ME498/599 Sensitivity Analysis and Uncertainty Quantification

University of Washington

Winter Quarter 2018

Class Web Site

We are interested in the simulation of heat transfer and fluid mechanics, from both a control volume (integral) approach, as well as a continuum mechanics approach. The control volume approach has been historically called thermal-network or fluid-network analysis/simulation. Broadly, a continuum mechanics approach utilizes a grid generation step, followed by the application of a discretization method. These methods can include (certainly not limited to) finite difference, finite volume or finite element approaches. The initial effort is to provide guidance and tools for sensitivity analysis and uncertainty quantification and propagation. Sensitivity analysis (local and global) is used to provide insight into how variations in the input parameters to a model affect the output parameters of the simulation. Uncertainty quantification and propagation is concerned with the determination of the uncertainty characteristics of model input parameters and then propagating them through the simulation to provide uncertainty estimates of the output.

The notes from the Spring Quarter 2017 class, which used the Dakota toolbox, can be found here.

Schedule

Date Lecture Topic Notes Remarks Homework Due Date Office Hours
Fri, Feb 16 The Cannon Problem Lecture Notes See the Introduction to Sensitivity Analysis section in [GHO17], as well as Global Sensitivity Analysis: The Primer, Saltelli, et al., 2008
Fri, Mar 2 Cannon Problem Analyis using Dakota Notes HW 10 Maximum Likelihood Parameter Estimation using the bbmle R package
Mon, Mar 5 Final Project Help Session

Software/Toolkit/Framework for Sensitivity Analysis, Parameter Estimation/Calibration, Uncertainty Quantification and Propagation

Summary Sources

Open Source

Commercial

R Project for Statistical Computing

Data Exchange with MATLAB

Sensitivity Analysis using R

We are interested in solving both steady and transient, nonlinear engineering math models, with multiple input parameters and one or more output parameters (quantities of interest, QoI). Per Section 4: Specialized R Software Packages, from the Introduction to Sensitivity Analysis chapter from the Handbook of Uncertainty Quantification, the following packages are recommended by the authors Bertrand Iooss and Andrea Saltelli:

Verification of Variance-based Global Sensitivity Analysis Methods

In order to study the properties of the various methods in the R sensitivity package, a number of math models (functions) are analytically solved for the Sobol' indices.

Math Model (function) has a Single Output

See Professor Emery's notes on Sensitivity Analysis and Sobol' Indices.

Function: \(f(x_1,x_2) = x_1^2 + x_1 x_2 + x_2\)

Function: \(f(x_1,x_2) = x_1 x_2\)

Math Model (function) has Multiple Outputs

Transient Math Model (function) has a Single Output

Transient Math Model (function) has Multiple Outputs

References

Standard Probability Distributions and Latin Hypercube Sampling with R

Density (d), cumulative distribution function (p), quantile function (q) and random variate generation (r) for many standard probability distributions are available in the core stats package (for a complete list of functions, use library(help = "stats")):

See Chapter 33, p. 1163 and Chapter 44, p. 1517, in Handbook of Uncertainty Quantification for discussions on sampling. Latin hypercube sampling (LHS) functions are available in the following packages:

library(DiceDesign)

# lhsDesign(n, dimension, randomized=TRUE, seed=NULL)

n <- 100
dimension <- 2

outF <- lhsDesign(n, dimension, randomized=FALSE, seed=38194)
outT <- lhsDesign(n, dimension, randomized=TRUE,  seed=38194)

par(mfrow=c(1,2))  # 1 row and 2 columns
plot(outT$design, main = "randomized=TRUE")
plot(outF$design, main = "randomized=FALSE")

par(mfrow=c(2,2))  # 2 rows and 2 columns

hist(outT$design[,1])
hist(outT$design[,2])
plot(density(outT$design[,1]))
plot(density(outT$design[,2]))

hist(outF$design[,1])
hist(outF$design[,2])
plot(density(outF$design[,1]))
plot(density(outF$design[,2]))

par(mfrow=c(1,1))  # return plot dev to single plot

library(lhs)

# transform a Latin hypercube sample to a normal and beta pdf

n = 100    # number of design points
d = 2      # number of design variables
set.seed(1976)   # set the seed
X <- randomLHS(n=n, k=d, preserveDraw=TRUE)

Y <- matrix(0, nrow=n, ncol=d)
Y[,1] <- qnorm(X[,1], mean=3, sd=0.1)      # normal distribution
Y[,2] <- qbeta(X[,2], shape1=2, shape2=2)  # beta distribution

par(mfrow=c(2,2))  # or could use layout() for more complex arrangements
hist(Y[,1])
hist(Y[,2])
plot(density(Y[,1]))
plot(density(Y[,2]))
par(mfrow=c(1,1))  # return plot dev to single plot

Sensitivity Analysis of the Cannon Problem using R

Parameter Estimation: Maximum Likelihood using R

A maximum likelihood parameter estimation of the velocity and drag parameter for the cannon problem, was performed using the mle2() function of the bbmle package. The script is: cannon_MLE.R

Parameter Estimation: Bayesian Inference using R

Introducing Monte Carlo Methods with R

Surrogate Models using R

See the Metamodel-Based Sensitivity Analysis: Polynomial Chaos Expansions and Gaussian Processes chapter from the Handbook of Uncertainty Quantification.

Gaussian Process

Polynomial Chaos Expansion

Links to Similar Courses

Books and References of Interest

2000
[HM00] Probability, Reliability, and Statistical Methods in Engineering Design, A. Haldar, S. Mahadevan, John Wiley & Sons, 2000, UW Library
2008
[SRACCGST08] Global Sensitivity Analysis: The Primer, Saltelli, et al., John Wiley & Sons, 2008
2009
[Ate09] Everyday Heat Transfer Problems: Sensitivities to Governing Variables, M. Kemal Atesmen, ASME, 2009, UW Library
[CS09] Experimentation, Validation and Uncertainty Analysis for Engineers, third edition, Coleman and Steele, John Wiley & Sons, 2009, UW Library
2014
[MR14] Applied Statistics and Probability for Engineers, sixth edition, D. Montgomery and G. Runger, John Wiley & Sons, 2014, textbook for UW IND E 315, UW Library
[Smi14] Uncertainty Quantification: Theory, Implementation, and Applications, Ralph Smith, SIAM, 2014, UW Library
[VE14] Forensic Metrology: Scientific Measurement and Inference for Lawyers, Judges, and Criminalists, Ted Vosk and Ashley Emery, CRC Press, 2014, UW Library
2017
[GHO17] Handbook of Uncertainty Quantification, Editors: Roger Ghanem, David Higdon and Houman Owhadi, Springer, 2017, UW Library

Journals

Organizations and Conferences

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