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Volume II

SENSITIVITY and

UNCERTAINTY ANALYSIS

Applications to Large-Scale Systems

Dan G. Cacuci Mihaela Ionescu-Bujor Ionel Michael Navon

Boca Raton London New York Singapore

Copyright © 2005 Taylor & Francis Group, LLC

Published in 2005 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742

© 2005 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group

No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1

International Standard Book Number-10: 1-58488-116-X (Hardcover) International Standard Book Number-13: 978-1-58488-116-2 (Hardcover) Library of Congress Card Number 2003043992

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use.

No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microﬁlming, and recording, or in any information storage or retrieval system, without written permission from the publishers.

For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-proﬁt organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.

Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identiﬁcation and explanation without intent to infringe.

Library of Congress Cataloging-in-Publication Data

Cacuci, D. G. Sensitivity and uncertainty analysis / Dan G. Cacuci. p. cm. Includes bibliographical references and index. Contents: v. 1. Theory ISBN 1-58488-115-1 (v. 1 : alk. paper) 1. Sensitivity theory (Mathematics) 2. Uncertainty (Information theory) 3. Mathematical

models—Evaluation. I. Title.

QA402.3.C255 2003 003’.5—dc21

2003043992

Taylor & Francis Group is the Academic Division of T&F Informa plc.

Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com

and the CRC Press Web site at http://www.crcpress.com

Copyright © 2005 Taylor & Francis Group, LLC

INTRODUCTION

Sensitivity and uncertainty analysis are becoming increasingly widespread in many fields of engineering and sciences, encompassing practically all of the experimental data processing activities as well as many computational modeling and process simulation activities. There are many methods, based either on deterministic or statistical concepts, for performing sensitivity and uncertainty analysis. Two of the modern deterministic methods, namely the Adjoint Sensitivity Analysis Procedure (ASAP) and the Global Adjoint Sensitivity Analysis Procedure (GASAP) were presented, in detail, in Volume I of this book. However, despite of this variety of methods, or perhaps because of it, a precise, unified terminology, across all methods, does not seem to exist yet, even though many of the same words are used by the practitioners of the various methods. For example, even the word “sensitivity” as used by analysts employing statistical methods may not necessarily mean or refer to the same quantity as would be described by the same word, “sensitivity,” when used by analysts employing deterministic methods. Care must be therefore exercised, since identical words may not necessarily describe identical quantities, particularly when comparing deterministic to statistical methods. Furthermore, conflicting and contradictory claims are often made about the relative strengths and weaknesses of the various methods.

Models of complex physical systems usually involve two distinct sources of uncertainties, namely: (i) stochastic uncertainty, which arises because the system under investigation can behave in many different ways, and (ii) subjective or epistemic uncertainty, which arises from the inability to specify an exact value for a parameter that is assumed to have a constant value in the respective investigation. Epistemic (or subjective) uncertainties characterize a degree of belief regarding the location of the appropriate value of each parameter. In turn, these subjective uncertainties lead to subjective uncertainties for the response, thus reflecting a corresponding degree of belief regarding the location of the appropriate response values as the outcome of analyzing the model under consideration. A typical example of a complex system that involves both stochastic and epistemic uncertainties is a nuclear reactor power plant: in a typical risk analysis of a nuclear power plant, stochastic uncertainty arises due to the hypothetical accident scenarios which are considered in the respective risk analysis, while epistemic uncertainties arise because of uncertain parameters that underlie the estimation of the probabilities and consequences of the respective hypothetical accident scenarios.

Sensitivity and uncertainty analysis procedures can be either local or global in scope. The objective of local analysis is to analyze the behavior of the system response locally around a chosen point (for static systems) or chosen trajectory (for dynamical systems) in the combined phase space of parameters and state variables. On the other hand, the objective of global analysis is to determine all of the system's critical points (bifurcations, turning points, response maxima, minima, and/or saddle points) in the combined phase space formed by the

Copyright © 2005 Taylor & Francis Group, LLC

parameters and dependent (state) variables, and subsequently analyze these critical points by local sensitivity and uncertainty analysis. The methods for sensitivity and uncertainty analysis are based on either statistical or deterministic procedures. In principle, both types of procedures can be used for either local or for global sensitivity and uncertainty analysis, although, in practice, deterministic methods are used mostly for local analysis while statistical methods are used for both local and global analysis.

To assist the reader set the various methods for sensitivity and uncertainty analysis in proper perspective, Chapter I of this Volume reviews and summarizes the salient features, highlighting relative strengths and weaknesses, of the most prominent screening design methods, statistical methods (local and global), and deterministic methods (local and global), as they are currently applied in practice. The following statistical procedures are discussed: sampling-based methods (random sampling, stratified importance sampling, and Latin Hypercube sampling), first- and second-order reliability algorithms (FORM and SORM, respectively), variance-based methods (correlation ratio-based methods, the Fourier amplitude sensitivity test, and Sobol’s method), and screening design methods (classical one-at-a-time experiments, global one-at-a-time design methods, systematic fractional replicate designs, and sequential bifurcation designs). It is important to note that all statistical uncertainty and sensitivity analysis methods first commence with the “uncertainty analysis” stage, and only subsequently proceed to the “sensitivity analysis” stage; this procedural path is the reverse of the procedural (and conceptual) path underlying the deterministic methods of sensitivity and uncertainty analysis, where the sensitivities are determined prior to using them for uncertainty analysis.

