Sensitivity Analysis in Chemical Kinetics

Ann. Rev. Phys. Chern. 1983. 34: 4I9-{)I Copyright © 1983 by Annual Reviews Inc. All rights reserved

SENSITIVITY ANALYSIS IN CHEMICAL KINETICS H. Rabitz, M. Kramer, 1 and D. Dacol

Department of Chemistry, Princeton University, Princeton, New Jersey 08544

INTRODUCTION

Complex mathematical models are increasingly being used as predictive tools and as aids for understanding the processes underlying observed chemical phenomena. The parameters appearing in these models, which may include rate constants, activation energies, thermodynamic constants, transport coefficients, initial conditions, and operating conditions, are seldom known to high precision. Thus, the predictions or conclusions of modeling endeavors are usually subject to uncertainty. Furthermore, regardless of uncertainty questions, there is always the overriding matter of which parameters control laboratory observations. Quantification of the role of the parameters in the model predictions is the traditional realm of sensitivity analysis. A significant amount of current research is directed at conceptualization and implementation of numerical techniques for determining parametric sensitivities for algebraic, differential, and partial differential equation models including those with stochastic character and nonconstant parameters. Recent studies have also served to extend the range of the conventional parametric analysis to address new questions, relevant to the process of model building and interpretation. This review attempts to present a full range of results and applications of sensitivity analysis, relevant to chemical kinetic modeling. We exclude a large class of literature that deals with system sensitivities from a control theory perspective (1). We further limit discussion of related subjects such as parameter identification (estimation of best parameter values for fitting data) and optimization, in which parametric sensitivities play only

1 Current address: Department of Chemical Engineering, Massachusetts Institute of

Technology, Cambridge, Massachusetts 02139.

419 0066-426Xj83jl101--0419$02.00

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420 RABITZ, KRAMER & DACOL

subsidiary roles. Most of the references cited herein are drawn from the chemical literature, but significant parallel literatures have developed in other scientific and engineering disciplines. We are aware of active research in biology and ecology (2), nuclear engineering (3), electrical engineering (4), and economics (5).

There have been, in recent years, several reviews of kinetic sensitivity analysis. Cukier, Levine & Shuler (6) discuss the Fourier (FAST) method for nonlinear sensitivity analysis. Tilden et al (7) present a more comprehensive review, covering 'Some local techniques [finite differences, direct differential methods, and adjoint (Green's function) methods] and global methods, including stochastic sensitivity analysis and the FAST method. Applications of sensitivity theory to molecular dynamics and kinetics have been summarized (8), utilizing nontraditional derived sensitivities, and trajectory feature analysis. Eno & Rabitz (9) have discussed sensitivity analysis as it applies to molecular dynamics.

The present review begins by discussing the sensitivity methodologies available for analysis of ordinary differential equation models. These methodologies can, in general, be divided into two categories based on the nature of the sensitivity measures they produce. Local methods, which produce sensitivity coefficients representing parametric gradients at single points in the parametric domain, are discussed first. Global methods, which produce measures of parametric importance over regions of parameter space, are then reviewed. This is followed by an analysis of stochastic systems (i.e. those containing fluctuations). The next section discusses sensitivity analysis of partial differential equation (space-time) models. We then discuss derived sensitivity analysis, which is intended to address certain questions that arise when depende:p.t-independent variable roles of the predictions and parameters are exchanged. This situation naturally occurs when certain experiments (possibly conceptual in nature) have been performed, and the response of the model parameters to this data is in question. The next section considers several aspects of sensitivity analysis relevant to the treatment of laboratory data. We then present a methodology for the decomposition of the elementary sensitivities into their basic structural components. This methodology is also useful for the construction of parametric scaling laws used to predict model solutions for values of the parameters away from the nominal parameter values. This is followed by consideration of cases in which the parameters are allowed to be functions of time and/or space, producing a sensitivity formalism based on functional derivatives. It is hoped that this review will provide a comprehensive update and a solid introduction for both theoretical modelers and experimentalists who use modeling for analysis and diagnosis of kinetic systems.

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SENSITIVITY ANALYSIS 421

TEMPORAL SYSTEMS: ELEMENTARY

SENSITIVITIES

In this section, we review numerical methods for determination of the parametric sensitivities associated with the purely temporal system

de dt

= f(c, IX, t), e(O) = co, 1.

where c is the n-vector of species concentrations or similar dependent variables, at is the m-vector of system parameters (which can include the initial conditions CO), and t (time) is the variable of integration. In this section, IX is considered constant, although this restriction is relaxed later. The sensitivity coefficients of Eq. 1 have been defined in a number of different ways and the associated methods can be grouped accordingly. Local methods calculate the partial derivatives oe(t)/olX, oe2(t)/oIX2, etc, about the nominal parameter value solution of Eq. 1. These sensitivities can be viewed as the gradients around the nominal solution in a multidimensional

parameter space. Global methods take into account, a priori, the range and/or statistical properties of the parameter uncertainty, and calculate certain averaged sensitivities, e.g. <8c(t )/8at), over the region of parameter uncertainty. These coefficients quantify the effect of simultaneous, large variation in IX, and thus are qualitatively different than the local sensitivities. Special methods are required when Eq. 1 is significantly influenced by stochastic variations. The appropriate sensitivities would in this case have the form 8<c)/81X, where <c) is the average over inherent concentration fluctuations. Such sensitivities may be required when the system is near a stability boundary, or for reactions with a very small number of participants, e.g., reactions in micelles. The choice of sensitivity methodology must be dictated by the characteristics of the particular system and parameters under consideration. As a general rule, local methods require less extensive calculation, and provide a higher level of detail, whereas global methods may be best for handling large variations in the system parameters.

Local Methods

FINITE DIFFERENCES The simplest conceptual route to calculating the local sensitivities 8kc(t)/OlXk is via finite difference approximations. Using this approach, Eq. 1 is solved using various sets of parameter values, and the sensitivity coefficients are calculated from an appropriate formula. For example the first order sensitivity to the jth parameter rt.j is

8c(t) � c(t,rt.j +L\rt.) -c(t,oc)

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with all other parameters held fixed. Using this strategy, m solutions ofEq. 1 are required to determine all first-order coefficients, m(m + 1)12 solutions for all second-order coefficients, etc, provided that the same l1!Xj suffices to estimate accurately OkC/O!X} for all i. It has been noted (10) that two conditions should be met in selecting l1!Xj, for the first order calculation. First,

( Oei(t)) l1!Xj � !cCi(t)/ fa o!Xj

,

where!c is the fractional error in e(t), and fa is the allowable fractional error in the sensitivity. This condition limits the round-off error in oejo!Xj. Second, l1!Xj should be as small as possible, so that the influence ofthe higher order corrections to the finite difference approximations are minimized. Therefore, l1!Xj should approach the value indicated by the equality in the above expression. To the extent that 0 In Ci(t)lO!Xj varies from species to species, and from time to time, a number of distinct l1!Xj may be required. Thus the work estimates of m and m(m + 1)/2 times the effort of. it single simulation may underestimate the true requirement. The advantage of this approach is that no extra numerical machinery is required, above that needed to solve Eq. 1 .

CURVE FITTING AND DERIVATIVE EXTRACTION This method involves reduction of global information (the solution c at several poirtts in parameter space), to local information (the sensitivity coefficients at a particular parametric point). The basic strategy is to fit an interpolating function to known points in the parametric domain, and then differentiate this function to determine the sensitivities at a given point. Frenklach (1 1 ) points out that sensitivities obtained by this method are adequate averages for the parametric region of interest, but caution is called for in always assuming that these values are the true sensitivities.

DIRECT DIFFERENTIAL METHODS This class of methods involves solution of the differential equations that govern the sensitivity coefficients. These equations are derived by differentiation of Eq. 1 with respect to the parameter of interest �j,

� � = J(t)� + of(t)

dt o!Xj o!Xj o!Xj' 2.

where J(t) = (of/oc)". The initial condition for Eq. 2 is 0, if!Xj is not an initial _ condition, and the vector (0, . . . , 0, 1, 0, ... , O)T, where 1 is in the ith position, if !Xj is the initial condition for species i. Equation 2 can be solved by coupling with Eq. 1 and solving this set of 2n equations, or stQring the solution to

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Eq. 1 in advance, and interpolating this data to provide J(t) and of(t)jortj as required in' the solution of Eq. 2. The latter approach has" been questioned in terms of its numerical stability (12); however, we h:lVe found that with careful interpolation, the rpethod performs well. Using the de�oupled methodology, the effort iIi solving Eq. 2 is approximately the same as that expended in solving the kinetic equations, because the two equations have the same Jacobian, and hence, the same stiffness properties. The advantage of the decoupled direct method as opposed, to finite differences is that the difficulty of sizing the variations I1rtj is eliminated. Thus, the total computational effort may be less than the finite difference approach.

Higher order sensitivities can also be calculated by the direct approach. For example, the governing equation for second order sensitivities, by differentiation of Eq. 2, is (13) :

3.

where L = oJ(t)/oc(t). This equation can, in theory, be solved by a ctecoupled method, with approximately the same computational effort as the first order sensitivity equation. A coupled formulation for r1.j "# r1.k would involve 4n equations, and would be much more expensive to solve.

THE GREEN'S FUNCTION METHOD (GFM) Direct solution of Eqs. 2 and/or 3 is not the optimum procedure, under many circumstances. Several researchers (e.g. 5, 13a,b, 14) have noted that because Eq. 2 is a linear, inhomogeneous equation with time-varying coefficients, it can be solved by first calculating the solution ofthe homogeneous part and then determining the particular solutions corresponding to each parameter. The homogeneous problem can be written

d dt

K(t, t') = J(t)K(t, t'), 4.

where K(t', t') = I and t > t'. K is an n x n matrix containing as its columns the n linearly-independent solutions admitted by Eq. 4, called the kernel or Green's function matrix. The full solution of Eq. 2 can be written

ve(t) = K(

') oe(t') it

K( ) of(s)

d ;) t, t ;) + t, s ;) s. uct.j ur1.j t' UlXj 5.

