3.1. Illustrating the Need for High-Order Sensitivity Analysis and Uncertainty Quantification: A Paradigm Reactor Physics Benchmark
The PERP benchmark is a spherical subcritical nuclear system driven by a source of spontaneous fission neutrons. The PERP benchmark comprises an inner sphere (designated as “material 1” and assigned to “zone 1”) which is surrounded by a spherical shell (designated as “material 2” and assigned to “zone 2”). The inner sphere of the PERP benchmark contains α-phase plutonium which acts as the source of particles; the radius is
= 3.794 cm. This inner sphere is surrounded by a spherical shell reflector made of polyethylene with a thickness of 3.81 cm; the radius of the outer shell containing polyethylene is
= 7.604 cm.
Table 1 specifies the constitutive materials of the PERP benchmark.
The neutron flux distribution within the PERP benchmark has been computed by using the deterministic software package PARTISN [
58], which solves the following multigroup approximation of the transport equation for the group fluxes,
:
where:
The PARTISN [
58] computations used the SOURCE4C code [
59] and the MENDF71X library [
60] which comprise 618 group cross sections. These cross sections were collapsed to
energy groups, with group boundaries,
, as indicated in
Figure 1. The MENDF71X library [
60] uses ENDF/B-VII.1 nuclear data [
61]. The group boundaries,
, are user-defined and are therefore considered to be perfectly well-known parameters.
The fundamental quantities of interest (i.e., “system responses”) for subcritical benchmarks (such as the PERP benchmark) are single-counting rates, double-counting rates, the leakage multiplication, and the total leakage. The total leakage from the outer surface of the PERP sphere was considered to be the paradigm response of interest for illustrating the need for high-order sensitivity analysis, because the total leakage does not depend on the detector configuration (as opposed to count rates) and is therefore more meaningful physically than the count rates.
The total neutron leakage from the PERP sphere, denoted as
, depends implicitly (through the neutron flux) on all model parameters and is defined as follows:
The numerical value of the PERP’s benchmark total leakage, computed by using Equation (26), is
neutrons/sec.
Figure 1 depicts the histogram plot of the leakage for each energy group for the PERP benchmark.
3.1.1. Overview of Results for the First, Second, Third, and Fourth-Order Sensitivities
The computations of the sensitivities, of all orders, were performed deliberately using a widely accessible desktop computer, namely a DELL AMD FX-8350 with an 8-core processor. A single adjoint (large-scale) computation is needed in order to obtain all 21,976 sensitives of the first-level adjoint function, which is subsequently used to compute all of the first-order sensitivities using inexpensive quadrature formulas. The CPU time for a typical PARTISN forward neutron transport computation with an angular quadrature of S32 is ca. 45 s. On the other hand, the CPU time for a typical adjoint neutron transport computation using the neutron transport solver PARTISN with an angular quadrature of S32 is ca. 24 s. The CPU time for computing the integrals over the adjoint function is ca. 0.004 s.
Thus, the exact computation of the 21,976 first-order sensitivities using the first-level adjoint function is (24 + 21,976 × 0.004) s = 112 s. By comparison, the approximate computation of the 21,976 first-order sensitivities using a two-point finite difference formula would require 21,976 × 2 × 45 s = 550 h. Most of the first-order relative sensitivities have been found to be negligibly small, having absolute values less than 0.1. The most important (i.e., largest) sensitivities of the leakage response are with respect to the group-averaged total microscopic cross sections, followed by the sensitivities of the leakage response with respect to the isotopic number densities. Only 16 first-order relative sensitivities have absolute values greater than 1.0; most of these involve the total cross sections of 1H (hydrogen) and 239Pu (plutonium-239).