In practice, sensitivities cannot be computed exactly by using statistical methods; this can be done only by using deterministic methods. Among deterministic methods, it is noted that the direct method and the Forward Sensitivity Analysis Procedure (FSAP) require at least as many modelevaluations as there are parameters in the model, while the ASAP requires a single model-evaluation of an appropriate adjoint model, whose source term is related to the response under investigation. The ASAP is the most efficient method for computing local sensitivities of large-scale systems, when the number of parameters and/or parameter variations exceeds the number of responses of interest. It appears that the only genuinely global deterministic method for sensitivity analysis, published thus far, is the global adjoint sensitivity analysis procedure (GASAP) that was presented in Chapter VI of Volume I. The GASAP uses both the forward and the adjoint sensitivity system to explore, exhaustively and efficiently, the entire phase-space of system parameters and dependent variables, in order to obtain complete information about the important global features of the physical system, namely the critical points of the response and the bifurcation branches and/or turning points of the system’s state variables. Notably, the adjoint sensitivity model can be developed using relatively modest additional resources, if it is developed simultaneously with the original model. However, if the adjoint sensitivity model is constructed

Copyright © 2005 Taylor & Francis Group, LLC

a posteriori, considerable skills may be required for its successful development and implementation.

Chapter II of this Volume presents applications of the ASAP to transient onedimensional two-phase flow problems modeled by well-posed quasi-linear partial differential equations. The chapter commences with the presentation of a self-contained formalism for applying the ASAP to functional-type responses associated with two-phase flow models that comprise equations describing conservation of mass, momentum, and energy, for practical one-dimensional, two-phase flow models. This theoretical presentation is followed by a presentation of the main aspects of implementing the ASAP into the RELAP5/MOD3.2 code system, which is a large-scale code that simulates the thermal-hydraulic characteristics of light water nuclear reactors. The thermalhydraulic part of the RELAP5/MOD3.2 code comprises a one-dimensional, nonequilibrium, non-homogeneous two-phase flow model, including conservation equations for boron concentration and non-condensable gases. Chapter II also highlights the fundamentally important aspect of consistency between the differential and the corresponding discretized equations used for sensitivity analysis. In particular, the following consistency correspondences must be assured: (i) the Discretized Forward Sensitivity Model must be consistent with the Differential Forward Sensitivity Model, if the FSAP is used; (ii) the Differential Adjoint Sensitivity Model must be consistent with the Discretized Adjoint Sensitivity Model, if the ASAP is used; and (iii) the Discretized (representation of the) Response Sensitivity must be consistent with the Integral (representation of the) Response Sensitivity for both the FSAP and the ASAP (in which the Integral and the Discretized Response Sensitivity are represented in terms of the corresponding adjoint functions).

From a historical perspective, in almost every field of scientific activity, the development of large-scale simulation models extended over many years, if not decades, and their respective development invariably involved large and sometimes changing teams of scientists. Furthermore, such complex models consist of many inter-coupled modules, each module simulating a particular physical sub-process. Since the ASAP has not been widely known in the past, most of the extant large-scale, complex simulation models were developed without having simultaneously developed and implemented the corresponding adjoint sensitivity model. Implementing a posteriori the ASAP for such largescale code systems is not trivial, and the development and implementation of the adjoint sensitivity model can seldom be executed all at once, in one fell swoop. Actually, an “all-or-nothing” approach for developing and implementing the complete, and correspondingly complex, adjoint sensitivity model for large-scale problems is at best difficult (and, at worst, impractical), and is therefore not recommended. Instead, the recommended strategy is a module-by-module implementation of the ASAP. In this approach, the ASAP is applied to each module, in turn, to develop a corresponding adjoint sensitivity system for each component module. The final step in this “modular” implementation of the ASAP is to “augment” (i.e., join together) the adjoint sensitivity systems for each of the

Copyright © 2005 Taylor & Francis Group, LLC

respective modules, avoiding redundant effort and/or loss of information, until all of the component adjoint sensitivity modules are judiciously connected together, accounting for all of the requisite feedbacks and liaisons between them. In view of its high importance for practical applications, Chapter III presents the theoretical foundation for the modular implementation of the ASAP for complex simulations systems, by starting with a selected code module, and then augmenting the size of the adjoint sensitivity system, module by module, until completing the entire system under consideration. The presentation of the general theory (i.e., the ASAP for augmented systems) is followed by an illustrative application of this theory to a large-scale system involving the augmentation of the adjoint sensitivity model corresponding to the two-fluid model in RELAP5/MOD3.2 (which was the subject of Chapter II) with the adjoint sensitivity model corresponding to the heat structure models in RELAP5/MOD3.2.

Often, the response functional of a physical system is located at a critical point (i.e., a maximum, minimum, saddle point, etc.) of a function that depends on the system’s state vector and parameters. In such situations, changes in the system’s parameter would affect not only the magnitude of the response, but also its location in phase-space, since the perturbed response would not only differ in value from the original response, but would also occur in a different spatial location, at a different point in time, etc. The general sensitivity theory, including the ASAP, for such responses, defined at critical points, is presented in Chapter IV. The practical application of the general theory is illustrated by means of a simple paradigm example (a simple particle diffusion problem), and also by means of a large-scale application to a paradigm transient scenario for a nuclear reactor system. The reactor’s transient behavior is simulated by using a largescale code system that solves equations describing the following phenomena: (a) thermal-hydraulics equations describing the conservation of thermal energy, mass, and momentum for the average channel fuel pin and surrounding singlephase coolant in the reactor’s core; (b) neutron point-kinetics equations describing the time-dependent behavior of the core-integrated neutron density; and (c) a loop-hydraulic equation that relates the core inlet and outlet coolant pressures.