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Note that in the integral the first argument of K is fixed, whereas the second is varying. This is the opposite situation to that in Eq. 4; thus, to evaluate the integral efficiently, an adjoint strategy is commonly employed. The adjoint Green's function Kt is defined by

6.

where Kt(t, t) = I and t' � t. To use Eq. 5, the identity Kt(t', t) = K(t, t') is applied, and propagation of K with time relies on the group property, K(t3' t1) = K(t3,t2)K(t2' t1)'

Edelson et al (15) examined two alternatives for solution of Eq. 6. One method handles each row of Kt independently, thus breaking Eq. 6 into a set ofn, n-vector ordinary differential equations (ODEs) following earlier work (13a,b, 16). Because each vector problem has the same Jacobian, there is a certain amount of redundancy to this method, and Edelsop anticipated improvements by handling the problem in matrix form. The expected efficiencies were not realized, however, because over twice the number of time steps had to be taken in the matrix calculation. The following remarks assume that the algorithm of Hwang et al (13a,b) is used.

The advantage of the GFM versus the direct differential method is as follows. Equation 6 requires solution of n2 differential equations; additionally, Eq. 5 requires evaluation of n integrals for each parameter. These integrals may usually be handled by quadrature [although this is sometimes inadequate (17)J, which is computationally less expensive than solving differential equations, particularly compared to the solution of stiff differential equations. The work of the GFM is therefore somewhat greater than n single solutions of Eq. 1. The work required by the direct method is about m times that of a single kinetic solution. Therefore, the GFM is preferable over the direct method if the number of parameters is greater than the number of species, i.e., m > n. This is the usual case in chemical kinetics problems. The GFM is particularly suited if the sensitivities of one concentration (or one linear superposition of concentrations) to several parameters are to be determined; in this case, only one row of Kt needs to be calculated, and the total effort is on the order of one kinetjc solution. The direct method, by contrast, would be best suited if the effect of one parameter on all species is to be determined.

The GFM can, in theory, be used for higher order sensitivities, because the second order direct method equation has essentially the same form as Eq. 2; only the inhomogeneity differs. The two equation�, therefore, have the same kernel, and the second order sensitivities can be written ( 13a,b)

02C(t) 02C(t) it -

0 0 = K(t, t')-

o- + K(t, s)Xjk(s) ds,

IX j IXk OIX j IXk t' 7.

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where Xjk is the inhomogeneity of Eq. 3. Equation 7 represents a highly significant improvement over Eq. 3. The effort in evaluation of Eq. 7 lies almost solely in the quadrature evaluation of the integral, a small expense, whereas Eq. 3 requires an effort equivalent to solution of Eq. 1 for each parameter set jk. In practice, the difficulty of obtaining a sufficient grid of points for evaluation of the integral in Eq. 7 makes calculation of higher order coefficients problematic. Recent algorithmic improvements ( 18a,b) have been suggested to facilitate this calculation.

THE AIM METHOD The analytically integrated Magnus (AIM) modification of the Green's function method (17) improves upon the original GFM in two areas. First, a specialized technique, the piecewise Magnus method (PMM), is used for solution of the kernel Eq. 4, which handles the problem as a matrix, rather than a succession of n, n-vector ODEs. The PMM approximates the solution to Eq. 4 by a matrix exponential:

C '+ll.t K(t+At,t) � exp Jr J(s) ds

The exponential is handled by decomposing

1 Cr+ll.t At Jr J(s) ds

8.

into the diagonal matrix D of the eigenvalues, the matrix Z containing the corresponding eigenvectors, and Z-1, and then applying (19)

K (t+At, t)';' exp(ZDAtZ-1) = Z (exp DAt)Z-l. 9.

The total effort in the PMM solution of Eq. 4 is proportional to n3• By contrast, the effort expended in solution of Eq. 4 (or the adjoint Eq. 6), by column (row) application of vector-ODE techniques such as Runge-Kutta or Gear, is proportional to »4. The additional factor of n significantly affects the relative efficiencies of the two approaches, favoring AIM for large problems.

The second innovation in AIM is in evaluation of the integral in Eq. 5. The PMM allows stepsizes At far too large for simple quadrature to be effective, although this can be useful if only the Green's function is desired. Equation 9, however, can be integrated analytically to give a closed-form expression that automatically accounts for the curvature in K, i.e.

cr+l1r Jr K(t+�t,s) ds � ZD-1 (exp D6t - I)Z-1 .

A similar form may be calculated for any polynomial approximation to of/o!J.j' and thus analytical approximations to the sensitivity integral are produced. Because the Green's function form is used, AIM also has the

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efficiency benefits for higher order coefficients discussed in the previous section.

A computer code for the AIM algorithm is in distribution (20). A reaction interpreter (2 1), which accepts chemical kinetics problems in almost formatfree form, has been designed to accompany the code. The interpreter sets up the pertinent equations automatically, and calls AIM for the calculation of the linear sensitivities.

THE STEADY-STATE: ALGEBRAIC SYSTEMS In this section we consider the problem of the sensitivity of the algebraic system f(c,�) = 0, which is the steady state of the ODE system Eq. 1 . We do not, however, assume that the dynamic relationship is known. For instance, the equations may be a thermodynamic description of chemical equilibria, containing no rate data whatsoever. The algebraic equations may be differentiated with respect to aj to give of/oaj+Joc/oaj = 0, which implies

� = _J-1�. 10. oa j oa j

Irwin & O'Brien (22) note that if Newton iteration is used to solve the original equations, J -1, or the LV decomposition of J, is available, and Eq. 1 0 can be solved at the expense of a single matrix multiplication.

A slightly more complicated calculation may be undertaken if the algebraic system is the t ->- 00 limit of a dynamic system. In this case, the sensitivity coefficients are dynamic in nature, governed by Eq. 2. The Jacobian and of/arxj are time invariant in this context, so the sensitivities can be written (M. A. Kramer, unpublished)

ac [ of OC ] of - (t) == J -1-;- + a(O) exp(Jt)-J -la. O(1,j Vaj rxj aj 1 1 .

The time profiles of the sensitivity coefficients give the dynamic response of the system to a differential change in (Xj around the steady state.

COMPARISON OF LOCAL METHODS In this section, we quote results of a detailed computational study (10), which used a n = 1 1 species, m = 52 parameter model of CO oxidation as a basis for comparing local sensitivity methods. In terms of total computation time, the AIM method was fastest, by factors of 2,2.6, and 5.5 over the GFM, direct method (decoupled), and finite difference (FD) approaches, respectively. The efficiency of the GFM was markedly reduced by increasing the number of output times, to the extent that an increase from one to ten output times increased computational effort in the kernel calculation by 250%. Second-order coefficients via the decoupled direct method were estimated to be five times as expensive as those calculated by AIM.

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POSSIBLE FUTURE IMPROVEMENTS Several authors have proposed to solve Eq. 2 by combining the numerical schemes used to solve the model and the sensitivities. These approaches depend on the fact that the local temporal linearization of Eq. 2 is the same as that for Eq. 1 (i.e. the two systems have the same Jacobian). Once J is calculated and inverted, this quantity can be applied to the sensitivity equation, and by simply introducing a new inhomogeneity the sensitivities may be propagated. Vemuri & Raefsky (23), Irwin & O'Brien (22), and Lojek (24) have developed various aspects of this approach. A question with this method concerns the fact that most stiff ODE solvers (25) will use the same Jacobian for several time steps. Although this is adequate in predictor-corrector algorithms for solving the kinetic equations, an accurate Jacobian is needed for the sensitivity equations since it enters as a coefficient matrix in the equations. Research remains to show if this is a serious practical matter calling for an updated Jacobian at each time step. It is also worth noting that this method without this complication can be implemented for steady state problems (see the discussion below).

A method of direct determination of the sensitivities of objective functions is given by Segneur et al (30) and Cacuci (3). These approaches are variants of the Green's function method. It should be noted that the sensitivity of any differentiable function or functional of c can be calculated from the elementary sensitivities. Another promising method has recently been proposed by J.-T. Hwang (private communication, 1982). By this approach, the sensitivity equation is solved by temporal collocation and fitting by Lagrange polynomials. It is clear that the high degree of interest in sensitivity analysis will continue to motivate algorithmic studies, and further improvements may be in the offing.

SAMPLE APPLICATIONS AND DISCUSSION The following are common applications of local sensitivity information (10) :

1. identification of governing/extraneous parameters and processes (8) ; 2. identification of missing model components (8, 26); 3. quantification of extent and sources of error (27) ; 4. mapping of parameter space (8, 28); 5. fitting of a model t6 data (29) ; 6. identification of steepest descent/ascent direction for optimization

routines (29); 7. sensitivity of objective functions (3, 30). Most of these applications are conceptually simple. The magnitude of the log-normalized sensitivity Ol'f/c/iJ['f/1% indicates the relative importance of the parameters to the concentrations. Thus the governing processes and those

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unimportant to the result can be identified, and information can be used for model reduction, by eliminating those reactions whose rate constants have little or no effect on c, at all times (31-33). One can assess possible model expansion by including the candidate processes in the sensitivity analysis while setting the parameters associated with the expansion to zero. The nominal solution to the model is undisturbed, but the sensitivities can still be determined. If the linear sensitivities have large magnitudes, the candidate process is probably an important addition to the model. Edelson (32) used the sensitivities to pinpoint the rate constants in an alkane pyrolysis model that contributed most to discrepancies between model and experiment. This information is very useful in experimental planning, and for assessing the significance of a modeling result. Sensitivity information can also contribute directly to parameter identification-finding the parameter values that best fit given data-or to optimization of systems with controllable parameters. Both of these problems are pervasive in science and engineering, and have extensive literatures of their own. It should be noted that algorithmic improvements in sensitivity methods have a direct impact on these very important fields.

Global Methods Given an uncertainty in the parameter vector IX, the resulting uncertainty in the concentration vector c(t) can be obtained using local sensitivity analysis only if the variations of IX about the reference value are small enough so that a low order Taylor expansion of c(t) is valid (2,27). For large variations one has to use global methods. The simplest of these consists of repeated solution of the model equations using various values for the parameters, to construct the solution surface in parameter space, that is c(t, IX) as a function of IX (parametrized by t). The choice of the parameters to be varied in this procedure can be made in a systematic way by using factorial design techniques and response surface methods (33). However, as the number of species and parameters increases this procedure becomes cumbersome and time-consuming. In such cases an approach based on a combination of both global and local sensitivity analysis might be more effective. Two methods for global sensitivity analysis are discussed here: stochastic sensitivity analysis and the FAST method. The related topic of scaling local sensitivity information is taken up later in the review.