After having computed all of the first-order sensitivities and ranked their importance in the order of the magnitudes of the corresponding first-order relative sensitivities, it is imperative to compute the second-order sensitivities in order to quantify and compare their impact to the impact of the first-order sensitivities. The theoretical framework for computing efficiently and exactly second-order sensitivities has therefore been developed and applied to the PERP benchmark 2
nd-CASAM-L, and the computational results obtained for all of the second-order sensitivities of the PERP model’s leakage response with respect to the benchmark’s parameters listed in
Table 2 were reported in [
19,
20,
21,
22,
23,
24]. The computation of these second-order sensitivities was performed in the order of priority indicated by the values of the first-order sensitivities. Hence, the highest priority was given to computing the total of
second-order sensitivities of the PERP model’s leakage response to the model’s group-averaged total microscopic cross sections. It was thus found [
19] that 720 of these elements have relative sensitivities greater than 1.0, and many of the second-order sensitivities are much larger than the corresponding first-order ones. The largest second-order sensitivities involve the total cross sections of
239Pu and
1H. The overall largest element is the unmixed second-order relative sensitivity
, which occurs in the lowest-energy group for
1H.
The next largest sensitivities of the PERP leakage response were found to be [
19,
20,
21,
22,
23,
24] the mixed second-order sensitivities involving the isotopic number density of
239Pu and the microscopic total, scattering, or fission cross sections for the 12
th or 30
th energy groups of
239Pu or
1H, respectively. On the other hand, the numerical results obtained in [
19,
20,
21,
22,
23,
24] indicated that the second-order sensitivities of the PERP leakage response with respect to the remaining parameters (including scattering cross sections, fission cross sections, average number of neutrons per fission, and source parameters) are much smaller than the corresponding first-order ones, so that the effects of uncertainties in these parameters on the expected value, variance, and skewness of the response distribution were negligible by comparison to the corresponding effects stemming from uncertainties in the total cross sections.
Neglecting the second-order sensitivities would cause an erroneous reporting of the response’s expected value and also a very large non-conservative error by under-reporting of the response variance. For example, if the parameters were uncorrelated and had a uniform standard deviation of 10%, neglecting second-order (and higher) sensitivities would cause a non-conservative error by under-reporting of the response variance by a factor of 947%. If the cross sections were fully correlated, neglecting the second-order sensitivities would cause an error as large as 2000% in the expected value of the leakage response, and up to 6000% in the variance of the leakage response. In all cases, neglecting the second-order sensitivities would erroneously predict a Gaussian distribution in parameter space (for the PERP leakage response) centered around the computed value of the leakage response. In reality, the second-order sensitivities cause the leakage distribution in parameter space to be skewed towards positive values relative to the expected value, which, in turn, is significantly shifted to much larger positive values than the computed leakage value.
The above-mentioned results for the second-order sensitivities indicated that the third-order sensitivities of the PERP leakage response with respect to the total cross sections (which were the most important parameters affecting the PERP leakage response) needed to be computed in order to assess their impact on the response and compare this impact to the corresponding impact of the second-order sensitivities. Recall that the PERP leakage response admits 180 first-order sensitivities and 32,400 s-order sensitivities (of which 16,290 are distinct from one another) with respect to the total cross sections. It follows that there are 5,832,000 third-order sensitivities (of which 988,260 are distinct from each other) of the leakage response (just) with respect to the total cross sections. The results for the third-order sensitivities of the PERP leakage response and their effects on the response’s expected value, variance, and skewness were reported in [
26,
27]. These results indicated that many third-order sensitivities were considerably larger than the corresponding second-order ones, which underscored the need to compute the fourth-order sensitivities corresponding to the largest third-order ones. There are 1,049,760,000 fourth-order sensitivities (of which 45,212,895 are distinct from each other) of the leakage response with respect to the total cross sections. The general methodology for computing fourth-order sensitivities exactly and efficiently while overcoming the curse of dimensionality has been developed by Cacuci [
28] and applied to the PERP benchmark. It was found [
29,
30] that the microscopic total cross sections of isotopes
1H and
239Pu are the most important parameters affecting the PERP benchmark’s leakage response, since they are involved in all of the large values (i.e., greater than 1.0) of the first-, second-, third- and fourth-order unmixed relative sensitivities. In particular, the largest unmixed fourth-order sensitivity is
with respect to
for the 30
th energy group of isotope
1H. The 30
th energy group comprises thermalized neutrons in the energy interval from 1.39 × 10
−4 eV to 0.152 eV. This sensitivity is about 90 times larger than the corresponding largest third-order relative sensitivity, about 6350 times larger than the corresponding largest second-order sensitivity, and about 291,000 times larger than the corresponding largest first-order relative sensitivity. The largest mixed fourth-order relative sensitivity is
, which involves the 30
th energy group of the microscopic total cross sections for isotopes #6 (
1H) and 5 (
12C). Thus, among the 180 microscopic total cross sections, the parameter
, i.e., the total cross sections of isotope #6 (
1H) in the lowest energy group (30
th), is the single most important parameter affecting the PERP benchmark’s leakage response, as it has the largest impact on the various response moments. The comparative impact of the sensitivities of various orders, from first order to fourth order, are summarized in
Section 3.1.2 below.