Chapters V and VI are devoted to applications of the ASAP for performing efficient sensitivity analysis of paradigm large-scale models used for numerical weather prediction and climatic research. Our understanding of atmospheric processes relies on the use of mathematical models to test the consequences of various physical assumptions. An essential part of weather prediction and climatic research consists of interpreting the results of large-scale simulation models. For example, the current concern about the climatic impact of CO2 stems from the sensitivity that climatic models exhibit to the atmospheric concentration of CO2. A further example is the occurrence of atmospheric blocks, which strongly affect the variability in predictive skills of numerical weather prediction (NWP) models; it is therefore important to understand the model errors associated with blocking situations. As mathematical models increase in

Copyright © 2005 Taylor & Francis Group, LLC

sophistication, though, the reasons for the results they give become less clear, making the results more difficult to interpret. A quantitative procedure to help interpret the results of a mathematical model is to perform a sensitivity analysis, i.e., to investigate how the results of the model change when parameters in the model are varied. For example, the ice-albedo feedback mechanism corresponds to the observed negative sensitivity of surface air temperature to surface albedo. Furthermore, sensitivities quantify the extent that uncertainties in parameters contribute to uncertainties in results of models. For example, sub-grid processes need to be parameterized, but such parameterizations are highly simplified approximations of complex processes, so the uncertainties in the parameters involved can be large. If the corresponding sensitivities are also large, then the results of the model will have large uncertainties.

Chapter V presents paradigm applications of the ASAP to a radiativeconvective model (RCM) for climate simulation and, respectively, to a two-layer isentropic primitive equation model for numerical weather prediction. The RCM contains the nonlinear phenomena characteristic of radiatively-coupled processes, and includes 312 variable parameters. The ASAP is applied to derive the adjoint sensitivity equations, to compute efficiently the response sensitivities (in terms of the adjoint functions) to all parameters, and to illustrate the use of sensitivities. Notably, the adjoint functions themselves can be interpreted as the sensitivity of a response to instantaneous perturbations of the model’s dependent variables. Furthermore the adjoint functions can be used to reveal the time scales associated with the most important physical processes in the model. In particular, the adjoint functions for the RCM reveal the three time scales associated with: (i) convective adjustment; (ii) heat transfer between the atmosphere and space; and (iii) heat transfer between the ground and atmosphere. Calculating the eigenvalues and eigenvectors of the matrix of derivatives occurring in the set of adjoint equations reveals similar physical information without actually needing to solve the adjoint sensitivity model. An illustrative use of the ASAP for evaluating the sensitivity to feedback mechanisms, is also presented. The paradigm response considered is the increase in the average surface air temperature which occurs after the atmospheric CO2 concentration in the model is doubled, while the paradigm feedback is the surface albedo feedback.

Chapter V continues with a paradigm application of the ASAP to more complex, operator-valued, responses, by considering a paradigm two-layer isentropic NWP model that simulates the nonlinear life cycles of baroclinic waves, including the occurrence of so-called “blocks.” The variability in predictive skills of NWP models is strongly related to the occurrence of such blocks; this occurrence is indicated by the so-called “blocking indices.” From a mathematical point of view, blocking indices are operator-valued responses. The ASAP is applied to perform a paradigm sensitivity analysis of a time-dependent blocking index to model parameters. This illustrative example underscores the fact that the exceptional computational efficiency of the ASAP yields quantitative results that could not have been obtained, in practice, by any other sensitivity analysis method, because of prohibitive computational costs.

Copyright © 2005 Taylor & Francis Group, LLC

Chapter VI sets forth the presentation of paradigm applications of the ASAP to large-scale models used for numerical weather prediction, by considering the following models: (i) the diagnostic equations underlying the nonlinear radiation model used in a version of the National Center for Environmental Prediction model for medium-range weather forecasting; (ii) the Florida State University Global Spectral Model; and (iii) the Relaxed Arakawa Schubert scheme in the NASA Goddard Earth Observing System-1 (GEOS-1) general circulation model, developed by the then Data Assimilation Office (now Global Modeling and Assimilation Office) at the NASA Goddard Space Flight Center. A particularly important implication of the paradigm sensitivity analysis results presented in this chapter is that accurate data for temperature, moisture and surface pressure are essential for an accurate evaluation of cumulus cloud effects, especially at the most influential vertical levels that were identified by sensitivity analysis. Of course, this is because small perturbations at such influential locations tend to exert a stronger impact on the responses than similar perturbations at other, less influential, locations. Therefore, data quality is particularly important at those levels and areas with positive feedback between cloud activities and the environment, since small errors tend to grow through positive feedback mechanisms. Such sensitivity analysis results also underscore the importance of the ASAP for variational data assimilation. For example, in variational assimilation of precipitation data, in which moist convection is the dominant process, the difference between model output rainfall and the observed rainfall is taken as input to the adjoint sensitivity model. Such information also indicates the regions where additional adaptive observations should be taken.