STOCHASTIC SENSITIVITY ANALYSIS If one assumes that the spread in values of the parameters due to uncertainties about their "true" values can be represented by means of a probability distribution, then the sensitivity analysis problem consists of determining the probability distribution for the values of the concentrations induced by the probability distribution for

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the parameters. Seinfeld & co-workers (34a) have addressed this problem by studying the Liouville-like equation satisfied by the probability distribution for the concentrations. A small change in the form of the equations for the concentrations allows for a unified treatment of uncertainties in the initial conditions and in the parameters (like rate constants) that enter in the kinetic equations themselves. Consider the chemically reacting system described by Eq. 1, which can be rewritten as

!: = F(x, t), x(O) = xc, 12.

where x is the n + m vector whose components are the n species concentrations and the m parameters entering in f(c, IX, t) in Eq. 1 (i.e. Xi = Ci for i = 1, . . . , n and Xi+n = (Xi for i = 1, . . . , m). Also Fi(X, t) = /;(c, IX, t) for i = l,oo . ,n and Fi+n = 0 for i = 1,oo.,m. Thus dxi+Jdt = 0 for i = 1,oo . ,m,

which is a statement of the fact that the parameters are time independent and therefore Xi+n(t) = x?+m i = 1 , ... , m. The hypothesis that the uncertainties regarding the optimal values of the parameters (including initial conditions) can be represented by probability distributions implies that the initial conditions XO in Eq. 12 are random variables with a probability distribution P o(XO). Let P(x, t) be the probability distribution of x(t) [notice that x(t) is also a random variable since it is a function of the random variable x°]. This probability distribution is to be used to compute averages of functions of x(t):

<g(x(t))) = f dx P(x, t)g(x). 13.

It can be shown (34a,b) that P(x, t) satisfies the following partial differential equation

oP(x, t) -a-

t

- = - V x • (F(x, t)P(x, t)), P{x,O) = P o{x). 14.

The boundary conditions are obtained by demanding that J dx P(x, t) = 1. Once P(x, t) is calculated, sensitivity information can be obtained by

studying the surface defined by P(x, t) (t fixed) in x space. This can be a difficult task for (n+m) � 1, and therefore it is useful to study reduced probability distributions obtained by integrating P(x, t) over a subset of the components of x.

THE FAST METHOD The Fourier Amplitude Sensitivity Test (FAST) method is closely related to stochastic sensitivity analysis. The basic idea was introduced by Bortner & Hochstim (35) and 'its present form is due to Shuler, Cukier & co-workers (6, 36, 37). In the FAST method the

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uncertainty in the parameters is also represented by assuming that the values of the parameters are distributed according to a given probability distribution function. Each parameter is assumed to be statistically independent from all others. The parameters are written in the form

rxp = rx�O) exp(llp)

where the independent random variables IIp' p = 1, ... , m, have a probability distribution

m

P(Il) = n P iflp)· p;l

The concentrations are functions of the random variables flp and the average concentration can be expressed as an integral weighted by the probability distribution P(Il). By writing flp = Gp (sin(wps)), where the wp are a set of incommensurate frequencies, it is possible to reduce the averaging over the ensemble of fl'S to an average over the s variable. The incommensurability of the frequencies wp assures that curve flp =

G p (sin(wps)) is space-filling; that is, it passes arbitrarily close to any preselected point in parameter space for which the joint distribution

m

does not vanish. The transformation functions G p must be chosen so that the fraction of the total arc length of the curve that lies between the values flp and flp + dflp is equal to P p(flp) dflp. It is found that G ix) is uniquely determined by P p through the following equation

dG n(1-x2)PiGix))

dxP = 1, GiO) = O.

In practice, questions of computational feasibility prevent the use of incommensurate frequencies. Schuler and co-workers used widely disparate integer frequencies that are chosen to mimic as closely as possible a space-filling curve. The advantage in using integer frequencies is that the concentrations are then periodic functions of s with period 2n and thus the s-average is reduced to an integral over s with 0:::;; s < 2n. However, in this case the curve flp = G p (sin(wps)) cannot fill the fl-space, as it is by necessity a closed curve. The curve flp = G P (sin(wps)) is called a search curve and s is called a search parameter. This procedure can be used to construct the response surface for the averaged concentrations in the space formed by the rxl�), p = 1, ... , m. The FAST method thus provides a pattern search procedure that requires a smaller number of sampling points than the more

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conventional Monte Carlo methods (7). Schuler and co-workers use the fact that now the concentrations are periodic functions of s to define the following Fourier coefficients

1 f2" (i) _ Anw (t) - - ds dt, p(s)) cos(nwps), p n 0

1 f2" (i) _ . Bnw (t) - - ds Ci(t, p,(S)) sm(nwps), p n 0

n = 0,1, ... 15.

n = 1,2, . . . 16.

These Fourier coefficients are the major sensitivity measures in the FAST method. Thus, for example, if A�i1p and B�i1p are zero for all p = 1, 2, . . . , then the concentration Ci is insensitive to the values of the pth parameter at the nth harmonic wp' The interpretation of the Fourier coefficients as sensitivity measures rests on the assumption that the Fourier coefficients of a given fundamental and all its harmonics separate the effects of each parameter uncertainty on the concentrations. For integer frequencies this is only approximately true. However, the errors made by using integer frequencies can be quantitatively predicted and controlled. The labor of performing such calculations can be considerable due to the need for repeated solutions of the kinetic equations at a large number of points in order to evaluate the oscillating integrals in Eqs. 15 and 16.

Stochastic Systems

The concentrations of participating species in a chemically reacting system are fluctuating quantities due to the aleatory nature of the intermolecular processes. The magnitude of the concentration fluctuations usually is ofthe order of the inverse of the volume of the system. Thus, for macroscopic systems, the fluctuations are often negligible and the deterministic kinetic equations provide an accurate description of the behavior of the concentrations. However, there are situations in which, even for macroscopic systems, fluctuations are important. A typical case occurs when chemical instabilities develop in the system. In this case large-scale fluctuations, spanning macroscopic volumes comparable to the volume of the system, can be amplified and cause a transition to a state distinct from the initial one (38). If the system possesses multiple steady states and is initially very close to an unstable steady state then the transition to one of the stable steady states can be driven by the fluctuations (39). Another case of considerable practical importance is in the study of kinetic processes in micellar systems where the reacting molecules are characterized by lengths that are not orders of magnitude smaller than a characteristic dimension of thc volume in which the chemical reactions take place (40). It is important to

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distinguish these stochastic phenomena and their use of averaging from the parameter distribution averaging methods of the last section.

A convenient framework for the description of fluctuations in chemical kinetics involves the use of stochastic differential equations (39). These equations are similar in structure to the deterministic ones, except for the addition of noise terms. The noise terms attempt to describe the changes in the concentrations that happen on a much faster time scale than the relaxation times for the chemical processes under study. The strength of the noise depends on the instantaneous values of the concentrations, because the probability of a given reaction occurring in the system depends on the concentrations of the species involved. The quantities of interest are not the solutions of the equations themselves, but the average values, variances, and correlations of the concentrations. The formalism presented here is not restricted to kinetic problems, but provides a framework for sensitivity studies of any system whose state variables obey stochastic differential equations with either additive or multiplicative white noise.

The kinetic and sensitivity analysis problem in stochastic chemical kinetics can be addressed as follows. Suppose that for a certain set of reference values of the parameters all the expectation values of products of the concentrations can be determined. These expectation values are correlation functions that can be obtained by solving the stochastic differential equations of chemical kinetics and averaging appropriate products of solutions. For a well-stirred mixture of n species the equations for the concentrations ci(t) are

ddCi = He, 17.) + ± piie, or:)�it), c;(O) = c�, t j= 1

1 7.

where ei(t) is a delta correlated Gaussian stochastic process (41) (i.e. white noise), with

18.

The matrix p(e, 17.) can be found from the fluctuation-dissipation analysis of Keizer (42) and Grossmann (43),

n L Pik(C,I7.)Pjk(C,I7.) = %{c,I7.), 19.

k=l

where q is a symmetric, positive semidefinite matrix that can be written in terms of the forward and backward rates for the elementary reactions included in the rate vector fin Eq. 17. Assuming that the concentrations are expressed in (number ofmolecules)j(unit volume) and that the forward and backward rates nk and nb for the kth elementary reaction are measured in

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(molecules/unit volume)/(unit time), the rate vector f and the variance matrix q can be written as (42)

M fi(c,oc) = L vik[nk(c,oc)-nk(c,oc)], 20. k=1 21 .

where M i s the number of elementary reactions, Vik i s the stoichiometric coefficient of species i in the reaction k, and n is the volume containing the chemically reacting system under study.

From a mathematical point of view the stochastic process ,(t) induces a new stochastic process c(t) through the mediation of Eq. 17. The statistical properties of ,(t) plus the dynamical properties incorporated in Eq. 17 (including initial conditions) uniquely determine the statistical properties of c(t), which allows for the computation of average values, variances, and correlations of the concentrations. A rigorous link has been established between this model and another popular phenomenological description of fluctuations, the birth and death formalism (44). Essentially, it can be shown that Eq. 17 constitutes a diffusion approximation to a corresponding master equation in the birth and death formalism (45). The larger the system is, the better is the approximation.

In stochastic chemical kinetics the objects of interest are of the form <F[c]), where F[c] is, in general, a functional of the concentration vector and the brackets indicate an ensemble average. Particularly important quantities are the average value and the variance of ci(t),

22.