3.1.2. Overview of the Impacts of the First-, Second-, Third-, and Fourth-Order Sensitivities on Uncertainties Induced in the PERP’s Leakage Response
The effects of the first-, second-, third-, and fourth-order sensitivities on the expected value of the PERP benchmark’s leakage response are presented in
Table 3, considering that the total cross sections are normally distributed (indicated by the superscript “N”) and uncorrelated (indicated by the superscript “U”). As shown in
Table 3, the effects of the second-order and fourth-order sensitivities through
and
to the expected response value
are negligibly small when considering a small relative standard deviation of 1% for each of the uncorrelated microscopic total cross sections of the isotopes included in the PERP benchmark. Considering a moderate (5%) relative standard deviation for each of the microscopic total cross sections, the results presented in
Table 3 show that
indicate that the contributions from the second-order sensitivities to the expected response amount to ca. 65% of the computed leakage value
, and contribute ca. 17% to the expected value
of the leakage response. Furthermore, the fourth-order sensitivities contribute around 56% to the expected value
, since
. Hence, if the computed value,
, is considered to be the actual expected value of the leakage response, neglecting the fourth-order sensitivities would produce an error of ca. 210% for 5% relative standard deviations for uncorrelated total cross sections.
The rightmost column in
Table 3 presents the results when considering a relative standard deviation of 10% for each of the uncorrelated microscopic total cross sections. These results indicate that
, which implies that contributions of the second-order term are about 2.6 times larger than the computed leakage value
, and contribute around 7% to the expected value
. For the fourth-order term, the results in
Table 3 show that
, indicating that the contributions from the fourth-order sensitivities to the expected response are around 34 times larger than the computed leakage value
and contribute about 90% to the expected value
. Thus, for 10% relative standard deviations for uncorrelated total cross sections, the computed value
would be ca. 3400% in error by comparison to the actual expected value of the leakage if the fourth-order sensitivities were neglected.
In summary, the impact of the fourth-order sensitivities on the expected value of the leakage response varies with the value of the standard deviation of the uncorrelated microscopic total cross sections. Generally, the larger the standard deviations of the microscopic total cross sections, the higher the impact of the fourth-order sensitivities will be on the expected value. For a small relative standard deviation of 1% for the parameters under consideration, the impact of the fourth-order sensitivities on the expected response value is smaller than the impact of the lower-order sensitivities. However, for a moderate relative standard deviation of 5%, the contributions from the fourth-order sensitivities are around 56% of the expected value. When the relative standard deviation is increased to 10%, the contributions from the fourth-order sensitivities to the expected value increase to nearly 90%. Notably, for the “RSD = 10%” case, neglecting the fourth-order sensitivities would cause a large error (ca. 3400%) if the computed value were considered to be the actual expected value of the leakage response.
The effects of the first-, second-, third-, and fourth-order sensitivities on the
variance of the leakage response are illustrated by the results presented in
Table 4. These effects are obtained when considering parameter relative standard deviations of 1%, 5%, and 10%, respectively, and considering that the total cross sections are normally distributed and uncorrelated. For uniform relative standard deviations of 1% for the uncorrelated microscopic total cross sections, the results presented in
Table 4 indicate that
,
, and
. These results indicate that, for very small relative standard deviations (e.g., 1%), the contributions from the first-order sensitivities to the response variance are significantly larger (ca. 70%) than those from higher order sensitivities.
For uniform relative standard deviations of 5% for the uncorrelated microscopic total cross sections, the results presented in
Table 4 indicate that
. Thus, the contributions from the third- and fourth-order sensitivities to the response variance are remarkably larger than those from the first- and second-order ones:
,
, and
. Hence, neglecting the fourth-order sensitivities would cause a significant error in quantifying the standard deviation of the leakage response for the PERP benchmark.
