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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
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THE NATIONAL ACADEMIES PRESS 500 Fifth Street, NW Washington, DC 20001
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, conclu -
sions, 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 Statisti -
cal 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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The National Academy of Sciences is a private, nonprofit, self-perpetuating society of distinguished scholars engaged in sci -
entific and engineering research, dedicated to the furtherance of science and technology and to their use for the general welfare.
Upon the authority of the charter granted to it by the Congress in 1863, the Academy has a mandate that requires it to advise the
federal government on scientific and technical matters. Dr. Ralph J. Cicerone is president of the National Academy of Sciences.
The National Academy of Engineering was established in 1964, under the charter of the National Academy of Sciences, as
a parallel organization of outstanding engineers. It is autonomous in its administration and in the selection of its members,
sharing with the National Academy of Sciences the responsibility for advising the federal government. The National Academy
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recognizes the superior achievements of engineers. Dr. Charles M. Vest is president of the National Academy of Engineering.
The Institute of Medicine was established in 1970 by the National Academy of Sciences to secure the services of eminent
members of appropriate professions in the examination of policy matters pertaining to the health of the public. The Institute acts
under the responsibility given to the National Academy of Sciences by its congressional charter to be an adviser to the federal
government and, upon its own initiative, to identify issues of medical care, research, and education. Dr. Harvey V. Fineberg
is president of the Institute of Medicine.
The National Research Council was organized by the National Academy of Sciences in 1916 to associate the broad com -
munity of science and technology with the Academy’s purposes of furthering knowledge and advising the federal government.
Functioning in accordance with general policies determined by the Academy, the Council has become the principal operating
agency of both the National Academy of Sciences and the National Academy of Engineering in providing services to the gov -
ernment, the public, and the scientific and engineering communities. The Council is administered jointly by both Academies
and the Institute of Medicine. Dr. Ralph J. Cicerone and Dr. Charles M. Vest a re chair and vice chair, respectively, of the
National Research Council.
www.national-academies.org
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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
v
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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
vi
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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
vii
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viii ACKNOWLEDGMENTS
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
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Contents
SUMMARY 1
1 INTRODUCTION 7
1.1 Overview and Study Charter, 7
1.2 VVUQ Definitions, 8
1.3 Scope of This Study, 9
1.3.1 Focus on Prediction with Physics/Engineering Models, 9
1.3.2 Focus on Mathematical and Quantitative Issues, 9
1.4 VVUQ Processes and Principles, 10
1.4.1 Verification, 10
1.4.2 Validation, 11
1.4.3 Prediction, 11
1.4.4 Uncertainty Quantification, 12
1.4.5 Key VVUQ Principles, 13
1.5 Uncertainty and Probability, 13
1.6 Ball-Drop Case Study, 14
1.6.1 The Physical System, 16
1.6.2 The Model, 16
1.6.3 Verification, 16
1.6.4 Sources of Uncertainty, 16