The sensitivity coefficient of <F[c]) with respect to a parameter IXp is defined as

F a Sp = ;:,<F[c]). 23. utl.p

It can be shown (46) that

/oF[c] I ) {f"" n } s� = \ ----a;:; c + .��+ 0 ds J1 <F[c]�js+e)bjp(c(s),�(s),oc» , 24.

where

� -1 [ ofi(s) � 0Pik(S) ] bjp(c(s), ,(s), oc) = i�1 Pji (s) Aip£5(s) + OlXp + k�1 ----a;:;�k(S) . 25.

In Eq. 25 Aip = 1 if tl.p = cf and AiP = 0 otherwise. The first term on the right-

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hand side of Eq. 24 is zero when F[c] is not explicitly dependent on Of. For the special case F[c] = Ci(t), Eq. 24 reduces to

S� = s��+ {-S ds (Ct(t) it eis+e)bJic(s), �(s), at)' 26.

The evaluation of the sensitivity coefficient S� does not require the solution of any other differential equation besides Eq. 17. This is to be contrasted with the deterministic case in which the evaluation of the sensitivity coefficients requires not only the solution of the differential equations for the concentrations but also the solution of the differential equations obeyed by the sensitivity coefficients. A particular realization of the stochastic variable �(t) is denoted �a(t), where a is an index labeling the sample trajectory in question. The sample trajectories �a(t) can be obtained by means of a random number generator. For each sample trajectory �a(t) numerical integration of Eq. 1 7 yields a sample trajectory ea(t), which is a particular realization of the stochastic variable e(t). Thus, if G[e,�] is a functional of the stochastic variables e and �, then

1 N <G[e,�]) = lim - L G[ca,�'J. N-OCi N a=l

27.

Therefore, after generating the �'(t) and obtaining the c'(t) from Eq. 1 7, <F[c]) and S� can be obtained by means of Eq. 27. A quasilinear approximation to the above theory can be obtained by

expanding in powers of the inverse system volume and keeping only the lowest order terms. In this approach one can calculate <cJt) and <C;(t)cit') as well as their sensitivities without performing any stochastic trajectories simultation (46). In the quasilinear approximation it is possible to obtain this information in terms of the deterministic concentrations c and of the deterministic Green's function K.

SPACE-TIME SYSTEMS: ELEMENTARY

SENSITIVITIES

Basic Equations

There are a variety of fundamentally and practically important chemically reacting systems in which the kinetics are intrinsically coupled to transport or mixing process. Examples can be found in atmospheric chemistry, combustion processes, biological systems, industrial design, etc. As an illustration of these problems we confine ourselves to diffusion in a chemically reacting system. Thus, in a one-dimensional chemically reacting and diffusing system, the concentrations must satisfy the following kinetic

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equations as an extension of Eq. 1,

ac. a2

at' = Di ax2 Ci + h(e, IX).


28.

In more general circumstances the diffusion constant Di for the ith species may also be a function of space, time, and species concentration (47). Assuming the range of the problem is Xl � X � X2 the concentrations are taken to fulfil the following linear initial and boundary conditions.

e(x, 0) = eO(x),

The elementary sensitivity coefficients are defined as

a Siix, t) = :;--c,(x, t). v(Xp

29.

30.

31.

In the above equation !Xp indicates any parameter appearing in Eqs. 28-30. Differentiation of Eqs. 28-30 leads to the following equation for Sip(X, t):

a a2 ali -a SiP(X, t) = Di-a 2 Siix, t) + L -a Sjp(X, t) + 9iix, t), t X j Cj

S. ( 0) = aci(x) ,px, a ' IXp

32.

33.

34.

Thus the sensitivity coefficient Siix, t) satisfies a linear, inhomogeneous, partial differential equation with initial condition given by Eq. 33 and boundary conditions defined by Eq. 34. The right-hand side ofEq. 34 can be written as

The inhomogeneous term in Eq. 32 is

aD. a2c. a giix, t) = -a ' c'\ ; + -;-f.{e, IX). IXp vX VIXp

35.

36.

The linearity of the equation for the sensitivity coefficient allows for writing its solution in terms of a Green's function (48). The Green's function of

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interest here obeys the following equations:

8 ( " ) 82 ( " ) � 8 ;=;-Kij\x,t ; x ,t = Di;=;-zKiJ\x,t ; x ,t + L. ;=;--ut uX k=l uCk x /;(c(x, t), rx)Kkix, t ; x', t') + !5ij!5(x -x')!5(t -t'), 37.

J1 = 1,2,

lim Kiix,t+e ; x',t) = !5ij!5(x-x'). 8- 0+

38.

39.

Besides the boundary conditions given by Eq. 38, K;ix, t; x', t') also satisfies similar conditions with respect to the variable x'. Using the Green's function one obtains the following expression for the sensitivity coefficient : it IX2 n

Sip(X, t) = dt' dx' .L Kiix, t; x', t')gjp(X', t'), ° x, J= 1

IX2 n ocO(x') + dx' L Kiix,t;x',O)-!;-

x, j= 1 !Xp

+ I dt' Ilt ( - l)ll[t Ki}(X,t;xwt')h;iX/L't')} In the last term of the above equation one has

_ 1 Kiix, t ; xl" t') = -1 Kiix, t ; Xw t'),

Ai" if At" =j. ° and Al" = 0,

KiJ{x, t ; x/l.' t') = - �["a, KiJ{x, t ; x', t')] , Aj/l. uX X'=Xp

if Alll =j. 0 and Afll = 0.

40.

41 .

42.

If both AIIl and Arll are nonzero then any of the above expressions for Kiix, t ; xll' t) can be used because in this case these two expressions will be identical due to the boundary conditions satisfied by KiJ{x, t ; x', t') with respect to its second variable x'.

The Green's function Kiix, t ; x', t') like the one used in the temporal case satisfies a group property of considerable practical importance:

n IX2 K . (x t · x' t') = " K· (x t · x" t")K (x" t'" x' t') dx" IJ' , " L..J r.k ' " kJ" " ,

k= 1 X, 43.

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where the intermediate time t" is arbitrary and assumes values in the interval t > til > t'.

The sensitivity coefficients Sip(X, t) have similar meaning, interpretation, and uses, along with their counterparts in the purely temporal case. Higher order sensitivity coefficients can be obtained by further differentiation of the equations for the Sip(X, t); their meanings and uses are also analogous to their counterparts in the temporal case.

Thus far little computational work has been performed on sensitivity analysis in space-time systems (49), and it is not clear whether the formal Green's function solution presented in Eq. 40 represents the best approach. The direct method could be employed, but even more promising may be an extension of implicit integration techniques based on the material presented below.

The Steady State Limit

The steady state limit of Eq. 28 arises in a number of physical situations. Furthermore, many numerical algorithms for solving space-time partial differential equations utilize the implicit integration method, which entails effectively solving a steady spatial boundary value problem at each time step. For all of these reasons, it is useful to examine carefully the characteristics of sensitivity analysis in steady kinetic systems.

The steady state kinetic equations are simply achieved by setting the lefthand side of Eq. 28 equal to zero,

a2 Di8x2 Ci+.h(C,rx) = O. 44.

The boundary conditions have the same form as those in Eq. 30. Problems defined this way have been frequently solved by returning to Eq. 28 and numerically integrating in time through the transient period into the steady state limit. It is our purpose here to consider that Eq. 44 be directly approached rather than its transient forebears. The sensitivity analysis of Eq. 44 can be achieved as usual by taking appropriate variations of the equation. The resulting equations for the first-order sensitivity coefficients have the following form:

82 ali Di�Sip(X) + I :;-Sjp(X) + gip(X) = 0

uX j UCj

with boundary conditions

1 (} 2 Ai" ax Sip (X,,) + Ai"Sip(X,,) = hip(x,,), J1 = 1,2

45.

46.

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where hip and gip have the same form indicated in Eqs. 35 and 36. Assuming that the solution to Eq. 44 for the species concentration vectors is available, the solution to Eq. 45 could be sought by a spatial Green's function technique. Although this method is especially valuable in the pure temporal case, it appears most efficient to take a different approach in the present circumstances as described below.

The optimum way of solving Eq. 45 is closely connected with the choice of procedure for solving Eq. 44. Ideally, one would like bqth solution algorithms to be chosen simultaneously for joint optimization. In this regard, there is a happy coincidence, and for these reasons we first describe the particular way of solving Eq. 44 before dealing directly with Eq. 45. Because Eq. 44 is typically nonlinear, these equations must be locally linearized in order to take advantage of available iterative numerical algorithms. For example, suppose we have an approximate solution vector en at the nth level of iteration and we seek an improved vector at the n + 1 level. Equation 44 is first linearized about en to produce

D a2 A n " ( ali ) A n R" i GX2 tlCi + 7 GC] tlCj = i 47.

where Aci = ci+1(x)-ci(x) and the residual on the right-hand side is

a2 Ri = Di GX2 ci + li(en, a) 48.

The iteration set up in Eqs. 47 and 48 is referred to as the Newton method, and algorithms are available for its implementation when the equations are discretized (e.g. by finite differences) (50). These latter discretized equations can be compactly written as

.

/nAcn = Rn 49.

where �n is the discretized form of the linear differential operator on the left-hand side ofEq. 47. It is important to note that the full Jacobian /" will generally depend on the nth approximation to the solution vector. In principle this Jacobian needs to be updated at each step of the iteration, but in practice efficient algorithms may only do this at a minimum number of times in order to cut computational costs. However, a Jacobian evaluated at the final level of solution will be required.

The significance of the Newton algorithm outlined above is clearly apparent upon comparing Eq. 45 for the sensitivity coefficients and Eq. 47. The operator of both of these equations is identical and they only differ in their inhomogeneities. An actual operating procedure would consist of first solving Eq. 49 to an acceptable level of accuracy followed by solving the analogous discretized form of the sensitivity equation in Eq. 45.

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Achievement of the task of solving the latter equation can be accomplished at virtually no overhead beyond the original cost of solving Eq. 44 due to the identical form of the sensitivity and linearized kinetic equations. In particular, at the last iteration step n = N for Eq. 49, the Jacobian fN and its decomposition (essentially for inversion) will be available for use in treating the sensitivity equations. Since formation of the Jacobian and its decomposition constitute almost all of the computational effort, the solution of the sensitivity equations is in hand from the process of solving the original kinetic system. The system spatial Green's function can also.be obtained by this procedure, which requires just calculating (fN) -1.