The results presented in the last column of
Table 4 indicate that, for relative standard deviations of 10% for the uncorrelated microscopic total cross sections,
,
, and
. Thus, the contributions from the third- and fourth-order sensitivities amount to ca. 99% of the total contribution to the response variance, while the contributions stemming from the first- and second-order sensitivities are negligibly small by comparison. The diverging trend displayed by the standard deviations of the leakage response as the order of sensitivity increases, namely,
, for the “10%” case is similar to the “5% case” above, but with much larger amplitudes.
The effects of the first-, second-, third-, and fourth-order sensitivities on the standard deviations of the leakage response when considering 10% relative standard deviations for the uncorrelated microscopic total cross sections are illustrated in
Figure 2, which presents plots for
,
,
,
, and
. Very importantly, the negative value of
shown in
Figure 2 indicates that considering the computed value
to be the expected value of the leakage response would produce unphysical results for 10% relative standard deviations for the microscopic total cross sections.
In summary, the impact of the sensitivities of various orders on the standard deviation of the leakage response is as follows: (i) for small (1%) relative standard deviations for the microscopic total cross sections, the uncertainty of the leakage response arising solely from the first-order sensitivities are significantly larger than the uncertainties arising solely from the second-, third-, and fourth-order sensitivities, respectively, but following an oscillating pattern: ; (ii) when considering moderate (5%) and larger (10%) relative standard deviations for the microscopic total cross sections, the standard deviation of the leakage response increases substantially as successively higher-order sensitivities are considered, leading to the following sequence of inequalities, , which indicates the need to verify the convergence/divergence of the Taylor series which underlies the expression of the response’s standard deviation.
3.1.3. Effects of the Fourth-Order Sensitivities on the Third-Order Response Moment and Skewness
The results obtained for the third-order response moment and skewness when considering relative standard deviations of 1%, 5%, and 10%, respectively, for the microscopic total cross sections (considered to be normally distributed and uncorrelated) are summarized in
Table 5.
Considering 1% relative standard deviations for the uncorrelated microscopic total cross sections, the results presented in
Table 5 indicate that
,
, and
. Thus, for small (1%) relative standard deviations, the contributions to the third-order response moment
stemming from the second-order sensitivities are the largest (e.g., around 53% in this case), followed by the contributions stemming from the third-order sensitivities, while the contributions stemming from the fourth-order sensitivities are the smallest.
Considering 5% relative standard deviations for the uncorrelated microscopic total cross sections, the results in
Table 5 show that
,
, and
. In this case, the contributions from the third-order sensitivities are the largest (e.g., around 60%), followed by the contributions from the fourth-order sensitivities, which contribute about 30%; the smallest contributions stem from the second-order sensitivities.
Considering 10% relative standard deviations for the uncorrelated microscopic total cross sections, the results in
Table 5 indicate that
,
, and
. These results display the same trends as the results for the “RSD = 5%” case, but the magnitudes of the respective contributions are significantly larger by comparison to the corresponding results for the “RSD = 5%” case.
For the response skewness,
, the results shown in in the last row of
Table 5 indicate that all the second-, third-, and fourth-order sensitivities produce a positive response skewness, which causes the leakage response distribution to be skewed towards the positive direction from its expected value
. The results shown in the last row of
Table 5 also indicate that as the relative standard deviation of the uncorrelated microscopic total cross sections increases from 1% to 10%, the value of the skewness decreases, thus causing the leakage response distribution to become increasingly more symmetrical about the mean value
. Neglecting the fourth-order sensitivities would cause a significant error in the skewness. For example, for the case “RSD = 5%”, if the fourth-order sensitivities were neglected, the contributions from the first-, second-, and third-order sensitivities to the skewness would have the value
, which would be 135% smaller than the more accurate result obtained by including the contributions of the fourth-order sensitivities. The fourth-order sensitivities produce a positive response
skewness, causing the leakage response distribution to be skewed towards the positive direction from its expected value. The impact of the fourth-order sensitivities on the skewness of the leakage response also changes with the value of the standard deviation of the microscopic total cross sections: larger standard deviations (for the parameters) tend to decrease the value of the skewness, causing the leakage response distribution to become more symmetrical about the mean value
.
