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ORIGINAL RESEARCH article

Front. Energy Res., 05 September 2024
Sec. Nuclear Energy
This article is part of the Research Topic Novel Nuclear Reactors and Research Reactors View all 11 articles

nth-order feature adjoint sensitivity analysis methodology for response-coupled forward/adjoint linear systems: II. Illustrative application to a paradigm energy system

  • Department of Mechanical Engineering, University of South Carolina, Columbia, SC, United States

This work presents a representative application of the newly developed “nth-order feature adjoint sensitivity analysis methodology for response-coupled forward/adjoint linear systems” (abbreviated as “nth-FASAM-L”), which enables the most efficient computation of exactly obtained mathematical expressions of arbitrarily high-order (nth-order) sensitivities of a generic system response with respect to all of the parameters (including boundary and initial conditions) underlying the respective forward/adjoint systems. The nth-FASAM-L has been developed to treat responses of linear systems that simultaneously depend on both the forward and adjoint state functions. Such systems cannot be considered particular cases of nonlinear systems, as illustrated in this work by analyzing an analytically solvable model of the energy distribution of the “contributon flux” of neutrons in a mixture of materials. The unparalleled efficiency and accuracy of the nth-FASAM-L stem from the maximal reduction in the number of adjoint computations (which are “large-scale” computations) for determining the exact expressions of arbitrarily high-order sensitivities since the number of large-scale computations when applying the nth-FASAM-N is proportional to the number of model features as opposed to the number of model parameters (which are considerably more than the number of features). Hence, the higher the order of computed sensitivities, the more efficient the nth-FASAM-N becomes compared to any other methodology. Furthermore, as illustrated in this work, the probability of encountering identically vanishing sensitivities is much higher when using the nth-FASAM-L than other methods.

1 Introduction

The accompanying work (“part I”) has presented the newly developed mathematical framework, the “nth-order feature adjoint sensitivity analysis methodology for response-coupled forward/adjoint linear systems” (abbreviated as “nth-FASAM-L”), conceived by Cacuci (2024c). This work illustrates the application of the nth-FASAM-L to a representative energy-dependent neutron-slowing down model of fundamental importance to reactor physics. The physical considerations underlying this model are presented in Section 2, which briefly reviews the concept of “contributon-flux density response” and particularizes this concept within the modeling of neutron slowing down in a mixture of materials. This physical model is of fundamental importance in nuclear reactor physics and enables the derivation of exact closed-form results for the application of the nth-FASAM-L. Section 2 also defines the “features” inherent to this model, which enable the advantageous application of the nth-FASAM-L. By definition, there are considerably fewer “feature functions” of the primary model parameters than there are primary model parameters.

Section 3 presents the first-order adjoint sensitivity analysis of the contributon flux with respect to the features and primary model parameters of the slowing-down model, comparing the application of the 1st-FASAM-L versus the first-order comprehensive adjoint sensitivity analysis methodology for response-coupled forward/adjoint linear systems (1st-CASAM-L). Using either the 1st-FASAM-L or 1st-CASAM-L involves solving the same operator equations and boundary conditions within the respective 1st-LASS but with differing source terms. For the computation of the first-order sensitivities, the 1st-FASAM-L enjoys only a slight computational advantage since it requires only one quadrature per component of the feature function, whereas the 1st-CASAM-L requires one quadrature per primary model parameter.

Section 4 presents the second-order adjoint sensitivity analysis of the contributon flux with respect to the features and primary model parameters of the slowing-down model, comparing the application of the 2nd-FASAM-L versus the 2nd-CASAM-L. It is shown that the 2nd-FASAM-L requires as many large-scale “adjoint” computations as there are non-vanishing first-order response sensitivities with respect to the components of the feature functions, whereas the 2nd-CASAM-L requires as many large-scale computations as there are non-vanishing first-order response sensitivities with respect to the primary model parameters. Hence, the 2nd-FASAM-L is inherently more efficient than the 2nd-CASAM-L. In particular, one of the three distinct second-order sensitivities with respect to the model’s features vanishes identically within the 2nd-FASAM-L but none of the ca. 100 second-order sensitivities with respect to the primary model parameters vanish within the 2nd-CASAM-L.

Section 5 presents the third-order adjoint sensitivity analysis of the contributon flux with respect to the features and primary model parameters of the slowing-down model, comparing the application of the 3rd-FASAM-L versus the 3rd-CASAM-L. For computing the exact expressions of the third-order contributon-response sensitivities, the 3rd-FASAM-L requires only two large-scale computations, whereas the 3rd-CASAM-L would require hundreds of large-scale computations.

