The response
admits six non-zero sensitivities, as obtained in Equations (58)–(63). In principle, each of these non-zero sensitivities would be considered a response. Each of these responses would give rise to a 2nd-level adjoint sensitivity system (2nd-LASS), which means that, in principle, there would be six such systems to be solved to obtain the 2nd-level adjoint sensitivity function that would correspond to the respective response. However, writing Equation (53) in the following form:
reveals that the indirect-effect terms for all of the 2nd-order sensitivities will arise from only two functionals that depend on the state functions, namely
, which underlies the three non-zero first-order sensitivities with respect to the parameters comprising the initial power
, and
, which underlies the remaining three non-zero first-order sensitivities. As the number of large-scale computations arises from solving the 2nd-LASS and as each indirect-effect term gives rise to a 2nd-LASS, it follows from Equation (64) that the number of 2nd-LASS that would need to be solved can be reduced to just two (from the possible total of six), thereby reducing the computational work by a factor of three.
4.1. Second-Order Sensitivities Stemming from the First-Order Sensitivities ,
In this work, the index
will be used to enumerate the 1st-order sensitivities of
with respect to the six model parameters. Recalling the definitions of the first three model parameters, i.e.,
,
, and
, it follows from Equation (64) that the 1st-order sensitivities of the released energy
to the parameters underlying the reactor’s initial power
can be written in the following form:
The 2nd-order sensitivities which stem from the 1st-order ones defined in Equation (65) are obtained from the first-order G-differential
, which is obtained, by definition, as follows:
where
The direct-effect term defined in Equation (67) can be computed immediately. The functional
defined in Equation (68) can be determined only after having computed the variational function
, which is the solution of the system of equations obtained by G-differentiating the 1st-LASS defined by Equations (51) and (52). Performing the G-differentiation of the 1st-LASS yields the following equations:
Concatenating Equations (69) and (70) with the 1st-LVSS for
defined in Equations (49) and (50) yields the following 2nd-Level Variational Sensitivity System (2nd-LVSS) for the 2nd-level variational function
:
where
The need for solving the 2nd-LVSS is circumvented by deriving an alternative expression for the functional
defined in Equation (68), in which the variational function
is replaced by a 2nd-level adjoint function which will be denoted as
and which will be the solution of a 2nd-Level Adjoint Sensitivity System (2nd-LASS) to be constructed by applying the 2nd-CASAM-N. The 2nd-LASS is constructed in a Hilbert space, denoted as
, which comprises as elements vectors of the same form as
, and is endowed with the following inner product of two vectors
and
:
The inner product defined in Equation (76) is used to construct the 2nd-Level Adjoint Sensitivity System (2nd-LASS) for the 2nd-level adjoint function , as follows:
(i) Using Equation (76), form the inner product of
with Equation (71) to obtain the following relation:
The notation for has the following significance: (i) the letter “A” indicates “adjoint”; (ii) the superscript “(2)” indicates “second-level”; (iii) the first argument, i.e., “2”, indicates that this vector has 2 components; (iv) the second argument, i.e., “”, indicates that this adjoint vector will correspond to the first three 1st-order sensitivities under consideration, in this case , . In the most general case, when all sensitivities have distinct “indirect-effect terms”, there will be a distinct 2nd-level adjoint sensitivity vector of the same type as , corresponding for each 1st-order sensitivity. Each of the components , , of are scalar-valued functions of time. The index will be omitted, for simplicity, in the derivations to follow below, but will be reinstated after obtaining the final closed-form expressions for the components .
(ii) Eliminate the boundary terms on the right side of Equation (77) and require the term on the right side of the second equality in Equation (77) to represent the functional
, by imposing the following relations:
where:
The relations represented by Equations (78) and (79) constitute the 2nd-LASS for the 2nd-level adjoint function .
(iii) Use the relations provided in Equations (68) and (77) together with the 2nd-LASS to obtain the following expression for the functional
in terms of the 2nd-level adjoint function
:
where
In Equation (81), the 1st-level adjoint sensitivity function
is the solution of the 1st-LASS comprising Equations (51) and (52), while the 2nd-level adjoint sensitivity function
is the solution of the 2nd-LVSS comprising Equations (78) and (79). Notably, the 2nd-LASS comprises Equations (78) and (79) is independent of parameter variations, so it needs to be solved just once to obtain the 2nd-level adjoint sensitivity function
. Furthermore, the 2nd-LASS is an upper-triangular system, so the equations need not solved simultaneously, but can be solved sequentially, first for the component
and subsequently for the component
. This procedure yields the following closed-form expressions for the components of the 2nd-level adjoint sensitivity function:
The components of are to be evaluated at the nominal parameter values, but the notation has been omitted for simplicity.