A “first-order sensitivity and uncertainty quantification” will always produce an erroneous third moment (and, hence, skewness) of the predicted response distribution, unless the unknown response distribution happens to be symmetrical. At least second-order sensitivities must be used in order to estimate the third-order moment (and, hence, the skewness) of the response distribution. With pronounced skewness, standard statistical inference procedures such as constructing a confidence interval for the mean (expectation) of a computed/predicted model response will be not only incorrect, in the sense that the true coverage level will differ from the nominal (e.g., 95%) level, but the error probabilities will be unequal on each side of the predicted mean. Thus, the truncation of Taylor expansion of the response (as a function of parameters) depends both on the magnitudes of the response sensitivities to parameters and the parameter uncertainties involved: if the uncertainties are small, then a fourth-order expansion suffices, in most cases, for obtaining relatively accurate results. In any case, the truncation error of a convergent Taylor series can be quantified a priori. If the parameter uncertainties are large, the Taylor series may diverge, so one would need to reduce such uncertainties, e.g., by performing additional measurements of the respective parameters. Of course, if the parameter uncertainties are large, all statistical methods are doomed to produce unreliable results for large-scale, realistic problems involving many uncertain parameters.
Obtaining the exact and complete set of first-order sensitivities of responses to model parameters is therefore of paramount importance for any analysis of a computational model. The second-order sensitivities contribute the leading correction terms to the response’s expected value, causing it to differ from the response’s computed value. The second-order sensitivities also contribute to the response variances and covariances. If the parameters follow a normal (Gaussian) multivariate distribution, the second-order sensitivities contribute the leading terms to the response’s third-order moment. In particular, the skewness of a single response, , is customarily denoted as , and is defined as follows: where denotes the third central moment of the response distribution. Thus, neglecting the second-order response sensitivities to normally distributed parameters would nullify the third-order response correlations and hence would nullify the skewness of a response. Hence, a “first-order sensitivity and uncertainty quantification” will always produce a zero-valued skewness of the predicted response distribution, which would be erroneous unless the unknown response distribution happens to be symmetrical. At least second-order sensitivities must be used in order to estimate the third-order moment and, hence, the skewness of the response distribution. The skewness provides a quantitative measure of the asymmetries in the respective distribution, indicating the direction and relative magnitude of a distribution’s deviation from the normal distribution. With pronounced skewness, standard statistical inference procedures such as constructing a confidence interval for the mean (expectation) of a computed/predicted model response will be incorrect, in the sense that the true coverage level will differ from the nominal (e.g., 95%) level. Furthermore, the error probabilities will be unequal on each side of the predicted mean.
3.2. Overview of the nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward and Adjoint Linear Systems (nth-CASAM-L)
The n
th-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward and Adjoint Linear Systems (n
th-CASAM-L), which was originally presented by Cacuci [
36], is briefly summarized in this subsection. As was mentioned in
Section 3.1, the mathematical/computational model of a physical system comprises model parameters which are imprecisely known. In addition to the
TP model parameters
described in
Section 3.1, the mathematical model of a physical system is considered to comprise
independent variables which will be denoted as
, and are considered to be the components of a
-dimensional column vector denoted as
, where the sub/superscript “
” denotes the “
Total (number of)
Independent variables” and where the components
, can be considered, without loss of generality, to be real quantities. The vector
of independent variables is considered to be defined on a phase-space domain, denoted as
,
, the boundaries of which may depend on (some of) the model parameters
. The lower boundary point of an independent variable is denoted as
(e.g., the inner radius of a sphere or cylinder, the lower range of an energy variable, the initial time value, etc.), while the corresponding upper boundary point is denoted as
(e.g., the outer radius of a sphere or cylinder, the upper range of an energy variable, the final time value, etc.). The boundary of
is denoted as
; it comprises the set of all of the endpoints
,
.