Although K is not being used here to calculate the general sensitivities, it is still useful to examine since elements of the matrix are of importance for system stability considerations. Finally, it is useful to point out that this overall algorithm depends critically on using a Newton iteration method for the initial kinetic equations in order that the coefficient matrix coincide with exactly the natural Jacobian in the sensitivity equations. In addition, the Jacobian fN must be of sufficient accuracy to assure good quality sensitivity information.

The procedure outlined above is a variant of what is referred to above as the direct method. The idea of joining sensitivity analysis with Newton iteration solvers for the original kinetic equations has been discussed a number of times recently (22,51, 52), (c. L. Irwin, private communication; A. Dunker (49); Y. Reuven, H. Rabitz, M. Smooke, to be published). This approach may well lead to the fastest algorithms available for sensitivity analysis, because it draws on the intimate workings of the original kinetic equation numerical procedure. This observation also implies that fundamental changes in how the original kinetic equations are solved would probably have an immediate impact on how to treat the sensitivity equations.

DERIVED SENSITIVITIES

Conventionally the concentrations are regarded as dependent variables and the parameters as independent variables. However, there are situations in the modeling of particular kinetic experiments in which an interchange of the role of dependent and independent variables among the concentrations and parameters leads to new insights on the chemistry of the system under study. For example, when a particular species concentration is easily monitored and a particular rate constant is to be determined, a natural choice is to assign the concentration of that species at a particular time as an independent variable and the rate constant as the dependent one. The derived sensitivity coefficients are the sensitivities of the new set of

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dependent vari�bles with respect to the new set of independent variables (13a,b). The decision of which concentrations and parameters to interchange is a most important one and should be carefully tailored to the particular experiment or modeling calculation of interest. Often it is useful to investigate the sensitivity of a particular rate constant that is to be determined with respect to other rate constants or monitored species concentrations. For example, a derived sensitivity like OIXp/OIXq, where IXp is one of the new dependent variables and IXq is one of the new independent variables, gives information on the correlation of the system parameters.

Suppose that several species concentrations, Ci, i = 1, . . . , s, may be easily monitored in a particular experiment and several of the parameters, IXp, p = 1, . . . , s, are treated as unknown. One can then use the derived sensitivities to investigate the relationships between the unknown parameters and the monitored concentrations and known parameters.

The differential dci(t) of Ci(t) (as a function of IX at time t) can be written as m

de;(t) = L Sip(t) dIXp- 50. p= l

To obtain the derived sensitivities the new set of independent variables is taken to be {Ci(t), i = 1, . . . , s ; lXp, P = s + I, . . . , m} and the new set of dependent variables is {lXp, P = 1, . . . , s ; e;(t), i = s + 1, ... , n}. Notice that it is necessary that s ,;;;; n and s ,;;;; m. A straightforward application of the rules of multivariate calculus yields the derived sensitivities. The procedure is entirely analogous to the one used to exchange variables in thermodynamics (53). For the derived sensitivity coefficients one first obtains

(a) i = 1, . . . , s, p = 1, . . . , s

D p;(t) = (S - l(t» pi, 51 .

where Dpi(t) is the partial derivative of lXp (a dependent variable) with respect to Ci (an independent variable). The matrix S - let) is the inverse of the s x s

matrix Set) whose elements are the relevant elementary sensitivity coefficients Sip(t). The parameters and concentrations that are to be interchanged must be chosen such that S has an inverse. The remaining derived sensitivities are

(b) p = I , . . . , s, q = s + I, . . . , m

s Dpit) = - L (S - l(t» piSiit), t = 1 52.

where Dpit) is the partial derivative of lXp (a dependent variable) with respect to lXq (an independent variable).

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(c) j = s + I, . . . , n, i = I, . . . , s

Dj;(t) = L Sjp(t) (S - 1(t))p;, p= 1


53.

where Dji(t) is the partial derivative of cj (a dependent variable) with respect to Ci (an independent variable).

(d) j = s + 1, . . . , n, q = s + 1, . . . , m

Djq(t) = Sjq(t)- L L Sjp(t) (S - l(t))PiSiq(t), 54. p = 1 i= 1

where Djit) is the partial derivative of cj (a dependent variable) with respect to IXq (an independent variable).

This formulation of the derived sensitivity problem can also be extended to the case of space-time systems such as the ones discussed in the last section. In this case the derived sensitivity coefficients are parametrically dependent on both space and time coordinates and Eqs. 5 1-54 hold with SivCt) replaced by Sip(X, t) in Eq. 3 1 . The case in which the parameters themselves are functions of the space and time variables is discussed below. With space and time dependent parameters one can go beyond the present discussion by interchanging dependent and independent variables over the whole space-time domain of the problem.

The concept of derived sensitivity can be further generalized by considering as new independent variables, not some of the concentrations at a particular point in time, but averages of these concentrations with carefully chosen weight functions. More explicitly one can define (R. Guzman, H. Rabitz, 1982, in preparation) the quantities

Cij = f dt ci(t)¢it), 55.

where each ¢j (t) is typically defined to vanish outside a finite time range. For example, the ¢j (t) might be the characteristic function over a sequence of time intervals [tj, tj+ 1], that is ¢j (t) = 1 if t E [tj, tj+ 1] and ¢j (t) = 0 otherwise for j = 1, . . . , 1. A similar definition can be introduced for space time systems. Derived sensitivities can then be introduced by treating some of the parameters (Xp as dependent variables and an equal number of the Cij as independent variables. The derived sensitivities are then obtained from

dCij = f dt¢it) pt Sip(t) dIXp 56.

The procedure is entirely analogous to the previous discussion, with Sdt¢j (t)SivCt) playing the role that SiP(t) played there. This "tile function

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method" allows for probing the interrelationship of parameters and monitored concentrations by using a much larger data set than is possible to utilize in the case of the previously defined derived sensitivities. This is so because in that case one uses the values of some concentrations at a particular instant of time, whereas in the tile function method one uses information about the behavior of the concentrations over finite time intervals (analogous statements are, of course, true for space time systems). The familiar least square technique is a related, but different, approach to bringing in a large data set for extraction of system parameter information (see Eq. 60 below and the following discussion).

The full physical meaning and utility of derived sensitivities remains to be established. Thus far two kinetic studies [(13a,b) R. Yetter, E. Eslava, and H. Rabitz, in preparation)] have been performed. One of these studies (13a,b) concerned the modeling and sensitivity analysis of the oxidation of CO in the presence of formaldehyde. The work was motivated by the experimental work of Vardanyan and co-workers (54), which aimed to establish rates for the reactions

k9 OH + CO � C02 + H '

k l o H02 + CO � C02 + 0H

out of a multistep mechanism. The actual measurements were of [H02] and [C02] and the desired statistical variance of kiO (and similarly for k9) is given by

" (ilklO)2 var(klO) � '7 ilf3i

var(f3;), 57.

where fJ == ([H02], [C02], k1, · · . , ks, kl l , • • • , k2S), and the derivative sensitivity is to be calculated by keeping all f3i,j ::f 1, fixed. In this way the error in k9 and kt 0 was estimated based on errors in the measurements and the parameters in the model. Derived sensitivities deserve careful further study, as they may have significant application to a number of experimental and theoretical problems, including experimental design.

SENSITIVITY CONSIDERATIONS RELEVANT TO

THE ANALYSIS OF LABORATORY DATA

Sensitivity analysis can be valuable for modeling in chemical kinetics, but ultimately there is the need to connect models with real laboratory or field situations. The natural question therefore arises concerning how to best

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utilize sensitivity analysis for joining of mathematical modeling and laboratory observations. The last section on derived sensitivities deals with an aspect of this topic, but the full answer to this question is not yet available. The three subsections below treat additional aspects of this matter.

Feature Sensitivity Analysis

For the sake of illustration this section is confined to consideration of purely temporal systems, although an exactly parallel analysis may be considered for space or joint space-time problems. To understand the role of feature sensitivity analysis, consider the schematic concentration profile shown in Figure 1, where the chemical species concentration exhibits a maximum as a function of time. This behavior, for example, would be typical of a transient reactive radical species in many systems. In addition to the concentration profile, we also have the ability to calculate a full sensitivity curve «(JeRkl) as a function of time for any appropriate parameter aj. Such a detailed sensitivity curve often can obscure experimentally interesting simple features in a profile such as in Figure 1. For example, the location of the concentration maximum t*, its amplitude h, and the width w all represent simple characterizing features of the profile. Such characteristic features are often adequate for describing the essential aspects of laboratory observable profiles, and each concentration is expected to have its own particular set of feature parameters. The general point is that any physically meaningful observable feature (including those requiring a convolution with a laboratory resolution function) can be considered for subsequent feature sensitivity analysis.

Just as we have referred to the collection of input kinetic parameters as IX,

the vector of characteristic output feature parameters will be called fl. The

c: o c:> �

c Q) u c: o <.>

t i me Figure 1 A schematic illustration of a concentration profile as a function of time. Also shown

are the feature parameters defined as the time t* locating the concentration maximum, the

amplitude h of the concentration maximum, and the width W ofthe maximum feature. Because

the profile is asymmetric with respect to t*, both a left and right half width could also be

defined.

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number of interesting features, and hence the length of the fJ-vector, need not have any specific relation to the number of input variables associated with the ac-vector. However, the implication behind this analysis is that the concentration profiles have two equivalent forms

e;(t, IX) == Ci(t, fJ)· 58.

This equation implies a functional relationship between the input and output vectors

59.

The need to consider different characteristic parameters at the input and output level of a kinetic problem reflects the fact that both ends of the problem have their own natural variables.

In practice, the dependence of fJ on the input parameters ac cannot be fully established since the kinetic calculations are always performed at some operating point acO in the input parameter space. However, we can readily ask for the fJ-vector at this reference point fJ = fJ(acO) and its gradient (8PJ8cx.)r.o. Thus far, two distinct procedures have been developed to obtain

the feature sensitivity coefficients (of3Jorx). The first procedure (55) is based on fitting with an a priori functional form and the second method (56) is referred to as a pointwise analysis. Both approaches are discussed separately below.