The concluding discussion presented in Section 6 emphasizes the fact that the unparalleled efficiency of the nth-FASAM-N increases as the order of computed sensitivities increases, and the probability of encountering vanishing sensitivities is much higher when using the nth-FASAM-L rather than any other methodology. Both the nth-FASAM-L and nth-CASAM-L overcome the limitation of dimensionality in the sensitivity analysis of linear systems, being incomparably more efficient and more accurate than any other method (statistical, finite differences, etc.) for computing exact expressions of response sensitivities (of any order) with respect to the uncertain parameters, boundaries, and internal interfaces of the model.

2 Modeling the contributon flux in a paradigm neutron slowing-down model

Fundamentally important responses of linear models depend simultaneously on both the forward and adjoint state functions governing the respective linear model, which makes it necessary to treat linear models/systems in their own right since such responses cannot be treated as particular cases of responses of nonlinear models. Typical examples of such responses arise in the modeling of self-diffusion processes in which the interaction mean free path is independent of the phase-space density. Such processes are modeled by linear equations of the Lorentz–Boltzmann type, and they occur in neutron, electron, and photon transport through media, as well as in certain types of transport processes in gas or plasma dynamics. Numerically solving such time-dependent integro-differential equations, albeit linear, is representative of “large-scale” computations and will be used in the sequel for illustrating the application of the nth-FASAM-L. In particular, the distribution of neutrons in a medium is modeled by the following standard form of the linear Boltzmann equation:

Lr,E,Ω,tφr,E,Ω,t=Qr,E,Ω,t,(1)

where the linear integro-differential operator Lr,E,Ω,t is defined below:

Lr,E,Ω,tφr,E,Ω,t1vφr,E,Ω,tt+Ω·φr,E,Ω,t+Σtr,Eφr,E,Ω,t0EfdE4πdΩΣsr,EE,ΩΩφr,E,Ω,t0EfdE4πχr,EEνΣr,Eφr,E,Ω,tdΩ.(2)

The quantities that appear in the standard notation used in Equation 2 are defined as follows:

(i) r denotes the three-dimensional position vector in space; E denotes the energy-independent variable; the directional vector Ω denotes the scattering solid angle; t denotes the time-independent variable; and v denotes the neutron particle speed.

(ii) φr,E,Ω,t denotes the flux of particles (i.e., particle number density multiplied by the particle speed) in the energy range dE about E and volume element dr about r, with directions of motion in the solid angle element dΩ about Ω.

(iii) Qr,E,Ω,t denotes the rate at which particles are produced in the same element of phase space from sources that are independent of the flux.

(iv) Σtr,E denotes the macroscopic total cross section.

(v) Σsr,EE,ΩΩ denotes the macroscopic scattering transfer cross section from energy E to energy E and from a scattering angle through angle Ω·Ω.

(vi) ν denotes the number of particles emitted isotropically 1/4π per fission.

(vii) Σfr,E denotes the macroscopic fission cross section.

(viii) χr,EE denotes the fraction of fission particles appearing in energy dE about E from fissions in dE about E.

The adjoint Boltzmann transport equation is formulated in the Hilbert space denoted as HB and is endowed with the following inner product, denoted as φr,E,Ω,t,ψr,E,Ω,tB, between two elements φr,E,Ω,tHB and ψr,E,Ω,tHB:

φ,ψB0tfdt0dE4πdΩVdVφr,E,Ω,tψr,E,Ω,t.(3)

In the Hilbert space HB, the generic adjoint Boltzmann transport equation is as follows:

L*r,E,Ω,tψr,E,Ω,t=Q*r,E,Ω,t,(4)

where the (adjoint) linear integro-differential operator L*r,E,Ω,t is defined below:

L*r,E,Ω,tψr,E,Ω,t1vψr,E,Ω,ttΩ·ψr,E,Ω,t+Σtr,Eψr,E,Ω,t0EfdE4πdΩΣsr,EE,ΩΩψr,E,Ω,tνΣr,E0EfdE4πχr,EEψr,E,Ω,tdΩ.(5)

By construction, the forward and adjoint transport equations satisfy the following relation:

φ,L*ψBψ,LφB=Pφ,ψ=φ,Q*Bψ,QB,(6)

where Pφ,ψ denotes the bilinear concomitant evaluated on the boundary of the phase-space domain under consideration. The “generalized reciprocity relation” expressed by Equation 6 relates the bilinear concomitant, which is a functional of the forward and adjoint fluxes at the initial and final times along the incoming and outgoing directions at the surface of the medium, to the fluxes in the interior of the medium comprising fixed sources. This reciprocity relation provides a physical interpretation of the adjoint flux as an “importance function,” which quantifies the contribution of a source to a detector and enables transport problems to be posed either in the forward or adjoint descriptions. These reciprocity relations also restrict the combination of forward and adjoint boundary conditions to those that ensure both the forward and adjoint formulations are mathematically “well posed.” The reciprocity relation expressed by Equation 6 is extensively used in the so-called “source-detector” problems in steady-state subcritical systems, where Q*r,E,Ω models the detector properties (cross section) in the sub-region occupied by the respective detector.