To keep this volume to a reasonable size, several important topics (e.g., methods of data adjustment and data assimilation in the presence of uncertainties; optimal control of fluid flow) have been deferred for presentation in subsequent volume(s). Nevertheless, by addressing computational issues and highlighting the major challenges that still remain to be resolved, the material presented in this Volume is also intended to provide a comprehensive basis for further advancements and innovations in the field sensitivity and uncertainty analysis. Two outstanding issues, whose solution would greatly advance the state of overall knowledge, would be: (i) to develop the adjoint sensitivity analysis procedure (ASAP) for problems describing turbulent flows, and (ii) to combine the GASAP with global statistical uncertainty analysis methods, striving to perform, efficiently and accurately, global sensitivity and uncertainty analyses for large-scale systems.

In closing, the authors would like to acknowledge the essential contributions made the editorial staff of Chapman & Hall / CRC. We are particularly grateful to Ms. Helena Redshaw for keeping the publication schedule on track with her friendly e-mails. Last but not least, our special thanks go to Bob Stern, Executive Editor, whose patience and unwavering support made it ultimately possible to bring this book to our readers. One of the authors (I. M. Navon) would also like to acknowledge support of NSF and NASA grants for his research.

Copyright © 2005 Taylor & Francis Group, LLC

TABLE OF CONTENTS

I.

A COMPARATIVE REVIEW OF SENSITIVITY

AND UNCERTAINTY ANALYSIS METHODS

FOR LARGE-SCALE SYSTEMS

1

A. STATISTICAL METHODS

5

1. Introduction

5

2. Sampling-Based Methods

6

3. Reliability Algorithms: FORM and SORM

16

4. Variance-Based Methods

16

5. Design of Experiments and Screening Design Methods

21

B. DETERMINISTIC METHODS

26

1. Deterministic Methods for Local Sensitivity Analysis

26

2. Deterministic Methods for Local Uncertainty Analysis

29

3. Deterministic Methods for Extending the Use of

Local Sensitivities; Global Deterministic

Sensitivity Analysis

31

C. COMPUTATIONAL CONSIDERATIONS

32

II.

APPLICATIONS OF THE ADJOINT SENSITIVITY

ANALYSIS PROCEDURE (ASAP) TO TWO-PHASE

FLOW SYSTEMS

37

A. ASAP FOR GENERIC TWO-PHASE FLOW PROBLEMS

38

1. Basic One-Dimensional Two-Phase Flow Equations

39

2. The Adjoint Sensitivity Analysis Procedure (ASAP)

40

3. Characteristic of the Adjoint Sensitivity System

46

4. Illustrative Example: The Homogeneous Equilibrium

Model (HEM) for Two-Phase Flow

47

B. ASAP FOR THE RELAP5/MOD3.2 TWO-FLUID

MODEL (REL/TF)

55

1. The RELAP5/MOD3.2 Two-Fluid Model

57

2. ASAP of the Two-Fluid Model in RELAP5/MOD3.2

65

3. Consistency between the Differential/Integral

And the Discretized Equations/Models for

Sensitivity Analysis

80

4. Validation of the Adjoint Sensitivity Model

(ASM-REL/TF)

82

5. Sensitivities of Thermodynamic Properties of Water

91

Copyright © 2005 Taylor & Francis Group, LLC

III. FORWARD AND ADJOINT SENSITIVITY

ANALYSIS PROCEDURES FOR AUGMENTED

SYSTEMS

123

A. THEORETICAL BASIS FOR THE ASAP FOR

AUGMENTED SYSTEMS

124

1. Sensitivity Analysis of the Primary

(Nonaugmented) System

124

2. Sensitivity Analysis of the Augmented System

131

3. Discussion: Constructing the Augmented Adjoint

Sensitivity Model from the Original Adjoint

Sensitivity Model and Viceversa

143

B. ILLUSTRATIVE EXAMPLE: ASAP FOR THE COUPLED

TWO-FLUID WITH HEAT STRUCTURES MODEL

IN RELAP5/MOD3.2 (REL/TF+HS)

148

1. ASM-REL/TFH: The Augmented Two-Fluid/Heat

Structure Adjoint Sensitivity Model

148

2. Summary Description of the QUENCH-04 Experiment

158

IV. FORWARD AND ADJOINT SENSITIVITY

ANALYSIS PROCEDURES FOR RESPONSES

DEFINED AT CRITICAL POINTS

171

A. FSAP AND ASAP FOR RESPONSES AT CRITICAL

POINTS: GENERAL THEORY

172

1. The Forward Sensitivity Analysis Procedure (FSAP)

174

2. The Adjoint Sensitivity Analysis Procedure (ASAP)

177

3. Discussion

181

4. Illustrative Example: A Simple Particle

Diffusion Problem

183

B. ILLUSTRATIVE EXAMPLE: ASAP FOR THE MAXIMUM

CLAD TEMPERATURE PREDICTED BY A REACTOR

SAFETY CODE

194

V.

