Session: Methods for Uncertainty Quantification, Sensitivity Analysis, and Prediction 2

Paper Number: 157955

157955 - Tseuqlib: A Python Library for Conducting Uncertainty Quantification With the Taylor Series Expansion Surrogate Model 

Abstract: 

The Taylor series expansion has shown to be an efficient and accurate surrogate model. Due to rapid advances in derivative computation in simulations and mathematical models, the Taylor series expansion is becoming more accessible. Therefore, a comprehensive numerical library that can exploit the convenience of the Taylor series for Uncertainty Quantification (UQ) is needed and provided herein. This talk presents an open-source Python library called TSEUQLib (Taylor Series Expansion Uncertainty Quantification Library) that was developed to construct arbitrary-order Taylor series expansions and conduct UQ and global sensitivity analysis. With this library, the user provides the distribution (or central moments) of each random variable and the partial derivatives of the high-fidelity model with respect to each random variable, evaluated at the mean of the input variables. TSEUQLib is used to construct a Taylor series expansion and compute statistical moments, the High-Dimensional Model Representation (HDMR), Sobol' indices, total Sobol' indices, and Shapley effects of the Taylor series. An approximation of the error in the Taylor series is provided using a truncated remainder term, which does not require additional realizations of the high-fidelity model.  Additionally, derivatives of these metrics with respect to distribution parameters of the random variables can also be computed. These derivatives are useful for conducting sensitivity analysis on the effect of variable mean, standard deviation, bounds, etc. on the expectation, variance, Sobol' indices, error, etc. of the output. The methodology and implementation of the functions in TSEUQLib are presented. TSEUQLib was implemented in Python using OTIlib (Order Truncated Imaginary algebra library). OTIlib was originally developed to compute arbitrary-order partial derivatives of mathematical models as part of the Hypercomplex Automatic Differentiation (HYPAD) methodology using OTI numbers and OTI algebra. OTI numbers are imaginary numbers with multiple imaginary bases, where the imaginary bases are a representation of the Taylor series expansion. OTIlib was repurposed in this work to represent the structure of the Taylor series expansion as well as perform algebraic operations applied to the Taylor series. To demonstrate and numerically validate the library, TSEUQLib was applied to several numerical examples. An application to a multivariate polynomial function was performed to numerically validate the implementation of the library. A high-dimensional problem was performed to evaluate the computational cost of computing statistical metrics with TSEUQLib as the number of variables and order of expansion increases. Variance approximation of the Ishigami function was performed to demonstrate the use of the remainder term to obtain sufficient error in the Taylor series expansion. Due to its computational efficiency, TSEUQLib can be used to conduct UQ on spatial and time dependent problems by repeatedly calling the library for each point. This is shown on a transient two-dimensional finite element simulation of heat transfer in a thermal fin.

Presenting Author: Matthew Balcer Los Alamos National Laboratory

Presenting Author Biography: Postdoctoral research associate at Los Alamos National Laboratory in the Verification and Analysis group.

Authors:

Matthew Balcer Los Alamos National Laboratory
Mauricio Aristizabal The University of Texas at San Antonio
Samuel Roberts Los Alamos National Laboratory
Christy Joy Tupas The University of Texas at San Antonio
Harry Millwater The University of Texas at San Antonio