A linear physical system is generally modeled by a system of coupled linear operator equations which can be generally represented in block-matrix form, as follows:
where
is a
-dimensional column vector of dependent variables and where the sub/superscript “
” denotes the “
Total (number of)
Dependent variables.” The functions
, denote the system’s “dependent variables” (also called “state functions”). The components
,
, of the
-dimensional matrix
are operators that act linearly on the dependent variables
and also depend on the uncertain model parameters
. The components of the
-dimensional column vector
represent inhomogeneous source terms which are considered to depend (nonlinearly) on the model parameters. Since the sources on the right side of Equation (27) may comprise distributions, all equalities that involve differential equations are considered to hold in the weak (i.e., “distributional”) sense.
Since the complete mathematical model is considered to be linear in
, the boundary and/or initial conditions needed to define the domain of
; when
contains differential operators, it must also be linear in
and can be represented as follows:
In (28), the components , of the -dimensional matrix operator act linearly on and nonlinearly on . The total number of boundary and initial conditions is denoted as . The operator is a -dimensional vector comprising inhomogeneous boundary source terms that act nonlinearly on . The subscript “” in Equation (28) indicates boundary conditions associated with the forward state function .
Physical problems modeled by linear systems and/or operators are naturally defined in Hilbert spaces. The dependent variables
, for the physical system represented by Equations (27) and (28) are considered to be square-integrable functions of the independent variables, belonging to a Hilbert space which is denoted as
(where the subscript “zero” denotes “zeroth-level“ or “original”). The Hilbert space
is considered to be endowed with the following inner product, denoted as
, between two elements
and
:
The product notation
in Equation (29) denotes the respective multiple integrals. The linear operator
admits an adjoint operator, which is denoted as
and which is defined through the following relation for a vector
:
In (30), the formal adjoint operator
is the
matrix comprising elements
obtained by transposing the formal adjoints of the forward operators
, i.e.,
Hence, the system adjoint to the linear system represented by (27) and (28) can generally be represented as follows:
When comprises differential operators, the operations (e.g., integration by parts) that implement the transition from the left side to the right side of Equation (30) give rise to boundary terms which are collectively called the “bilinear concomitant.” The domain of is determined by selecting adjoint boundary and/or initial conditions so as to ensure that the bilinear concomitant vanishes when the selected adjoint boundary conditions are implemented together with the forward boundary conditions given in Equation (28). The adjoint boundary conditions thus selected are represented in operator form by Equation (33), where the subscript “” indicates adjoint boundary and/or initial conditions associated with the adjoint state function . In most practical situations, the Hilbert space is self-dual. Solving Equations (27) and (28) at the nominal parameter values yields the nominal value of the forward state function, while solving Equations (32) and (33) at yields the nominal value of the adjoint state function.
The relationship shown in Equation (30), which is the basis for defining the adjoint operator, also provides the following fundamental “reciprocity-like” relation between the sources of the forward and the adjoint equations, cf. Equations (27) and (32), respectively:
The functional on the right side of Equation (34) represents a “detector response”, i.e., a reaction rate between the particles and the medium represented by , which is equivalent to the “number of counts” of particles incident on a detector of particles that “measures” the particle flux . In view of the relation provided in Equation (34), the source term in the adjoint Equation (32) is usually associated with the “result of interest” to be measured and/or computed, which is customarily called the system’s “response.” In particular, if , then , which means that, in such a case, the right side of (34) provides the value of the dependent variable (particle flux, temperature, velocity, etc.) at the point in the phase space where the respective measurement is performed.
Since the linear operators that underly linear systems admit adjoint operators, important model responses of linear systems can be functions of both the forward and the adjoint state functions. Examples of such responses are the Schwinger [
37] and Roussopoulos [
38] functionals, as well as the Raleigh quotient [
39] for computing eigenvalues of linear systems. Such responses cannot be encountered in nonlinear problems since nonlinear operators do not admit adjoint operators. In this sense, linear systems differ fundamentally from nonlinear systems and cannot always be considered to be particular cases of nonlinear systems but need to be treated comprehensively in their own right. Thus, the fundamental form of a response computed using a linear model is an operator which acts nonlinearly on the model’s forward and adjoint state functions, as well as on the imprecisely known model parameters, directly and indirectly through the state functions, having the following generic form:
where
denotes a suitably Gateaux (G)-differential function of the indicated arguments. In general,
is nonlinear in
,
, and
, and the components of
are considered to also include parameters that may specifically appear only in the definition of the response under consideration (but which might not appear in the definition of the model). Thus, the (physical) “system” is defined and understood to comprise both the system’s computational model and the system’s response. Due to the response’s implicit dependence on the model parameters through the forward and/or adjoint state functions, the response of a linear system is a nonlinear function of the model’s parameters, except perhaps in particularly trivial situations. The nominal value of the response,
, is determined by using the nominal parameter values
, the nominal value
of the forward state function, and the nominal value
of the adjoint state function.
