ASSESSING THE RELIABILITY
OF COMPLEX MODELS

MATHEMATICAL AND STATISTICAL FOUNDATIONS OF VERIFICATION,
VALIDATION, AND UNCERTAINTY QUANTIFICATION

Committee on Mathematical Foundations of Verification, Validation, and Uncertainty Quantification

Board on Mathematical Sciences and Their Applications

Division on Engineering and Physical Sciences

NATIONAL RESEARCH COUNCIL
OF THE NATIONAL ACADEMIES

THE NATIONAL ACADEMIES PRESS
Washington, D.C.
www.nap.edu

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NOTICE: The project that is the subject of this report was approved by the Governing Board of the National Research Council, whose members are drawn from the councils of the National Academy of Sciences, the National Academy of Engineering, and the Institute of Medicine. The members of the committee responsible for the report were chosen for their special competences and with regard for appropriate balance.

This project was supported by the Department of Energy under Contracts DE-AT01-07NA78285 and DE-DT0001857, Task Order 9, and by the Air Force Office of Scientific Research under Contract FA9550-10-1-0435. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the views of the organizations or agencies that provided support for the project.

Cover image credit: The galaxy image at the bottom is of white dwarf stars in globular cluster NGC 6397—Hubble Space Telescope. Courtesy of NASA, ESA, and H. Riches, University of British Columbia.

International Standard Book Number-13: 978-0-309-25634-6
International Standard Book Number-10: 0-309-25634-8

Additional copies of this report are available from the National Academies Press, 500 Fifth Street, NW, Keck 360, Washington, DC 20001; (800) 624-6242 or (202) 334-3313; http://www.nap.edu.

Suggested citation: National Research Council. 2012. Assessing the Reliability of Complex Models: Mathematical and Statistical Foundations of Verification, Validation, and Uncertainty Quantification. Washington, D.C.: The National Academies Press.

Copyright 2012 by the National Academy of Sciences. All rights reserved.

Printed in the United States of America

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COMMITTEE ON MATHEMATICAL FOUNDATIONS OF VERIFICATION, VALIDATION, AND UNCERTAINTY QUANTIFICATION

MARVIN L. ADAMS, Texas A&M University, Co-Chair

DAVID M. HIGDON, Los Alamos National Laboratory, Co-Chair

JAMES O. BERGER, Duke University

DEREK BINGHAM, Simon Fraser University

WEI CHEN, Northwestern University

ROGER GHANEM, University of Southern California

OMAR GHATTAS, University of Texas at Austin

JUAN MEZA, University of California, Merced

ERIC MICHIELSSEN, University of Michigan

VIJAYAN N. NAIR, University of Michigan

CHARLES W. NAKHLEH, Sandia National Laboratories

DOUGLAS NYCHKA, National Center for Atmospheric Research

STEPHEN M. POLLOCK, University of Michigan (retired)

HOWARD A. STONE, Princeton University

ALYSON G. WILSON, Institute for Defense Analyses

MICHAEL R. ZIKA, Lawrence Livermore National Laboratory

Staff

NEAL GLASSMAN, Study Director

MICHELLE SCHWALBE, Associate Program Officer

BARBARA WRIGHT, Administrative Assistant

BOARD ON MATHEMATICAL SCIENCES AND THEIR APPLICATIONS

C. DAVID LEVERMORE, University of Maryland, Chair

TANYA STYBLO BEDER, SBCC, Inc.