Collecting the results obtained in Equations (67), (68) and (81) yields the expressions for the 2nd-order sensitivities, which stem from the first-order sensitivities , , as presented below.
4.1.1. Second-Order Sensitivities Stemming from
Collecting the results for
in Equations (67), (68), (81) and using the expressions provided in Equations (31) and (32) yields the following expression for the 2nd-order partial differential stemming from
:
Collecting the terms that multiply the same parameter variations on the left side and, respectively, right side of Equation (85) yields the following expressions for the respective 2nd-order partial sensitivities:
4.1.2. Second-Order Sensitivities Stemming from
Collecting the results for
in Equations (67), (68), (81) and using the expressions provided in Equations (31) and (32) yields the following expressions for the 2nd-order partial differential stemming from
:
Collecting the terms that multiply the same parameter variations on the left side and, respectively, right side of Equation (92) yields the following expressions for the respective 2nd-order partial sensitivities:
4.1.3. Second-Order Sensitivities Stemming from
Collecting the results for
in Equations (67), (68), (81), and using the expressions provided in Equations (31) and (32) yields the following expressions for the 2nd-order partial differential stemming from
:
Collecting the terms that multiply the same parameter variations on the left side and, respectively, right side of Equation (99) yields the following expressions for the respective 2nd-order partial sensitivities:
4.2. Second-Order Sensitivities Stemming from the First-Order Sensitivities ,
Recalling the definitions of the first three model parameters, i.e.,
,
, and
, it follows from Equation (64) that the 1st-order sensitivities
,
and
can be written in the following form:
The 2nd-order sensitivities which stem from the 1st-order ones defined in Equation (106) are obtained by applying the definition of the first-order G-differential to Equation (106), which yields the following relations:
where
The direct-effect term defined in Equation (108) can be computed immediately. The functional defined in Equation (109) can be determined only after solving the 2nd-Level Variational Sensitivity System (2nd-LVSS) defined by Equations (71) and (72) to obtain the 2nd-level variational function .
As before, the need for solving the 2nd-LVSS is circumvented by deriving an alternative expression for the functional , in which the variational function is replaced by a 2nd-level adjoint function denoted as , which will be the solution of a 2nd-Level Adjoint Sensitivity System (2nd-LASS) to be constructed by applying the 2nd-CASAM-N. The 2nd-LASS is constructed in the same Hilbert space, which was denoted as in the previous subsection, and which is endowed with the inner product defined in Equation (76). This inner product is used to construct the 2nd-Level Adjoint Sensitivity System (2nd-LASS) for the 2nd-level adjoint function , as follows:
(i) Using Equation (76), form the inner product of
with Equation (71) to obtain the following relation, which has the same form as shown in Equation (77), namely:
where the operator
has the same expression as defined in Equation (80).
(ii) Eliminate the boundary terms on the right side of Equation (110) and require the term on the right side of the second equality in Equation (110) to represent the functional
, by imposing the following relations:
The relations represented by Equations (111) and (112) constitute the 2nd-LASS for the function, the 2nd-level adjoint function . Notably, the 2nd-LASS is independent of parameter variations, so it needs to be solved just once to obtain . Furthermore, the 2nd-LASS is an upper-triangular system, so the equations need not be solved simultaneously but can be solved sequentially, first for the component and subsequently for the component .
(iii) Use the relations provided in Equations (110) and (109) together with Equations (111) and (112) to obtain the following expression for the functional
in terms of the 2nd-level adjoint function
:
where
Solving Equations (111) and (112) yields the following closed-form expressions for the components of the 2nd-level adjoint sensitivity function
:
Collecting the results obtained in Equations (108), (109), and (113) yields the expressions for the 2nd-order sensitivities, which stem from the first-order sensitivities , , as presented below.