1.6.5 Propagation of Input Uncertainties, 17
1.6.6 Validation and Prediction, 17
1.6.7 Making Decisions, 17
1.7 Organization of This Report, 18
1.8 References, 18
2 SOURCES OF UNCERTAINTY AND ERROR 19
2.1 Introduction, 19
2.2 Projectile-Impact Example Problem, 20
ix
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x CONTENTS
2.3 Initial Conditions, 23
2.4 Level of Fidelity, 24
2.5 Numerical Accuracy, 24
2.6 Multiscale Phenomena, 25
2.7 Parametric Settings, 26
2.8 Choosing a Model Form, 26
2.9 Summary, 26
2.10 Climate-Modeling Case Study, 27
2.10.1 Is Formal UQ Possible for Truly Complex Models?, 28
2.10.2 Future Directions for Research and Teaching Involving UQ for Climate Models, 29
2.11 References, 30
3 VERIFICATION 31
3.1 Introduction, 31
3.2 Code Verification, 32
3.3 Solution Verification, 33
3.4 Summary of Verification Principles, 35
3.5 References, 36
4 EMULATION, REDUCED-ORDER MODELING, AND FORWARD PROPAGATION 37
4.1 Approximating the Computational Model, 38
4.1.1 Computer Model Emulation, 38
4.1.2 Reduced-Order Models, 39
4.2 Forward Propagation of Input Uncertainty, 41
4.3 Sensitivity Analysis, 42
4.3.1 Global Sensitivity Analysis, 43
4.3.2 Local Sensitivity Analysis, 44
4.4 Choosing Input Settings for Ensembles of Computer Runs, 46
4.5 Electromagnetic Interference in a Tire Pressure Sensor: Case Study, 46
4.5.1 Background, 46
4.5.2 The Computer Model, 46
4.5.3 Robust Emulators, 48
4.5.4 Representative Result, 49
4.6 References, 49
5 MODEL VALIDATION AND PREDICTION 52
5.1 Introduction, 52
5.1.1 Note Regarding Methodology, 54
5.1.2 The Ball-Drop Example Revisited, 57
5.1.3 Model Validation Statement, 58
5.2 Uncertainties in Physical Measurements, 59
5.3 Model Calibration and Inverse Problems, 60
5.4 Model Discrepancy, 63
5.5 Assessing the Quality of Predictions, 67
5.6 Automobile Suspension Systems Case Study, 70
5.6.1 Background, 70
5.6.2 The Computer Model, 70
5.6.3 The Process Being Modeled and Data, 70
5.6.4 Modeling the Uncertainties, 71
5.6.5 Analysis and Results, 72
5.7 Inference from Multiple Computer Models, 74
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xi
CONTENTS
5.8 Exploiting Multiple Sources of Physical Observations, 75
5.9 PECOS Case Study, 75
5.9.1 Overview, 75
5.9.2 Verification, 76
5.9.3 Code Verification, 76
5.9.4 Solution Verification, 77
5.9.5 Validation, 78
5.10 Rare, High-Consequence Events, 79
5.11 Conclusion, 80
5.12 References, 83
6 MAKING DECISIONS 86
6.1 Overview, 86
6.2 Decisions Within VVUQ Activities, 86
6.3 Decisions Based on VVUQ Information, 87
6.4 Decision Making Informed by VVUQ in the Stockpile Stewardship Program, 88
6.5 Decision Making Informed by VVUQ at the Nevada National Security Site, 90
6.5.1 Background, 90
6.5.2 The Physical System, 91
6.5.3 Computational Modeling of the Physical System, 92
6.5.4 Parameter Estimation, 92
6.5.5 Making (Extrapolative) Predictions and Describing Uncertainty, 93
6.5.6 Reporting Results to Decision Makers and Stakeholders, 93
6.6 Summary, 93
6.7 References, 94
7 NEXT STEPS IN PRACTICE, RESEARCH, AND EDUCATION FOR VERIFICATION, 95
VALIDATION, AND UNCERTAINTY QUANTIFICATION
7.1 VVUQ Principles and Best Practices, 95
7.1.1 Verification Principles and Best Practices, 96
7.1.2 Validation and Prediction Principles and Best Practices, 97
7.2 Principles and Best Practices in Related Areas, 98
7.2.1 Transparency and Reporting, 98
7.2.2 Decision Making, 99
7.2.3 Software, Tools, and Repositories, 99
7.3 Research for Improved Mathematical Foundations, 100
7.3.1 Verification Research, 100
7.3.2 UQ Research, 101
7.3.3 Validation and Prediction Research, 102
7.4 Education Changes for the Effective Integration of VVUQ, 103
7.4.1 VVUQ at the University, 103
7.4.2 Spreading the Word, 106
7.5 Closing Remarks, 106
7.6 References, 106
APPENDIXES
A Glossary 109
B Agendas of Committee Meetings 120
C Committee Biographies 124
D Acronyms 130
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