PATRICIA FLATLEY BRENNAN, University of Wisconsin-Madison

GERALD G. BROWN, U.S. Naval Postgraduate School

LOUIS ANTHONY COX, JR., Cox Associates

BRENDA L. DIETRICH, IBM T.J. Watson Research Center

CONSTANTINE GATSONIS, Brown University

DARRYLL HENDRICKS, UBS Investment Bank

KENNETH L. JUDD, The Hoover Institution

DAVID MAIER, Portland State University

JAMES C. McWILLIAMS, University of California, Los Angeles

JUAN MEZA, University of California, Merced

JOHN W. MORGAN, Stony Brook University

VIJAYAN N. NAIR, University of Michigan

CLAUDIA NEUHAUSER, University of Minnesota, Rochester

J. TINSLEY ODEN, University of Texas at Austin

DONALD G. SAARI, University of California, Irvine

J.B. SILVERS, Case Western Reserve University

GEORGE SUGIHARA, University of California, San Diego

EVA TARDOS, Cornell University

KAREN VOGTMANN, Cornell University

BIN YU, University of California, Berkeley

Staff

SCOTT WEIDMAN, Director

NEAL GLASSMAN, Senior Program Officer

MICHELLE SCHWALBE, Associate Program Officer

BARBARA WRIGHT, Administrative Assistant

Acknowledgments

This report has been reviewed in draft form by individuals chosen for their diverse perspectives and technical expertise, in accordance with procedures approved by the National Research Council’s Report Review Committee. The purpose of this independent review is to provide candid and critical comments that will assist the institution in making its published report as sound as possible and to ensure that the report meets institutional standards for objectivity, evidence, and responsiveness to the study charge. The review comments and draft manuscript remain confidential to protect the integrity of the deliberative process. We wish to thank the following individuals for their review of this report:

Andrew Booker, Boeing Corporation

Donald Haynes, Los Alamos National Laboratory

Max Morris, Iowa State University

William Oberkampf, WLO Consulting

J. Tinsley Oden, University of Texas at Austin

Elaine Oren, Naval Research Laboratory

Naomi Oreskes, University of California, San Diego

Susan Sanchez, Naval Postgraduate School

Kaspar Willam, University of Colorado at Boulder

Although the reviewers listed above have provided many constructive comments and suggestions, they were not asked to endorse the conclusions or recommendations nor did they see the final draft of the report before its release. The review of this report was overseen by Ali Mosleh. Appointed by the National Research Council, he was responsible for making certain that an independent examination of this report was carried out in accordance with institutional procedures and that all review comments were carefully considered. Responsibility for the final content of this report rests entirely with the authoring committee and the institution.

The committee also acknowledges the valuable contribution of the following individuals, who provided input at the meetings on which this report is based:

Mark Anderson, Los Alamos National Laboratory

Bilal Ayyub, University of Maryland

Chris Barrett, Virginia Tech

Louis Anthony Cox, Cox Associates

Andrew Dienstfrey, National Institute of Standards and Technology

Michael Eldred, Sandia National Laboratories

Peter Gleckler, Lawrence Livermore National Laboratory

Thuc Hoang, National Nuclear Security Administration

Richard Klein, Lawrence Livermore National Laboratory

Alex Levis, George Mason University

J. Tinsley Oden, University of Texas at Austin

Mikel Petty, University of Alabama in Huntsville

Bruce Robinson, Los Alamos National Laboratory

Susan Sanchez, Naval Postgraduate School

Christopher Sims, Princeton University

Timothy Trucano, Sandia National Laboratories

The co-chairs also thank the following individuals for their helpful discussions over the course of this study:

Donald Estep, Colorado State University

Elizabeth Keating, Los Alamos National Laboratory

James McWilliams, University of California, Los Angeles

Robert Moser, University of Texas at Austin

Habib Najm, Sandia National Laboratories

Leonard Smith, London School of Economics

Karen Willcox, Massachusetts Institute of Technology

Contents

SUMMARY

  1  INTRODUCTION

  1.1  Overview and Study Charter

  1.2  VVUQ Definitions

  1.3  Scope of This Study

1.3.1   Focus on Prediction with Physics/Engineering Models

1.3.2   Focus on Mathematical and Quantitative Issues

  1.4  VVUQ Processes and Principles

1.4.1   Verification

1.4.2   Validation

1.4.3   Prediction

1.4.4   Uncertainty Quantification

1.4.5   Key VVUQ Principles

  1.5  Uncertainty and Probability

  1.6  Ball-Drop Case Study

1.6.1   The Physical System

1.6.2   The Model

1.6.3   Verification

1.6.4   Sources of Uncertainty

1.6.5   Propagation of Input Uncertainties

1.6.6   Validation and Prediction

1.6.7   Making Decisions

  1.7  Organization of This Report

  1.8  References

  2  SOURCES OF UNCERTAINTY AND ERROR

  2.1  Introduction

  2.2  Projectile-Impact Example Problem

  2.3  Initial Conditions

  2.4  Level of Fidelity

  2.5  Numerical Accuracy

  2.6  Multiscale Phenomena

  2.7  Parametric Settings

  2.8  Choosing a Model Form

  2.9  Summary

2.10  Climate-Modeling Case Study

2.10.1   Is Formal UQ Possible for Truly Complex Models?

2.10.2   Future Directions for Research and Teaching Involving UQ for Climate Models

2.11  References

  3  VERIFICATION

  3.1  Introduction

  3.2  Code Verification

  3.3  Solution Verification

  3.4  Summary of Verification Principles

  3.5  References

  4  EMULATION, REDUCED-ORDER MODELING, AND FORWARD PROPAGATION

  4.1  Approximating the Computational Model

4.1.1   Computer Model Emulation

4.1.2   Reduced-Order Models

  4.2  Forward Propagation of Input Uncertainty

  4.3  Sensitivity Analysis

4.3.1   Global Sensitivity Analysis

4.3.2   Local Sensitivity Analysis

  4.4  Choosing Input Settings for Ensembles of Computer Runs

  4.5  Electromagnetic Interference in a Tire Pressure Sensor: Case Study

4.5.1   Background

4.5.2   The Computer Model

4.5.3   Robust Emulators

4.5.4   Representative Result

  4.6  References

  5  MODEL VALIDATION AND PREDICTION

  5.1  Introduction

5.1.1   Note Regarding Methodology

5.1.2   The Ball-Drop Example Revisited

5.1.3   Model Validation Statement

  5.2  Uncertainties in Physical Measurements

  5.3  Model Calibration and Inverse Problems

  5.4  Model Discrepancy

  5.5  Assessing the Quality of Predictions

  5.6  Automobile Suspension Systems Case Study

5.6.1   Background

5.6.2   The Computer Model

5.6.3   The Process Being Modeled and Data

5.6.4   Modeling the Uncertainties

5.6.5   Analysis and Results

  5.7  Inference from Multiple Computer Models

  5.8  Exploiting Multiple Sources of Physical Observations

  5.9  PECOS Case Study

5.9.1   Overview

5.9.2   Verification

5.9.3   Code Verification

5.9.4   Solution Verification

5.9.5   Validation

5.10  Rare, High-Consequence Events

5.11  Conclusion

5.12  References

  6  MAKING DECISIONS

  6.1  Overview

  6.2  Decisions Within VVUQ Activities

  6.3  Decisions Based on VVUQ Information

  6.4  Decision Making Informed by VVUQ in the Stockpile Stewardship Program

  6.5  Decision Making Informed by VVUQ at the Nevada National Security Site

6.5.1   Background

6.5.2   The Physical System

6.5.3   Computational Modeling of the Physical System

6.5.4   Parameter Estimation

6.5.5   Making (Extrapolative) Predictions and Describing Uncertainty

6.5.6   Reporting Results to Decision Makers and Stakeholders

  6.6  Summary

  6.7  References

  7  NEXT STEPS IN PRACTICE, RESEARCH, AND EDUCATION FOR VERIFICATION, VALIDATION, AND UNCERTAINTY QUANTIFICATION

  7.1  VVUQ Principles and Best Practices

7.1.1   Verification Principles and Best Practices

7.1.2   Validation and Prediction Principles and Best Practices

  7.2  Principles and Best Practices in Related Areas

7.2.1   Transparency and Reporting

7.2.2   Decision Making

7.2.3   Software, Tools, and Repositories

  7.3  Research for Improved Mathematical Foundations

7.3.1   Verification Research

7.3.2   UQ Research

7.3.3   Validation and Prediction Research

  7.4  Education Changes for the Effective Integration of VVUQ

7.4.1   VVUQ at the University

7.4.2   Spreading the Word

  7.5  Closing Remarks

  7.6  References

APPENDIXES

A  Glossary

B  Agendas of Committee Meetings

C  Committee Biographies

D  Acronyms

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