The approach to feature sensitivity analysis based on fitting consists of choosing ajudicious functional form ci(t, fJ) explicitly containing the feature parameters and fitting this form by, for example, least squares to the kinetic profile over an appropriate time interval tl � t � t2 containing the features of interest

60.

The minimization of the residual Ri with respect to the feature parameters will determine these parameters at the nominal point fJ(IXO) in the input space of variables. Deviations about this norminal point can be explored by taking an additional derivative ofEq. 60 with respect to an input parameter rx,. This procedure will produce the following explicit solution for the desired feature sensitivity coefficients

6 1 .

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where


62.

63.

The choice of functional form ci(t, fJ) is central to this procedure. The form should be a balance between flexibility and simplicity in order to ensure a good fit as well as a unique meaning to the feature parameters. This overall procedure was implemented (55) for the system of oscillating chemical reactions suggested by Lotka in which the period, amplitude, and phase of the oscillator were examined.

The above analysis should be applicable to a number of chemical problems, but a more general and often simpler procedure (56) consists of the following pointwise analysis. Typically, the features in a kinetic system can be characterized by some point or points on a kinetic profile (or perhaps its integral transform). For example, the feature parameters shown in Figure 1 are of this type. An such feature parameters can ultimately be characterized in terms of a functional of the original kinetic profil�. it2 /3j = dt Fldt, ot:), t]

t , 64.

where the function Fj depends on the particular feature of interest. The desired feature sensitivity coefficient can be obtained directly by differentiation of the integral in Eq. 64

_J _ dt _J _' (0/3 .) it2 (OF .) (ac.)

oal - t, OCi oal ' 65.

In practice, the identification of the function Fj and evaluation of the integral in Eq. 65 can usually be done quite simply to make this a very transparent procedure. For example, if we desire the sensitivity of the time t* in Figure 1 with respect to the system variables then the function F becomes

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where the Dirac delta function ensures the constraint that the concentration extrema be identified by the concentration profile having zero time slope. Utilization of this expression in Eqs. 64 and 65 finally yields the feature sensitivity coefficient

Similar expressions can be obtained for most any feature of concern and this method has been illustrated for the wet oxidation of CO (57). The approach encompassed by Eqs. 64 and 65 can also be viewed as an objective function analysis. Such an objective function approach has been taken by Cacuci (3).

Sensitivity Analysis of Experimental Data

This section concerns the role of sensitivity analysis for unravelling the relationship between model input and actual observables generated in a real laboratory experiment. A number of matters arise when considering the comparison of a kinetic model with experimental data. The usual approach to this problem consists of fitting the model parameters, or some subset of them, to the available data by a least squares type procedure. In addition, when a partial set of experiments has been performed, a relevant question concerns which new experiment would be useful to augment the available set. Certain of the derived sensitivity concepts are related to this matter as described below.

The connection between experiment and model is made at the level of defining a minimizing functional. For purposes of illustration we can take a least squares functional of the same general form as Eq. 60

66.

where c�XP(t) is an experimental concentration profile. Demanding that the residual be minimized with respect to the input parameter vector IX implies a functional relationship IX = lX[ciXP(t)] between the system parameters and the experimental data. Information about this relationship can be explored by taking a functional derivative with respect to the data curve to produce the following result

� =

" B

-: 1 (OCi(t, IX)) (jc�XP(t) '-:- IJ aCt. . I J J 67.

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where

Bj/ = r t2 dt,{(OC;(t'� IX») (OC;(t" IX») J t l GIXJ GIXI

- [ciXP(t') - c;(t', IX)] (G��;��»)}-The similar structure of Eqs. 67 and 61 is quite evident. The functional derivative apparent in Eq. 67 is a direct measure of how a given parameter depends on the concentration profile at a particular point in time. This can be useful for refinement of experiments by identifying which time regions are important for further study.

A special class of derived sensitivities may be arrived at by utilizing the above model-experimental data sensitivities. In particular, from Eq. 67 we may deduce

8cit, at) ,, (8cit, IX») ( SIXI ) ociXP(t') = 7 � ociXP(t') ·

68.

The utility of this equation arises from considering theoretical concentration profiles cit, ex) that were in the model but not included in the least squares analysis of the available data. The question then arises concerning which of these additional theoretical profiles is least dependent on the available experimental curves. This identification from Eq. 68 would be helpful in a process of experimental design for choosing the next profile to determine in the laboratory (58).

Sensitivity Analysis and Model Parameter Uncertainty

A traditional role of sensitivity analysis has been its contribution to the error analysis of parameter determination when fitting experimental data to a model (1, 27). For our purposes here we assume that the numerical calculations involved in solving the model equations contain no statistical errors themselves. Under these conditions we can define the statistical contribution to the covariance matrix of the parameters

it2 it2 ( bIX ) COV(IX/IX!')lstat = � � dt dt' � �x:( ) ' J Ii Ii ue, t ( bIX!, ) X -- cov [c�XP(t)c�XP(t')] .

ciXP(t') ' J 69.

The sensitivities contained in the integrand are just those calculated from Eq. 67. In this equation, the covariance of the experimental data must be

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available. Typically, the nonlocal character of that information will not be available and it will be simply taken as diagonal with time. The nonfunctional analog of this equation is illustrated in Eq. 57.

Errors in the parameters can also be introduced due to the discrepancy between the experimental and model concentration profiles. We assume here that this discrepancy is due to a failing of the model and not to systematic errors in the experimental data. Utilizing this logic, one can generate an estimate of the parameter covariance matrix arising from model experimental discrepancies

it it2 ( orx ) COV(rx1rx1,)lmod = � � 2 dt dt' �( / ) I J " " uCI t, rt

70.

where ([)rx//fJc;(t, 0:)) is a special theoretical functional derivative obtained from solving Eq. 67 with

Note that these functional derivatives differ from those discussed below. In essence, the present set are not obtained as derived sensitivities but rather by a least squares type method. The total covariance for the model parameters is now seen to have two components arising from statistical errors in the data and errors introduced by the model.

STRUCTURAL ELEMENTS IN TEMPORAL

SENSITIVITIES

The discussions offeature sensitivity analysis in the last section provide the background for the present topics. Here we review some work covering special sensitivity aspects of oscillating and transient systems along with the topic of parameter space scaling.

Limit cycle systems are of interest for a number of physical reasons and they also exhibit a special aspect of sensitivity analysis. Quantities of natural interest for oscillating systems, such as the sensitivities of amplitude, period, and cycle phase, are not addressed by the elementary sensitivities 8c/8a.. The period and phase sensitivities both have the general form 8t/8rxj, and thus we are motivated to consider t as a function of system parameters. In a limit cycle, the steady-state periodic orbit is independent of the initial conditions c(O). A perturbation of ciO) can only affect the phase of the orbit,

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once the limit cycle is attained. This phase sensitivity is given by (59)

ot _

. t I /dek(tl) �) - hm -Kkit, t ) -d- ' 7l. uc},t /' -+ 00 t

The limit only need be observed to the extent that a problem-dependent transient period is passed. The sensitivity of the phase to noninitial condition parameters is obtained by again making t a dependent variable.

ot(t') _ ± ot OCj(t') OlXk - j= 1 oeit') --;;;;. 72.

The coefficient ot(t')/OlXk is a measure of the phase lead or lag, in time units, resulting from the change in the parameter IXk• This quantity is the coefficient of unbounded growth (secularity) of the sensitivity

oe; _ (oc; ) _

� de; - , 73. or:xj oaj </> or:xj dt

where the subscript cp indicates the secular free sensitivity. Tomovic (1) and others (e.g. 60) have identified ot/or:xj = tOlm/or:xj. where '0 is the period of the limit cycle. This linear growth of Ot/OlXj with time is an approximation ofEq. 72 that properly accounts for the overall accumulation of phase lead or lag from cycle to cycle. In either case, the secular free sensitivity is periodic for t' greater than the transient time of the sensitivity coefficient, and it was shown to represent accurately the sensitivity of the phase-space (timeimplicit) shape of the limit cycle. The phase sensitivity can be used to find the sensitivity of the cycle period, since the accumulated phase lead or lag after one revolution about the limit cycle is equal to this quantity (59, 61)

0'0 at I ot ,

- = -(t + '0) - -(t ) OlXj OlXj alXj

_ (oc; ( ') oCi ( , ))/OC;(t') - - t -- t +T --. oaj oaj ot 74.

Larter (62) has discussed the relationship between noninitial condition sensitivities and the subject of structural stability (63) of the limit cycle. An oscillation is said to be structurally stable if its "topological character" is insensitive to parameter variations. This question can only be addressed through the secular free sensitivities.

Nonlimit cycle and damped oscillators have many of the same sensitivity characteristics discussed above. In the purely transient situation the same secular behavior is present, but it does not manifest itself so dramatically. For example, it can be shown that (l/c;) (ociaaj) will typically exhibit

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secular growth in time. The behavior of transient kinetic systems encompasses a wide class of dependencies and no general secular free sensitivity analysis has thus far been accomplished, although certain specific models have been considered (64).

Structural Information in Parametric Scaling

It has been shown (64) that structural information can be used to great advantages in parametric scaling, i.e. prediction of c(a; + L\a;) from the nominal solution and the sensitivities. The Taylor series

75.

is the normal tool for the task. This series is usually truncated to low order and the range L\IX in which the approximation is valid may be quite small. Methods like Pade approximants (65) can be used, if a sufficient number of sensitivities are available. Before discussing the methods of "feature" scaling, it is necessary to discuss proper selection of scaling variables, without which the feature scaling may not be effective. It has been argued (64) that the Taylor series Eq. 75 is often a poor choice for scaling, because

unphysical states, e.g. negative concentrations or concentrations exceeding

mass conservation limits, may be predicted. The variable transformation z = lnlc - c/I, where c, is zero or another a priori known conservation limit, may therefore be helpful. Writing a first-order Taylor series in z and solving for c, one obtains

(OC L\IX ) c(O( + L\O() = c/ + (c(o() - c/) exp ;:;- ( )

. vo( c O( - C,

76.