USING THE ASAP TO GAIN NEW INSIGHTS

INTO PARADIGM ATMOSPHERIC

SCIENCES PROBLEMS

231

A. A PARADIGM RADIATIVE-CONVECTIVE MODEL (RCM)

OF THE ATMOSPHERE

232

B. APPLYING THE ASAP FOR EFFICIENT AND

EXHAUSTIVE SENSITIVITY ANALYSIS OF THE RCM

237

Copyright © 2005 Taylor & Francis Group, LLC

SENSITIVITY and

UNCERTAINTY ANALYSIS

Applications to Large-Scale Systems

Dan G. Cacuci Mihaela Ionescu-Bujor Ionel Michael Navon

Boca Raton London New York Singapore

Copyright © 2005 Taylor & Francis Group, LLC

Published in 2005 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742

© 2005 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group

No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1

International Standard Book Number-10: 1-58488-116-X (Hardcover) International Standard Book Number-13: 978-1-58488-116-2 (Hardcover) Library of Congress Card Number 2003043992

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use.

No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microﬁlming, and recording, or in any information storage or retrieval system, without written permission from the publishers.

For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-proﬁt organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.

Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identiﬁcation and explanation without intent to infringe.

Library of Congress Cataloging-in-Publication Data

Cacuci, D. G. Sensitivity and uncertainty analysis / Dan G. Cacuci. p. cm. Includes bibliographical references and index. Contents: v. 1. Theory ISBN 1-58488-115-1 (v. 1 : alk. paper) 1. Sensitivity theory (Mathematics) 2. Uncertainty (Information theory) 3. Mathematical

models—Evaluation. I. Title.

QA402.3.C255 2003 003’.5—dc21

2003043992

Taylor & Francis Group is the Academic Division of T&F Informa plc.

Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com

and the CRC Press Web site at http://www.crcpress.com

Copyright © 2005 Taylor & Francis Group, LLC

INTRODUCTION

Sensitivity and uncertainty analysis are becoming increasingly widespread in many fields of engineering and sciences, encompassing practically all of the experimental data processing activities as well as many computational modeling and process simulation activities. There are many methods, based either on deterministic or statistical concepts, for performing sensitivity and uncertainty analysis. Two of the modern deterministic methods, namely the Adjoint Sensitivity Analysis Procedure (ASAP) and the Global Adjoint Sensitivity Analysis Procedure (GASAP) were presented, in detail, in Volume I of this book. However, despite of this variety of methods, or perhaps because of it, a precise, unified terminology, across all methods, does not seem to exist yet, even though many of the same words are used by the practitioners of the various methods. For example, even the word “sensitivity” as used by analysts employing statistical methods may not necessarily mean or refer to the same quantity as would be described by the same word, “sensitivity,” when used by analysts employing deterministic methods. Care must be therefore exercised, since identical words may not necessarily describe identical quantities, particularly when comparing deterministic to statistical methods. Furthermore, conflicting and contradictory claims are often made about the relative strengths and weaknesses of the various methods.

Models of complex physical systems usually involve two distinct sources of uncertainties, namely: (i) stochastic uncertainty, which arises because the system under investigation can behave in many different ways, and (ii) subjective or epistemic uncertainty, which arises from the inability to specify an exact value for a parameter that is assumed to have a constant value in the respective investigation. Epistemic (or subjective) uncertainties characterize a degree of belief regarding the location of the appropriate value of each parameter. In turn, these subjective uncertainties lead to subjective uncertainties for the response, thus reflecting a corresponding degree of belief regarding the location of the appropriate response values as the outcome of analyzing the model under consideration. A typical example of a complex system that involves both stochastic and epistemic uncertainties is a nuclear reactor power plant: in a typical risk analysis of a nuclear power plant, stochastic uncertainty arises due to the hypothetical accident scenarios which are considered in the respective risk analysis, while epistemic uncertainties arise because of uncertain parameters that underlie the estimation of the probabilities and consequences of the respective hypothetical accident scenarios.

Sensitivity and uncertainty analysis procedures can be either local or global in scope. The objective of local analysis is to analyze the behavior of the system response locally around a chosen point (for static systems) or chosen trajectory (for dynamical systems) in the combined phase space of parameters and state variables. On the other hand, the objective of global analysis is to determine all of the system's critical points (bifurcations, turning points, response maxima, minima, and/or saddle points) in the combined phase space formed by the

Copyright © 2005 Taylor & Francis Group, LLC

parameters and dependent (state) variables, and subsequently analyze these critical points by local sensitivity and uncertainty analysis. The methods for sensitivity and uncertainty analysis are based on either statistical or deterministic procedures. In principle, both types of procedures can be used for either local or for global sensitivity and uncertainty analysis, although, in practice, deterministic methods are used mostly for local analysis while statistical methods are used for both local and global analysis.

To assist the reader set the various methods for sensitivity and uncertainty analysis in proper perspective, Chapter I of this Volume reviews and summarizes the salient features, highlighting relative strengths and weaknesses, of the most prominent screening design methods, statistical methods (local and global), and deterministic methods (local and global), as they are currently applied in practice. The following statistical procedures are discussed: sampling-based methods (random sampling, stratified importance sampling, and Latin Hypercube sampling), first- and second-order reliability algorithms (FORM and SORM, respectively), variance-based methods (correlation ratio-based methods, the Fourier amplitude sensitivity test, and Sobol’s method), and screening design methods (classical one-at-a-time experiments, global one-at-a-time design methods, systematic fractional replicate designs, and sequential bifurcation designs). It is important to note that all statistical uncertainty and sensitivity analysis methods first commence with the “uncertainty analysis” stage, and only subsequently proceed to the “sensitivity analysis” stage; this procedural path is the reverse of the procedural (and conceptual) path underlying the deterministic methods of sensitivity and uncertainty analysis, where the sensitivities are determined prior to using them for uncertainty analysis.