4.2.1. Second-Order Sensitivities Stemming from
Collecting the results for
in Equations (107)−(109), (113), and using the expressions provided in Equations (31) and (32) yields the following expressions for the 2nd-order partial differential stemming from
:
Collecting the terms that multiply the same parameter variations on the left side and, respectively, right side of Equation (117) yields the following expressions for the respective 2nd-order partial sensitivities:
4.2.2. Second-Order Sensitivities Stemming from
Collecting the results for
in Equations (107)–(109), (113), and using the expressions provided in Equations (31) and (32) yields the following expressions for the 2nd-order partial differential stemming from
:
Collecting the terms that multiply the same parameter variations on the left side and, respectively, right side of Equation (124) yields the following expressions for the respective 2nd-order partial sensitivities:
4.2.3. Second-Order Sensitivities Stemming from
Collecting the results for
in Equations (107)–(109), (113), and using the expressions provided in Equations (31) and (32) yields the following expressions for the 2nd-order partial differential stemming from
:
Collecting the terms that multiply the same parameter variations on the left side and, respectively, right side of Equation (131) yields the following expressions for the respective 2nd-order partial sensitivities:
4.3. Computational Advantages of Using the 2nd-CASAM-N
“Large-scale” computations are those needed to solve differential and/or integral equations, such as those underlying the original model and the 1st-LASS. By comparison, the computational effort involved in evaluating integrals by means of quadrature formulas are “small-scale”. The application of the 1st-CASAM-N has shown that a single large-scale computation needed to solve the 1st-LASS to obtain the 1st-level adjoint sensitivity function suffices to obtain all of the 1st-order sensitivities of a model response with respect to the underlying model parameters, which are computed using quadrature formulas to evaluate integrals involving the 1st-level adjoint sensitivity function. Using any other methods (e.g., statistical methods or finite-difference methods) would require at least as many large-scale computations—for solving the original model with altered parameter values—as there are model parameters.
Each of the first-order sensitivities becomes the “model response” for the application of the 2nd-CASAM-N. If all of the first-order sensitivities have differing (among each other) functional dependencies on the original state functions and 1st-level adjoint sensitivity functions, then there will be as many 2nd-Level Adjoint Systems to be solved as there are 1st-order sensitivities. Notably, all of these 2nd-LASS have the same left side; only the sources on the right sides of these 2nd-LASS would differ from each other, each source stemming from one of the distinct 1st-level sensitivities. Thus, the same software package would be used to invert the (matrix-valued) operator on the left side of the 2nd-LASS.
In most practical situations, however, the 1st-order sensitivities do share common expressions involving the original state functions and 1st-level adjoint sensitivity functions. For the paradigm Nordheim–Fuchs model, the sensitivities
,
, have in common the functional
, while the sensitivities
,
, have in common the functional
. In such cases, the number of 2nd-level adjoint sensitivity functions (and corresponding large-scale computations) is reduced considerably; in the case of the Nordheim–Fuchs model, the number of large-scale computations is reduced by a factor of 3 (from 6 to 2), as illustrated in
Section 4.2.
As has been illustrated in
Section 4.2, one adjoint computation for solving the 2nd-LASS for a selected 1st-order sensitivity provides the 2nd-level adjoint needed for computing all of the partial 2nd-order sensitivities stemming from the selected 1st-order sensitivities. The order of priority of computing the 2nd-order sensitivities should be established in the ranking order of the magnitude of the 1st-order sensitivities: thus, the 2nd-order sensitivities stemming from the largest (in absolute value) 1st-order sensitivity should be computed with the highest priority, the 2nd-order sensitivities stemming from the next-largest (in absolute value) 1st-order sensitivity should be computed next, and so on. The user may decide if any of the 1st-order sensitivities would be sufficiently insignificant to be neglected in this process.
As there are 6 first-order sensitivities, there will be 36 second-order sensitivities, of which 21 are distinct from one another. As illustrated by the results obtained in
Section 4.2, the unmixed 2ndorder sensitivities of the form
,
, have individually distinct expressions, each involving the 2nd-level adjoint sensitivity function corresponding to the originating 1st-order sensitivities. In contradistinction, the mixed 2nd-order sensitivities
,
, are obtained twice, using distinct (in general) 2nd-level adjoint systems. Occasionally, although obtained following two different computational paths, some of these mixed sensitivities have the same expression, as exemplified by the 2nd-order sensitivities, which involve the functionals
and
, respectively, as shown below:
In most cases, however, the mixed sensitivities have distinct expressions involving distinct adjoint functions, as is apparent from the results obtained in
Section 4.2. In all cases, though, the symmetry property
provides an intrinsic verification procedure for assessing the accuracy of computing the respective 2nd-level adjoint sensitivity functions. Furthermore, the user can select which of the alternative −but equivalent− expressions of the 2nd-order mixed sensitivity under consideration is computationally more advantageous to use. For example, the expressions in Equations (118) and (89) must be equivalent, i.e.,
Consequently, either of the two equivalent expressions above can be used to evaluate but the expression involving the integral over appears to be simpler to compute. Similar considerations apply to the remaining mixed 2nd-order sensitivities.
In summary, the application of the 2nd-CASAM-N has necessitated 2 large-scale computations (for solving the two 2nd-LASS) to obtain all of the 36 second-order sensitivities for the Nordheim–Fuchs reactor safety model. Using any other methods (e.g., statistical or finite-differences) would have required ca. 5 × 36 large-scale computations for solving the original model with altered parameter values, as needed for the respective finite-difference or statistical schemes.