Volume I SENSITIVITY and UNCERTAINTY ANALYSIS THEORY Dan G. Cacuci CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C. © 2003 by Chapman & Hall/CRC 9408 disclaimer Page 1 Friday, April 11, 2003 9:28 AM 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 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. 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Visit the CRC Press Web site at www.crcpress.com © 2003 by Chapman & Hall/CRC No claim to original U.S. Government works International Standard Book Number 1-58488-115-1 Library of Congress Card Number 2003043992 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper INTRODUCTION In practice, scientists and engineers often face questions such as: how well does the model under consideration represent the underlying physical phenomena? What confidence can one have that the numerical results produced by the model are correct? How far can the calculated results be extrapolated? How can the predictability and/or extrapolation limits be extended and/or improved? Such questions are logical and easy to formulate, but quite difficult to answer quantitatively, in a mathematically and physically well-founded way. Answers to such questions are provided by sensitivity and uncertainty analyses. As computer-assisted modeling and analysis of physical processes have continued to grow and diversify, sensitivity and uncertainty analyses have become indispensable investigative scientific tools in their own rights. Since computers operate on mathematical models of physical reality, computed results must be compared to experimental measurements whenever possible. Such comparisons, though, invariably reveal discrepancies between computed and measured results. The sources of such discrepancies are the inevitable errors and uncertainties in the experimental measurements and in the respective mathematical models. In practice, the exact forms of mathematical models and/or exact values of data are not known, so their mathematical form must be estimated. The use of observations to estimate the underlying features of models forms the objective of statistics. This branch of mathematical sciences embodies both inductive and deductive reasoning, encompassing procedures for estimating parameters from incomplete knowledge and for refining prior knowledge by consistently incorporating additional information. Thus, assessing and, subsequently, reducing uncertainties in models and data requires the combined use of statistics together with the axiomatic, frequency, and Bayesian interpretations of probability. As is well known, a mathematical model comprises independent variables, dependent variables, and relationships (e.g., equations, look-up tables, etc.) between these quantities. Mathematical models also include parameters whose actual values are not known precisely, but may vary within some ranges that reflect our incomplete knowledge or uncertainty regarding them. Furthermore, the numerical methods needed to solve the various equations introduce themselves numerical errors. The effects of such errors and/or parameter variations must be quantified in order to assess the respective model’s range of validity. Moreover, the effects of uncertainties in the model’s parameters on the uncertainty in the calculated results must also be quantified. Generally speaking, the objective of sensitivity analysis is to quantify the effects of parameter variations on calculated results. Terms such as influence, importance, ranking by importance, and dominance are all related to sensitivity analysis. On the other hand, the objective of uncertainty analysis is to assess the effects of parameter uncertainties on the uncertainties in calculated results. Sensitivity and uncertainty analyses can be considered as formal methods for evaluating data and models because they are associated with the computation of specific © 2003 by Chapman & Hall/CRC quantitative measures that allow, in particular, assessment of variability in output variables and importance of input variables. For many investigators, a typical approach to model evaluation involves performing computations with nominal (or base-case) parameter values, then performing some computations with parameter combinations that are expected to produce extreme responses of the output, performing computations of output differences to input differences in order to obtain rough guesses of the derivatives of the output variables with respect to the input parameters, and producing scatter plots of the outputs versus inputs. While all of these steps are certainly useful for evaluating a model, they are far from being sufficient to provide the comprehensive understanding needed for a reliable and acceptable production use of the respective model. Such a comprehensive evaluation and review of models and data are provided by systematic (as opposed to haphazard) sensitivity and uncertainty analysis. Thus, the scientific goal of sensitivity and uncertainty analysis is not to confirm preconceived notions, such as about the relative importance of specific inputs, but to discover and quantify the most important features of the models under investigation. Historically, limited considerations of sensitivity analysis already appeared a century ago, in conjunction with studies of the influence of the coefficients of a differential equation on its solution. For a long time, however, those considerations remained merely of mathematical interest. The first systematic methodology for performing sensitivity analysis was formulated by Bode (1945) for linear electrical circuits. Subsequently, sensitivity analysis provided a fundamental motivation for the use of feedback, leading to the development of modern control theory, including optimization, synthesis, and adaptation. The introduction of state-space methods in control theory, which commenced in the late 1950’s, and the rapid development of digital computers have provided the proper conditions for establishing sensitivity theory as a branch of control theory and computer sciences. The number of publications dealing with sensitivity analysis applications in this field grew enormously (see, e.g., the books by: Kokotovic, 1972; Tomovic and Vucobratovic, 1972; Cruz, 1973; Frank, 1978; Fiacco, 1984; Deif, 1986; Eslami, 1994; Rosenwasser and Yusupov, 2000). In parallel, and mostly independently, ideas of sensitivity analysis have also permeated other fields of scientific and engineering activities; notable developments in this respect have occurred in the nuclear, atmospheric, geophysical, socio-economical, and biological sciences. As has been mentioned in the foregoing, sensitivity analysis can be performed either comprehensively or just partially, by considering selected parameters only. Depending on the user’s needs, the methods for sensitivity analysis differ in complexity; furthermore, each method comes with its own specific advantages and disadvantages. Thus, the simplest and most common procedure for assessing the effects of parameter variations on a model is to vary selected input parameters, rerun the code, and record the corresponding changes in the results, or responses, calculated by the code. The model parameters responsible for the largest relative changes in a response are then considered to be the most © 2003 by Chapman & Hall/CRC important for the respective response. For complex models, though, the large amount of computing time needed by such recalculations severely restricts the scope of this sensitivity analysis method; in practice, therefore, the modeler who uses this method can investigate only a few parameters that he judges a priori to be important. Another method of investigating response sensitivities to parameters is to consider simplified models obtained by developing fast-running approximations of complex processes. Although this method makes rerunning less expensive, the parameters must still be selected a priori, and, consequently, important sensitivities may be missed. Also, it is difficult to demonstrate that the respective sensitivities of the simplified and complex models are