Since the model parameters
are imprecisely known quantities, their actual values (which are not known in practice) will differ from their nominal values (which are known in practice) by quantities denoted as
The variations
will cause corresponding variations
,
, in the forward and adjoint state functions, respectively. In turn, all of these variations will cause a response variation
around the nominal value
. The first-order sensitivities of the response with respect to the model parameters are contained within the first-order Gateaux (G)-variation, denoted as
, of the response
for arbitrary variations
in the model parameters and state functions, in a neighborhood
around
. By definition, the first-order Gateaux (G)-variation
is obtained as follows:
where the “direct-effect” term
depends only on the parameter variation
and is defined as follows:
and where the “indirect effect” term
depends only on the variations
and
in the state functions, and is defined as follows:
The notation
has been used in Equations (37) and (38) to indicate that the quantity within the brackets is to be evaluated at the nominal values of the parameters
and state functions. While the “direct effect” term
defined in Equation (37), which depends only on the parameter variations
, can be computed immediately, the “indirect effect” term
defined in Equation (38) can be computed only after having determined the values of the variations
and
. The variations
and
are the solutions of the system of equations obtained by taking the G-differentials of Equations (27), (28), (32), and (33), which can be written in the following matrix-vector form:
where
The matrices
and
, which appear on the right side of Equation (39), are defined as follows:
The system of equations comprising Equations (39) and (40) is called the “1st-Level Variational Sensitivity System” (1st-LVSS) and its solution, , is called the “1st-Level variational sensitivity function”, as indicated by the superscript “(1)”. Since the source term, , is a function of , it follows that the 1st-LVSS would need to be solved times in order to obtain the vector which would correspond to each parameter variation , . The need for computing the functions and can be avoided by applying the First-Order Comprehensive Adjoint Sensitivity Analysis Methodology (1st-CASAM-L), which enables the elimination of and from the expression of the indirect effect term defined in Equation (38). This elimination is achieved by implementing the following sequence of steps:
Introduce a Hilbert space, denoted as
, comprising square-integrable function vector-valued elements of the form
,
,
. The inner product between two elements,
,
, is denoted as
and is defined as follows:
In the Hilbert
, form the inner product of Equation (39) with a vector-valued function
to obtain:
Using the definition of the adjoint operator in the Hilbert space
, recast the left side of Equation (46) as follows:
where
denotes the bilinear concomitant defined on the phase-space boundary
, and where
is the operator formally adjoint to
, defined as follows:
Require the first term on right side of Equation (47) to represent the indirect effect term defined in Equation (38), to obtain the following relation:
where
Implement the boundary conditions given in Equation (40) into Equation (47) and eliminate the remaining unknown boundary values of the functions
and
from the expression of the bilinear concomitant
by selecting appropriate boundary conditions for the function
, to ensure that Equation (49) is well-posed while being independent of
unknown values of
,
, and
. The boundary conditions thus chosen for the function
can be represented in operator form as follows:
The selection of the boundary conditions for the adjoint function represented by Equation (51) eliminates the appearance of the unknown values of in the bilinear concomitant and reduces it to a residual quantity which will be denoted as and which contains boundary terms involving only known values of , , , , and . This residual quantity does not automatically vanish.
The system of equations comprising Equation (49) together with the boundary conditions represented in Equation (51) constitute the First-Level Adjoint Sensitivity System (1st-LASS). The solution of the 1st-LASS is called the First-level adjoint sensitivity function. The 1st-LASS is called “first-level” (as opposed to “first-order”) because it does not contain any differential or functional derivatives, but its solution, , is used below to compute the first-order sensitivities of the response with respect to the model parameters.
