.

Univariate Basis Approximations

Class hierarchy: BasisPolynomial.

This hierarchy provides sets of univariate basis functions in support of the multivariate BasisApproximation capabilities. Specific univariate basis functions options include:

• OrthogonalPolynomial: polynomial basis functions which are orthogonal with respect to a variety of continuous probability distribution function weightings:
  • HermiteOrthogPolynomial
  • LegendreOrthogPolynomial
  • LaguerreOrthogPolynomial
  • JacobiOrthogPolynomial
  • GenLaguerreOrthogPolynomial
  • ChebyshevOrthogPolynomial
  • NumericGenOrthogPolynomial
• InterpolationPolynomial: one-dimensional interpolation polynomials.
  • LagrangeInterpPolynomial: global value-based Lagrange interpolation polynomials
  • HermiteInterpPolynomial: global gradient-enhanced Hermite interpolation polynomials
  • PiecewiseInterpPolynomial: local value-based (linear) and gradient-enhanced (cubic) spline polynomials using a nodal basis formulation
  • HierarchPWInterpPolynomial: local value-based (linear) and gradient-enhanced (cubic) spline polynomials using a hierarchical basis formulation

Services

A variety of services are provided in PECOS for parallel computing, memory management, etc. An overview of the classes and member functions involved in performing these services is included below.

• Memory management: PECOS employs the techniques of reference counting and representation sharing through the use of letter-envelope and handle-body idioms (Coplien, "Advanced C++").
• Fast Fourier transform (FFT): Two FFT libraries will be supported. First, FFTW provides platform tuned and parallel FFT operations in C under a GNU GPL license. Second, DFFTPACK provides a reference FFT implementation in F77 in the public domain.

Code Examples

A number of test drivers are available in pecos/test that demonstrate interaction with PECOS capabilities. The following drivers demonstrate basis stochastic process sample generation from power spectral density (PSD):

• pecos_ifft_g.cpp: scalar stochastic process generation from PSD using inverse FFT in the method of Grigoriu.
• pecos_ifft_sd.cpp: scalar stochastic process generation from PSD using inverse FFT in the method of Shinozuka and Deodatis.

The following drivers demonstrate generation of grids for collocation methods:

• pecos_int_driver.cpp: generation of integration driver sample sets (Smolyak sparse grids, tensor-product quadrature, cubature, and local refinable grids)

• pecos_lhs_driver.cpp: generation of an LHS sample set

Finally, the following drivers demonstrate miscellaneous statistical utilities:

• boost_test_dist.cpp: tests Boost statistical distribution functions.
• boost_test_rng.cpp: tests Boost random number generators

Additional Resources

Additional development resources include:

• Project web pages will be maintained in the future at http://dakota.sandia.gov/packages/pecos