In practice, sensitivities cannot be computed exactly by using statistical methods; this can be done only by using deterministic methods. Among deterministic methods, it is noted that the direct method and the Forward Sensitivity Analysis Procedure (FSAP) require at least as many modelevaluations as there are parameters in the model, while the ASAP requires a single model-evaluation of an appropriate adjoint model, whose source term is related to the response under investigation. The ASAP is the most efficient method for computing local sensitivities of large-scale systems, when the number of parameters and/or parameter variations exceeds the number of responses of interest. It appears that the only genuinely global deterministic method for sensitivity analysis, published thus far, is the global adjoint sensitivity analysis procedure (GASAP) that was presented in Chapter VI of Volume I. The GASAP uses both the forward and the adjoint sensitivity system to explore, exhaustively and efficiently, the entire phase-space of system parameters and dependent variables, in order to obtain complete information about the important global features of the physical system, namely the critical points of the response and the bifurcation branches and/or turning points of the system’s state variables. Notably, the adjoint sensitivity model can be developed using relatively modest additional resources, if it is developed simultaneously with the original model. However, if the adjoint sensitivity model is constructed

Copyright © 2005 Taylor & Francis Group, LLC

a posteriori, considerable skills may be required for its successful development and implementation.

Chapter II of this Volume presents applications of the ASAP to transient onedimensional two-phase flow problems modeled by well-posed quasi-linear partial differential equations. The chapter commences with the presentation of a self-contained formalism for applying the ASAP to functional-type responses associated with two-phase flow models that comprise equations describing conservation of mass, momentum, and energy, for practical one-dimensional, two-phase flow models. This theoretical presentation is followed by a presentation of the main aspects of implementing the ASAP into the RELAP5/MOD3.2 code system, which is a large-scale code that simulates the thermal-hydraulic characteristics of light water nuclear reactors. The thermalhydraulic part of the RELAP5/MOD3.2 code comprises a one-dimensional, nonequilibrium, non-homogeneous two-phase flow model, including conservation equations for boron concentration and non-condensable gases. Chapter II also highlights the fundamentally important aspect of consistency between the differential and the corresponding discretized equations used for sensitivity analysis. In particular, the following consistency correspondences must be assured: (i) the Discretized Forward Sensitivity Model must be consistent with the Differential Forward Sensitivity Model, if the FSAP is used; (ii) the Differential Adjoint Sensitivity Model must be consistent with the Discretized Adjoint Sensitivity Model, if the ASAP is used; and (iii) the Discretized (representation of the) Response Sensitivity must be consistent with the Integral (representation of the) Response Sensitivity for both the FSAP and the ASAP (in which the Integral and the Discretized Response Sensitivity are represented in terms of the corresponding adjoint functions).

From a historical perspective, in almost every field of scientific activity, the development of large-scale simulation models extended over many years, if not decades, and their respective development invariably involved large and sometimes changing teams of scientists. Furthermore, such complex models consist of many inter-coupled modules, each module simulating a particular physical sub-process. Since the ASAP has not been widely known in the past, most of the extant large-scale, complex simulation models were developed without having simultaneously developed and implemented the corresponding adjoint sensitivity model. Implementing a posteriori the ASAP for such largescale code systems is not trivial, and the development and implementation of the adjoint sensitivity model can seldom be executed all at once, in one fell swoop. Actually, an “all-or-nothing” approach for developing and implementing the complete, and correspondingly complex, adjoint sensitivity model for large-scale problems is at best difficult (and, at worst, impractical), and is therefore not recommended. Instead, the recommended strategy is a module-by-module implementation of the ASAP. In this approach, the ASAP is applied to each module, in turn, to develop a corresponding adjoint sensitivity system for each component module. The final step in this “modular” implementation of the ASAP is to “augment” (i.e., join together) the adjoint sensitivity systems for each of the

Copyright © 2005 Taylor & Francis Group, LLC

respective modules, avoiding redundant effort and/or loss of information, until all of the component adjoint sensitivity modules are judiciously connected together, accounting for all of the requisite feedbacks and liaisons between them. In view of its high importance for practical applications, Chapter III presents the theoretical foundation for the modular implementation of the ASAP for complex simulations systems, by starting with a selected code module, and then augmenting the size of the adjoint sensitivity system, module by module, until completing the entire system under consideration. The presentation of the general theory (i.e., the ASAP for augmented systems) is followed by an illustrative application of this theory to a large-scale system involving the augmentation of the adjoint sensitivity model corresponding to the two-fluid model in RELAP5/MOD3.2 (which was the subject of Chapter II) with the adjoint sensitivity model corresponding to the heat structure models in RELAP5/MOD3.2.

Often, the response functional of a physical system is located at a critical point (i.e., a maximum, minimum, saddle point, etc.) of a function that depends on the system’s state vector and parameters. In such situations, changes in the system’s parameter would affect not only the magnitude of the response, but also its location in phase-space, since the perturbed response would not only differ in value from the original response, but would also occur in a different spatial location, at a different point in time, etc. The general sensitivity theory, including the ASAP, for such responses, defined at critical points, is presented in Chapter IV. The practical application of the general theory is illustrated by means of a simple paradigm example (a simple particle diffusion problem), and also by means of a large-scale application to a paradigm transient scenario for a nuclear reactor system. The reactor’s transient behavior is simulated by using a largescale code system that solves equations describing the following phenomena: (a) thermal-hydraulics equations describing the conservation of thermal energy, mass, and momentum for the average channel fuel pin and surrounding singlephase coolant in the reactor’s core; (b) neutron point-kinetics equations describing the time-dependent behavior of the core-integrated neutron density; and (c) a loop-hydraulic equation that relates the core inlet and outlet coolant pressures.