This formula was used with success by Dougherty et al ( 13a,b) and Hwang ( 18). An asymptotic approach to c, is predicted for large L\IX. Eslava et al (66) in another physical context offered a formula related to Eq. 76,

lX + illX -A ) Nne C(IX + L\IX) = C(o:)

(-0:-

alnIX . 77.

Both Eqs. 76 and 77 contain implicit higher order information, and thus they can provide much enhanced convergence properties relative to the Taylor series Eq. 75, by preventing unrealistic scaling predictions.

Feature scaling can significantly enlarge the parametric domain addressed by the local sensitivities. We have seen in Eq. 61 that in certain cases the elementary sensitivities can be decomposed into various structural elements, given by the feature sensitivities ofJJoO:j. Given Eq. 58, the

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SENSITIVITY ANALYSIS 45 1

"natural" scaling is given by

ofJ ) e(t, Ot + �Ot) = e(t, fJ + OOt

�Ot • 78.

This formula is advantageous because it automatically incorporates nonlinearities implicit in e(t, IX). By contrast, the Taylor series is globally linear in �IX. The extent to which this more general scaling can be realized is dependent on the extent of the knowledge of c(t, fJ). For example, we have already seen that the sensitivities associated with oscillatory systems may be decomposed by isolating the period ('[) "feature" (see Eq. 73). The '[-feature scaling law is developed as follows. To a first-order approximation, the Taylor series for e(t*, IX + �IX) can be written

de oc d (OC(t, IX») C(t*, IX + �IX) = C(t, IX) + dt

(t, IX)M + OIX

(t, IX)�IX + dt

-----a;- !:i1X�t

where 8t = t* - t. The increment 8t or t* may be chosen (64) so as to make the "secular" term, t(olm:/olX) dc/dt disappear ; specifically,

olm: �t = t��IX,

and therefore

e(t*, IX + �IX) = C(t, IX) +CC��IX»).�IX. 79.

This scaling law solves the secularity problem, because (oe/olX), is periodic, and gives very accurate parameter scaling for oscillatory problems (64). In the same work another example involving transient kinetics of C2H6 pyrolysis was considered using Eqs. 75-77 along with a related feature scaling approach.

FUNCTIONAL SENSITIVITY ANALYSIS

Elementary Sensitivity Densities

The discussion of the sensitivity analysis of space-time systems presented above was concerned with the calculation of the sensitivity coefficients, that is, the partial derivatives of the concentrations with respect to the parameters. In that case the parameters were considered to be independent of time and space coordinates. However, some of the parameters (or even all) can have space and time dependence. When the parameters are functions ofthe space and time coordinates the sensitivity analysis becomes more complex than when the parameters are constants. Porter (67) was

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apparently the first to consider this problem. He suggested that the parameters be represented by expansions in orthogonal basis functions, leading to a lumped parameter system, for which the sensitivity coefficients can be obtained in a standard manner. As pointed out by Seinfeld and coworkers (68) in the case of spatially and temporally varying parameters, the quantities of interest in sensitivity analysis are functional derivatives of the concentrations with respect to the parameters. The functional derivatives are defined such that the change in the concentrations, (jCj(x, t) due to small perturbations OlY.iX, t) in the parameters lY.iX, t) are

� ( ) f d ' f d , ;, Jcj(x, t) � ( ' ') 80 uCj x, t = x t 1..., J; (

' ') ulY.p x , t , . p= l ulY.p x , t with the elementary sensitivity densities defined as

" Jc;(x, t) S;ix, t; x , t ) = J; (

' ')

' ulY.p x , t 8 1 .

The name elementary sensitivity density is motivated by the relationship that exists between the objects defined by Eq. 8 1 and the elementary sensitivity coefficient (Eq. 3 1). When the parameters are independent of space and time, the sensitivity derivatives are related to the densities by the relation

S;p(x, t) = f dx' f dt' Sjp(x, t ; x', t'). 82.

It is important to realize that the elementary sensitivity densities can be obtained even when the parameters are not space and time dependent. Equations for the elementary sensitivity densities can be obtained from the reaction-diffusion equations for the concentrations by a procedure analogous to the one previously used to obtain the elementary sensitivity coefficients. On a different approach, Koda & Seinfeld (69) obtained the elementary sensitivity densities by using techniques of distributed parameter optimal control theory. Cacuci (3) provides a very general and mathematically rigorous discussion of the sensitivity theory for nonlinear systems. The definition of the "sensitivity of the system response" introduced by Cacuci is general enough to encompass the concepts of elementary sensitivity coefficients and elementary sensitivity densities. The formulation of the sensitivity theory presented in that work is centered on evaluating the Gateaux differential of the system response (i.e. a concentration). This quantity has been considered by Tomovic & Vukobratovic (1) to be "the most general measure ofthe sensitivity of a response to variations in the systems parameters." In the case of the concentrations for example,

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the Gateaux differential is closely related to the variation ()ci(X, t) introduced in Eq. 80. A proper definition of the Gateaux differential and its relation to the sensitivity analysis is beyond the scope of this review and the interested reader is referred to Cacuci's article (3) for further information.

We confine ourselves to the problem of a chemically reacting system in whieh diffusion occurs (70, 7 1). The system is assumed to occupy a volume n enclosed by a boundary surface a. It must be remarked here that this formulation does not exclude the case of an open system in which incoming and outgoing material flows exist. Such flows are specified by the boundary conditions (47). The species concentrations satisfy the following reactiondiffusion equations

oc · Tt = V ' (Di(x, t) · VCi) + ./i(c, k). 83.

The rate constants and other parameters entering in ./i are collectively labeled kix, t), p = 1, . . . , r. In addition /; could be an explicit function of x and t through externally imposed conditions. Di(x, t) is the space and time dependent diffusion tensor for the ith species. A diffusion tensor is used instead of a diffusion coefficient in order to take into account the possibility of spatially anisotropic media. Equation 82 is to be solved subject to the following initial and boundary conditions

C;(X, O) = c�(x), Ar(s, t)n ' Di(s, t) · VCi(S, t) + Ar(s, t)c/(s, t) = ai(S, t),

84a.

84b.

where - n . D i(S, t) · V Ci(S, t) is the component of the diffusion flux of the ith species normal to the boundary surface 0. The coefficients A:(s, t) and ai(s, t) may depend on time and may vary over the surface 0 ; this is indicated by their possible dependence on the vector s, which denotes a point on the surface a.

In problems involving moving boundaries (i.e. a time dependent 0) one usually can, at least for simple geometries, find a suitable time dependent coordinate transformation that accompanied by a corresponding redefinition ofthe diffusion and boundary condition coefficients preserves the form of Eqs. 82-84 but with a time independent enclosing surface (70). In the following it is assumed that in problems involving moving boundaries such a transformation has been carried out. This is necessary in order to apply the Green's function method to obtain the elementary sensitivity densities. Moving boundaries arise naturally in the study of vertical diffusion and chemical reaction in the atmosphere ; a typical example is provided by trajectory models describing simultaneous vertical diffusion

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and chemical reaction of air pollutants in an air parcel adjacent to the ground (68).

The set of all parameters entering in Eqs. 82-84 is denoted {cxp(x, t), P = 1, . . . , m}. This includes not only the kn(x, t) but also the components of D;(x, t) as well as the initial and boundary condition coefficients. An important point here is that the definition of the elementary sensitivity · density in Eq. 8 1 treats all parameters on an equal basis, and it considers all parameters to be space-time dependent. The freedom still exists to choose CiCXi(X, t) to have any arbitrary space and/or time dependence (or independence).

Since functional differentiation commutes with the partial derivatives in Eqs. 83-84 the elementary sensitivity densities are found to satisfy the following linear, nonhomogeneous, reaction-diffusion equations

n L L;ix, t)Sjp(x, t ; x', t') = g;p(x, t ; x', t'), j� 1

with initial conditions

Sip(X, O ; x', t') = Aip(X ; x',t'), and boundary conditions

Af(s, t)n . Di(s, t) . V Sip(S, t ; x', t') + Ar(s, t)Sip(S, t ; x, t') = h;p(s, t ; x', t').

The linear differential operator L is defined by

( 0 ) oj; L;J{x, t) = bij ot - V · (Di(X, t) · V) - OCj

'

The functions g, A, and h are

( I ') t7 (bDJX, t) V ( )) oj; �( ' ) �( ') gip X, t ; X , t = v ' � ( ' ) ' ci x, t + ;;-u x - x u t- t , ur:l.p X , t ur:l.p bci(x) A;p(X ; XI, t') = � ( ' ') ' ucxp x , t baJs, t) bANs, t) ) h;p(s, t ; x', t') = � ( ' ')

- � ( ' ') n ' Di(s, t urxp x , t urxp x , t

1 CiD;(s, t) . Vc;(s, t) -Ai (s, t)n " ( ) ucxp x', t' JAi(s, t) . V Ci(S, t) - " ( ' ') c;(s, t). urxp x , t

85.

86.

87.

88.

89.

90.

9 1 .

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The functional derivatives indicated in Eqs. 90 and 91 are to be computed as follows. In Eq. 90 consider cr(x) as the initial (t = 0) value of a subset of the parameter set, thus c�(x) = ap'(x, O) for Pl < P' < Pl + N - 1. Similarly, for the boundary condition terms, thus

ai(s, t) == ap"(s, t), P2 < P" < P2 + N - 1

and analogous relations for At(s, t) and Ar(s, t), Pl and P2 will depend on how one chooses to associate an ap(x, t) to an actual parameter ofthe system (the enumeration of the system parameters is, of course, arbitrary). With this convention it follows that

()c�(x) _ Dap'(x, O) _ 5: 5:( ') 5:( ') ----'--- - - u , u x - x u t ,

bap(x', t') bCXp(X', t') p p

bai(s, t) _ bap"(s, t) _ 5: 5:( _ ') 5:( _ ') --:----,----,-----::- - - u " u S X u t t .

bap(x', t') bap(x', t') p p

92.

93.

As in the cases studied �bove, the linear differential equation Eq. 85 for the elementary sensitivity densities can be solved (at least formally) with the help of an adequately defined Green's function. The Green's function of interest here is such that

I Liix, t)KkJ{x, t ; x', t') = bijb(X - X')b(t - t'), 94. k � l

with homogeneous boundary conditions

At(s, t)n ' Di(s, t) · V Kiis, t ; x', t) + Ar(s, t)Kiis, t ; x', t') = 0 95.