It follows from (46) and (47) that the following relation holds:
Recalling from (49) that the first term on the right side of (52) is the indirect effect term
, it follows from (52) that the indirect effect term can be expressed in terms of the first-level adjoint sensitivity function
as follows:
As indicated by the identity relation in Equation (53), the variations and have been eliminated from the original expression of the indirect effect term, which now instead depends on the first-level adjoint sensitivity function . Very importantly, the 1st-LASS is independent of parameter variations . Hence, the 1st-LASS needs to be solved only once (as opposed to the 1st-LVSS or the 1st-OFSS, which would need to be solved anew for each parameter variation) to determine the first-level adjoint sensitivity function . Subsequently, the indirect effect term is computed efficiently and exactly by simply performing the integrations over the adjoint function , as indicated by Equation (53).
Adding the expression of the direct effect term defined in Equation (37) to the expression of the indirect effect term obtained in Equation (53) yields the following expression for the total first-order G-differential
of the response
computed at the nominal values for the models’ parameter and state functions:
The expression obtained in Equation (54) no longer depends on the variations
and
but instead depends on the first-level adjoint sensitivity function
. In particular, the expression in Equation (54) reveals that the sensitivities of the response
to parameters that characterize the system’s boundary and/or internal interfaces can arise both from the direct effect and indirect effect terms. It also follows from Equation (54) that the quantity
has become a total differential in the parameter variation
so it can be expressed in the following form:
where the quantities
represent the first-order sensitivities (i.e., partial functional derivatives) of the response with respect to the model parameter
, evaluated at the nominal values of the state functions and model parameters. Furthermore, each of the first-order sensitivities
can be represented in integral form as follows:
In particular, if the residual bilinear concomitant does not vanish, the functions would contain suitably defined Dirac delta functionals for expressing the respective non-zero boundary terms as volume integrals over the phase space of the independent variables. Dirac delta functionals would also be needed to represent, within , the terms containing the derivatives of the boundary endpoints with respect to the model and/or response parameters.
Cacuci [
36] has obtained the general expression of the arbitrarily high-order (
nth-order, where
n is an arbitrary integer) sensitivities of the response
to model parameters by treating each sensitivity as a response, applying the principles outlined in items (1)−(9) to successively higher-order sensitivities and, ultimately, using mathematical induction to demonstrate that general expression thus obtained for the
nth-order sensitivity is correct. Each of the
(n − 1)th-order sensitivities
of the response
with respect to the parameters
, where
, are used in the role of a “model response”, for which the first-order G-differential is obtained by applying the principles outlined in items (1)−(9), thereby ultimately obtaining the general expression of the
nth-order sensitivities of the response
. The details of the n
th-CASAM-L methodology are provided by Cacuci [
36]. The end results produced by the n
th-CASAM-L methodology enable the exact and efficient computation—overcoming the curse of dimensionality in sensitivity analysis—of the n
th-order partial sensitivities of the response
with respect to the model parameters, using the expression provided below:
The various quantities which appear in Equation (57) are defined as follows:
- (1)
The quantity
denotes the n
th-order partial sensitivity of the response
with respect to the model parameters, evaluated at the nominal values of the parameters and state functions. When the symmetries among the various partial derivatives/sensitivities are considered, the respective indices have ranges as follows:
;
; …;
. In integral form, the n
th-order partial sensitivity of the response can be written as follows:
- (2)
The
nth-level state
, which appears in the arguments of
, is defined as follows:
- (3)
The inner product which appears in Equation (57) is defined in a Hilbert space denoted as
, comprising
-dimensional block vectors
having the following structure:
,
,
. The inner product between two elements,
and
, of the Hilbert space
, is denoted as
and is defined as follows:
- (4)
The adjoint operator
which appears on the left side of Equation (62) is constructed by using the inner product defined in Equation (60), via the following relation:
- (5)
The
nth-level adjoint sensitivity function
, which appears in the arguments of
, is the solution of the following
nth-level adjoint sensitivity system
nth-LASS, for
;
; …
:
where the vector
,
comprises
components defined for each
;
; …;
, as follows:
- (6)
The quantity
denotes the residual bilinear concomitant defined on the phase-space boundary
, evaluated at the nominal values of the model parameter and respective state functions.