Chapters V and VI are devoted to applications of the ASAP for performing efficient sensitivity analysis of paradigm large-scale models used for numerical weather prediction and climatic research. Our understanding of atmospheric processes relies on the use of mathematical models to test the consequences of various physical assumptions. An essential part of weather prediction and climatic research consists of interpreting the results of large-scale simulation models. For example, the current concern about the climatic impact of CO2 stems from the sensitivity that climatic models exhibit to the atmospheric concentration of CO2. A further example is the occurrence of atmospheric blocks, which strongly affect the variability in predictive skills of numerical weather prediction (NWP) models; it is therefore important to understand the model errors associated with blocking situations. As mathematical models increase in

Copyright © 2005 Taylor & Francis Group, LLC

sophistication, though, the reasons for the results they give become less clear, making the results more difficult to interpret. A quantitative procedure to help interpret the results of a mathematical model is to perform a sensitivity analysis, i.e., to investigate how the results of the model change when parameters in the model are varied. For example, the ice-albedo feedback mechanism corresponds to the observed negative sensitivity of surface air temperature to surface albedo. Furthermore, sensitivities quantify the extent that uncertainties in parameters contribute to uncertainties in results of models. For example, sub-grid processes need to be parameterized, but such parameterizations are highly simplified approximations of complex processes, so the uncertainties in the parameters involved can be large. If the corresponding sensitivities are also large, then the results of the model will have large uncertainties.

Chapter V presents paradigm applications of the ASAP to a radiativeconvective model (RCM) for climate simulation and, respectively, to a two-layer isentropic primitive equation model for numerical weather prediction. The RCM contains the nonlinear phenomena characteristic of radiatively-coupled processes, and includes 312 variable parameters. The ASAP is applied to derive the adjoint sensitivity equations, to compute efficiently the response sensitivities (in terms of the adjoint functions) to all parameters, and to illustrate the use of sensitivities. Notably, the adjoint functions themselves can be interpreted as the sensitivity of a response to instantaneous perturbations of the model’s dependent variables. Furthermore the adjoint functions can be used to reveal the time scales associated with the most important physical processes in the model. In particular, the adjoint functions for the RCM reveal the three time scales associated with: (i) convective adjustment; (ii) heat transfer between the atmosphere and space; and (iii) heat transfer between the ground and atmosphere. Calculating the eigenvalues and eigenvectors of the matrix of derivatives occurring in the set of adjoint equations reveals similar physical information without actually needing to solve the adjoint sensitivity model. An illustrative use of the ASAP for evaluating the sensitivity to feedback mechanisms, is also presented. The paradigm response considered is the increase in the average surface air temperature which occurs after the atmospheric CO2 concentration in the model is doubled, while the paradigm feedback is the surface albedo feedback.

Chapter V continues with a paradigm application of the ASAP to more complex, operator-valued, responses, by considering a paradigm two-layer isentropic NWP model that simulates the nonlinear life cycles of baroclinic waves, including the occurrence of so-called “blocks.” The variability in predictive skills of NWP models is strongly related to the occurrence of such blocks; this occurrence is indicated by the so-called “blocking indices.” From a mathematical point of view, blocking indices are operator-valued responses. The ASAP is applied to perform a paradigm sensitivity analysis of a time-dependent blocking index to model parameters. This illustrative example underscores the fact that the exceptional computational efficiency of the ASAP yields quantitative results that could not have been obtained, in practice, by any other sensitivity analysis method, because of prohibitive computational costs.

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Chapter VI sets forth the presentation of paradigm applications of the ASAP to large-scale models used for numerical weather prediction, by considering the following models: (i) the diagnostic equations underlying the nonlinear radiation model used in a version of the National Center for Environmental Prediction model for medium-range weather forecasting; (ii) the Florida State University Global Spectral Model; and (iii) the Relaxed Arakawa Schubert scheme in the NASA Goddard Earth Observing System-1 (GEOS-1) general circulation model, developed by the then Data Assimilation Office (now Global Modeling and Assimilation Office) at the NASA Goddard Space Flight Center. A particularly important implication of the paradigm sensitivity analysis results presented in this chapter is that accurate data for temperature, moisture and surface pressure are essential for an accurate evaluation of cumulus cloud effects, especially at the most influential vertical levels that were identified by sensitivity analysis. Of course, this is because small perturbations at such influential locations tend to exert a stronger impact on the responses than similar perturbations at other, less influential, locations. Therefore, data quality is particularly important at those levels and areas with positive feedback between cloud activities and the environment, since small errors tend to grow through positive feedback mechanisms. Such sensitivity analysis results also underscore the importance of the ASAP for variational data assimilation. For example, in variational assimilation of precipitation data, in which moist convection is the dominant process, the difference between model output rainfall and the observed rainfall is taken as input to the adjoint sensitivity model. Such information also indicates the regions where additional adaptive observations should be taken.