A �(s' t)n" D ?'(s' t') · V'K. {x t · s' t') + A�(s' t')K. (x t · s' t') = 0 96. J " J ' lJ\ , " J ' lJ" "

and initial condition

lim Kij(x, t + 8 ; x', t') = bijb(X -x'). , ... 0

The solution to Eq. 85 can then be written as

S· (x t · x' t') = It dt" i dx" .:;, K · {x t · x" t")g . (x" t" ' X' t') 'P , , , � IJ' " , lP , , , o n j= l

97.

+ It dt" f du" i Kiix, t ; s", t")hjp(s", t" ; x', t'), 98. o 0 j= 1

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where

_ 1 Kij(x, t ; s", t") =

A] (s", t,,)Kij(x, t ; s", t") if A](s", t") of 0

and

- 1 T K . .(x t · sIt tIt) = ( - n" · D · (s" tIt) IJ ' " A J( s", t") J '

. V"Kjx, t ; s", t"» if AJ(s", t") =f. 0

99.

100.

The above expressions for Kij assume that either AJ or A] is identically zero. If neither Ai; nor A� is identically zero then, due to the boundary conditions stated in Eq. 96, both expressions are identical and one can use Eq. 99 for Kij.

The Green's function Kij is a concentration response function. It gives the linear response of species i to the presence of a weak source of species j. If a small amount of species .i is deposited on point Xo at time to this can be represented by a source term of the form JeJijJ(x - xo)J(t - to) included in the right hand side of the reaction diffusion equation. Thus, to first order in A, the change in the concentrations will be Oci(x, t) = ..1.Kiix, t ; xo, to). Similarly the elementary sensitivity densities are generalized response functions. The importance of the concentration response function can be inferred from the fact that all other response functions are fundamentally dependent on it. This interpretation illuminates the physical meaning of sensitivity analysis and shows that the elementary sensitivity coefficients and densities belong to a broad class of mathematical objects of considerable interest in statistical mechanics.

The concept of elementary sensitivity density (a first-order functional derivative) can be extended to include the notion of arbitrary order functional sensitivity densities defined as arbitrary order functional derivatives of a chemical species concentration with respect to the space and time dependent parameters of the system (71). Going beyond the firstorder work described in this Section, the higher order sensitivity densities contain interesting information and allow for large excursions from the reference point in parameter space. The second derivatives provide information on the structure of the critical points (points where the first derivative vanishes) in parameter space (i.e. they determine whether a particular critical point is a local maximum, minimum, or saddle point). Using the Green's function method it has been shown that the higher order sensitivity densities can be obtained in a straightforward recursive manner from the lower order ones. This recursive property should be useful in practical numerical computations.

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Derived Sensitivity Densities


The elementary sensitivities densities are the mathematically and physically natural gradients to consider when the species concentrations are dependent variables and the system parameters are independent variables. However, circumstances can be envisioned in which some subset of the concentrations is assumed to actually be measured and hence "known." This latter subset of concentrations should now be considered as a part of the set of independent variables. It is assumed that there are s :::::; n such concentrations (C1, . . . , c., etc . . . ). Since the dynamical ties between the dependent and independent variables are still the same, an equivalent number of formerly independent variables must now be transferred to the set of dependent variables. This procedure is similar to the one discussed above, except that now it is desired to exchange the dependent variables c;(x, t) (i = 1, . . . , s) as functions of space and time coordinates with the independent variables r:J.p(x, t) (p = 1 , . . . , s). This is possible only if the exchanged objects belong to the same function space. Thus the parameters that are to be interchanged should be defined over the whole space time domain of the problem under study. This implies that these parameters are the ones that enter explicitly in the body of the reaction-diffusion equation itself since the initial condition and boundary condition coefficients are uniquely defined only at t = 0 and on the boundaries of the system, respectively. Suppose now that the new set of independent variables is taken to be {c1(x, t), . . . , c.(x, t), r:J.s+ 1 (x, t), . . . , r:J.m(x, t)} and that the new set of dependent variables is {r:J.1(X, t), . . . , r:J.s(x, t), cs +(x, t), . . . , cn(x, t)} with s :::::; m and s :::::; n. The derived sensitivity densities can be obtained by a procedure analogous to the one used before to obtain derived sensitivity coefficients. In this way one obtains (72)

(a) i = l, . . . , s, p = l, . . . , s Dp;(x, t ; x', () = s;;t(x, t ; x', t'), 101 .

where Dp;(x, t ; x', t') is the functional derivative of r:J.p(x, t) (p = 1 , . . . ,s, a dependent variable) with respect to c;(x', t') (i = 1 , . . . • s, an independent variable). S - 1 is the operator inverse to the operator S whose kernel consists of the elementary sensitivities S;p(x, t ; x', t'), i = 1, . . . , s. Thus, with p, p' = 1, . . . , s, one has

f dX f dt' � S-. l(X t · x' ()S· (x' ( · x" t") L... pI ' " IP " , ;= 1

= (jpp,(j(x -x")(j(t- t'). 102.

The parameters chosen to be interchanged with the concentrations must be

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458 RABITZ, KR AMER & DACOL

such that the above inverse exists. The remaining derived densities are given by the following equations.

(b) p = 1, . . . , S, q = S + 1, . . . , m

Dpix, t ; x", t") = -fdX1 fdtl ,± 1=1

x S-. 1(X t · x' t')S. (X' t" X" t") pI ' " lq , , , , 103.

where Dpix, t ; x", t") is the functional derivative of (Xp(x, t) (p = 1 , . . . , s, a dependent variable) with respect to (Xix", t") (q = S + 1, . . . , m, an independent variable).

(c) i = s + I , . . . , n, j = 1, . . . , s

Diix, t ; x", t") = f dx' f dt' Jl Sip(X, t ; x', t')S;/(x', t' ; x", t"), 104.

where D;ix, t ; x", t") is the functional derivative of ci(x, t) (i = s + 1, . . . , n, a dependent variable) with respect to cix", t") (j = 1 , . . . , s, an independent variable).

(d) i = s + I , . . . , n, q = s + 1, . . . , m

Diq(x, t ; x/, t') = Siq(X, t ; XI, t') -f dX1 f dt1

X f dX2 f dt2 pt1 it1 SiP(X, t ; Xl' t1)

x S;/(xh t1 ; x2, t2)SjiX2' t2 ; X', t') 105.

where Diix, t ; x', t') is the functional derivative of C;(x, t) (i = s + 1, . . . , n, a dependent variable) with respect to (Xq(x', t') (q = s + 1, . . . , m, an independent variable). There is an obvious structural similarity between these expressions for the derived sensitivity densities and the expressions obtained for the derived sensitivity coefficients (Eqs. 51-54). It is important to notice that, unlike the case for the elementary sensitivities, there is no simple relation between the derived sensitivity coefficients and the derived sensitivity densities. Although the elementary sensitivity coefficients can be shown to be equal to the integral over the space time domain of the corresponding elementary sensitivity densities (see Eq. 82), this is not true for the derived sensitivities.

In order to get the derived sensitivity densities from the expressions obtained above it is necessary to have obtained the elementary sensitivity densities and to be able to invert the operator S. However, it is also possible

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SENSITIVITY ANALYSIS 459 to obtain these derived sensitivity densities as solutions to certain partial differential equations related to the reaction-diffusion equation of interest. These equations for the derived sensitivity densities can be obtained by applying the operator Liix, t) to the above expressions for the derived sensitivity densities and then using the differential equations for the elementary sensitivity densities (73). These differential equations can be solved by a number of means, including the Green's function method, to yield the derived sensitivities without first calculating the elementary sensitivities. This approach was illustrated for a model reaction-diffusion system (73).

It is possible to introduce the concept of higher order derived sensitivity densities, defined as higher order functional derivatives of the new set of dependent variables with respect to the new set of independent variables. These objects allow for a more detailed probing ofthe relationships among the system parameters and monitored concentrations. The fact that the derived sensitivity densities satisfy linear algebraic and differential equations implies that the higher order derived sensitivity densities will also satisfy similar equations. As with higher order elementary sensitivity densities, the Green's function method can be used to show that the higher order derived sensitivity densities can be obtained in a straightforward recursive manner from the lower order ones.

CONCLUSION

This review has covered sensitivity analysis as it applies to the field of chemical kinetics. There are a number of points to keep in mind in order to appreciate fully the content of the review. First, although the subject of sensitivity analysis has a rather long history (1), it is quite new as a formally developed topic in the field of chemical kinetics. The past few years have seen a rapid development of many of the basic concepts along with numerical techniques. The conceptual developments have tended to outstrip actual practical applications, but the number of numerical calculations is continuing to grow. It is our belief that systematic sensitivity analysis should be adopted as a routine tool when performing kinetic modeling. The technique can not only provide an immediate quantitative measure of modeling error propagation, but perhaps more importantly it can lead to detailed and often subtle physical insights concerning mechanistic structure. Although it is true that a rough sense of many of these matters can be achieved by merely performing repeated calculations while "tuning up" the model, there is in fact no substitute for quantitative measures of this information. For example, without the quantitative and mathematically well-defined nature of sensitivity gradients, the rich variety

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of questions interrelating the system dependent and independent variables cannot be addressed. A thorough sensitivity analysis does not come free, but if one can afford to perform the original modeling calculations, then a sensitivity analysis can also be afforded. In regard to the latter point, there is still much room for improved numerical techniques, especially for space and space-time problems. Nevertheless, while these basic developments will continue to grow, the available numerical technology exists for performing practical calculations, especially for purely temporal systems. Finally, it is worth noting that one of the most exciting aspects of this subject is its general applicability to a wide variety of problems. Literally, any mathematically described physical problem can be characterized by appropriate input and output information connected by intermediate (dynamical) equations. From this perspective many of the tools described in this review have applications to other fields besides chemical kinetics.

ACKNOWLEDGMENT

The authors acknowledge support for this work from the Department of Energy and the Office of Naval Research.

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