To keep this volume to a reasonable size, several important topics (e.g., methods of data adjustment and data assimilation in the presence of uncertainties; optimal control of fluid flow) have been deferred for presentation in subsequent volume(s). Nevertheless, by addressing computational issues and highlighting the major challenges that still remain to be resolved, the material presented in this Volume is also intended to provide a comprehensive basis for further advancements and innovations in the field sensitivity and uncertainty analysis. Two outstanding issues, whose solution would greatly advance the state of overall knowledge, would be: (i) to develop the adjoint sensitivity analysis procedure (ASAP) for problems describing turbulent flows, and (ii) to combine the GASAP with global statistical uncertainty analysis methods, striving to perform, efficiently and accurately, global sensitivity and uncertainty analyses for large-scale systems.

In closing, the authors would like to acknowledge the essential contributions made the editorial staff of Chapman & Hall / CRC. We are particularly grateful to Ms. Helena Redshaw for keeping the publication schedule on track with her friendly e-mails. Last but not least, our special thanks go to Bob Stern, Executive Editor, whose patience and unwavering support made it ultimately possible to bring this book to our readers. One of the authors (I. M. Navon) would also like to acknowledge support of NSF and NASA grants for his research.

Copyright © 2005 Taylor & Francis Group, LLC

TABLE OF CONTENTS

I.

A COMPARATIVE REVIEW OF SENSITIVITY

AND UNCERTAINTY ANALYSIS METHODS

FOR LARGE-SCALE SYSTEMS

1

A. STATISTICAL METHODS

5

1. Introduction

5

2. Sampling-Based Methods

6

3. Reliability Algorithms: FORM and SORM

16

4. Variance-Based Methods

16

5. Design of Experiments and Screening Design Methods

21

B. DETERMINISTIC METHODS

26

1. Deterministic Methods for Local Sensitivity Analysis

26

2. Deterministic Methods for Local Uncertainty Analysis

29

3. Deterministic Methods for Extending the Use of

Local Sensitivities; Global Deterministic

Sensitivity Analysis

31

C. COMPUTATIONAL CONSIDERATIONS

32

II.

APPLICATIONS OF THE ADJOINT SENSITIVITY

ANALYSIS PROCEDURE (ASAP) TO TWO-PHASE

FLOW SYSTEMS

37

A. ASAP FOR GENERIC TWO-PHASE FLOW PROBLEMS

38

1. Basic One-Dimensional Two-Phase Flow Equations

39

2. The Adjoint Sensitivity Analysis Procedure (ASAP)

40

3. Characteristic of the Adjoint Sensitivity System

46

4. Illustrative Example: The Homogeneous Equilibrium

Model (HEM) for Two-Phase Flow

47

B. ASAP FOR THE RELAP5/MOD3.2 TWO-FLUID

MODEL (REL/TF)

55

1. The RELAP5/MOD3.2 Two-Fluid Model

57

2. ASAP of the Two-Fluid Model in RELAP5/MOD3.2

65

3. Consistency between the Differential/Integral

And the Discretized Equations/Models for

Sensitivity Analysis

80

4. Validation of the Adjoint Sensitivity Model

(ASM-REL/TF)

82

5. Sensitivities of Thermodynamic Properties of Water

91

Copyright © 2005 Taylor & Francis Group, LLC

III. FORWARD AND ADJOINT SENSITIVITY

ANALYSIS PROCEDURES FOR AUGMENTED

SYSTEMS

123

A. THEORETICAL BASIS FOR THE ASAP FOR

AUGMENTED SYSTEMS

124

1. Sensitivity Analysis of the Primary

(Nonaugmented) System

124

2. Sensitivity Analysis of the Augmented System

131

3. Discussion: Constructing the Augmented Adjoint

Sensitivity Model from the Original Adjoint

Sensitivity Model and Viceversa

143

B. ILLUSTRATIVE EXAMPLE: ASAP FOR THE COUPLED

TWO-FLUID WITH HEAT STRUCTURES MODEL

IN RELAP5/MOD3.2 (REL/TF+HS)

148

1. ASM-REL/TFH: The Augmented Two-Fluid/Heat

Structure Adjoint Sensitivity Model

148

2. Summary Description of the QUENCH-04 Experiment

158

IV. FORWARD AND ADJOINT SENSITIVITY

ANALYSIS PROCEDURES FOR RESPONSES

DEFINED AT CRITICAL POINTS

171

A. FSAP AND ASAP FOR RESPONSES AT CRITICAL

POINTS: GENERAL THEORY

172

1. The Forward Sensitivity Analysis Procedure (FSAP)

174

2. The Adjoint Sensitivity Analysis Procedure (ASAP)

177

3. Discussion

181

4. Illustrative Example: A Simple Particle

Diffusion Problem

183

B. ILLUSTRATIVE EXAMPLE: ASAP FOR THE MAXIMUM

CLAD TEMPERATURE PREDICTED BY A REACTOR

SAFETY CODE

194

V.

USING THE ASAP TO GAIN NEW INSIGHTS

INTO PARADIGM ATMOSPHERIC

SCIENCES PROBLEMS

231

A. A PARADIGM RADIATIVE-CONVECTIVE MODEL (RCM)

OF THE ATMOSPHERE

232

B. APPLYING THE ASAP FOR EFFICIENT AND

EXHAUSTIVE SENSITIVITY ANALYSIS OF THE RCM

237

Copyright © 2005 Taylor & Francis Group, LLC

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