When the boundary conditions for Equation 1 are homogeneous and there is no external source, i.e., when Qr,E,Ω,t=0, the stationary neutron transport problem becomes an eigenvalue problem. The largest (i.e., fundamental) eigenvalue in such a case is called the “effective multiplication factor” and, depending on its value, corresponds to a critical, subcritical, or supercritical physical system (e.g., nuclear reactor). This eigenvalue (multiplication factor) is an important system (model) response, and its mathematical expression is a functional (“Raleigh quotient”) of the forward and the adjoint fluxes. Additional important model responses that are functionals of both the forward and adjoint fluxes include the reactivity, generation time, and lifetime of the system, along with several other Lagrangian functionals used in variational principles for developing efficient Raleigh–Ritz type numerical methods (see, e.g., Lewins, 1965; Stacey, 1974; Stacey, 2001). Perhaps the simplest quantity that depends on both the forward and adjoint fluxes—and has important applications in particle transport (particularly in particle shielding)—is the so-called “contributon flux” (Williams and Engle, 1977), which arises as follows:

(i) Multiplying the stationary form of Equation 1 by ψr,E,Ω, multiplying the stationary form of Equation 4 by φr,E,Ω, subtracting the resulting equations from each other, and integrating the resulting equation over only the energy- and solid angle-independent variables yield the following relation:

·vcRr=SrS*r,(7)

where

Rcr1v0EfdE4πdΩφr,E,Ωψr,E,Ω,(8)
vc0EfdE4πdΩΩφr,E,Ωψr,E,Ω1v0EfdE4πdΩφr,E,Ωψr,E,Ω,(9)
Scr0EfdE4πdΩQr,E,Ωψr,E,Ω,(10)
Sc*r0EfdE4πdΩQ*r,E,Ωφr,E,Ω.(11)

(ii) The form of Equation 7 is the same as the mass continuity balance/equation for compressible flow, indicating that the “contributon response density” Rcr is conserved as it flows from the “contributon response source” Scr toward the “contributon response sink” Sc*r, with a “contributon response mean velocity” vc corresponding to the neutron speed v.

The application of the nth-FASAM-L is illustrated in this section by considering the simplified model of the distribution in the asymptotic energy range of neutrons produced by a source of neutrons placed in an isotropic medium comprising a homogeneous mixture of “M” non-fissionable materials having constant (i.e., energy-independent) properties. For simplicity, but without diminishing the applicability of the nth-FASAM-L, this medium is considered to be infinitely large. The simplified form of the Boltzmann neutron transport equation, as shown in Equation 1, that models the energy distributions of neutrons within a mixture of materials is called the “neutron slowing-down equation.” This equation is written using neutron lethargy (rather than the neutron energy) as the independent variable. Neutron lethargy is customarily denoted using the variable/letter “u” and is defined as ulnE0/E, where E denotes the energy variable and E0 denotes the highest energy in the system. Thus, the neutron slowing-down model (see, e.g., Meghreblian and Holmes, 1960; Lamarsh, 1966) for the energy distribution of the neutron flux in a homogeneous mixture of non-fissionable materials of infinite extent takes the following simplified form of Equation 1:

dφudu+Σaξ¯Σtφu=Suξ¯Σt;0<uuth;(12)
φ0=0;atu=0.(13)

The quantities that appear in Equation 12 are defined as follows.

(i) The lethargy-dependent neutron flux is denoted as φu; uth denotes a cut-off lethargy, usually corresponding to the thermal neutron energy (ca. 0.0024 electron volts).

(ii) The macroscopic elastic scattering cross section for the homogeneous mixture of “M” materials is denoted as Σs and is defined as follows:

Σsi=1MNmiσsi,(14)

where σsi,i=1,...,M denotes the elastic scattering cross section of material “i,” and the atomic or molecular number density of material “i” is denoted as Nmi,i=1,...,M and is defined as NmiρiNA/Ai, where NA is Avogadro’s number 0.602×1024nuclei/mole, while Ai and ρi denote the mass number and density of the material, respectively.

(iii) The average gain in lethargy of a neutron per collision is denoted as ξ¯ and is defined as follows for the homogeneous mixture:

ξ¯1Σsi=1MξiNmiσsi;ξi1+ailnai1ai;aiAi1Ai+12.(15)

(iv) The macroscopic absorption cross section is denoted as Σa and is defined as follows for the homogeneous mixture:

Σai=1MNmiσγi,(16)

where σγi,i=1,...,M denotes the microscopic radiative-capture cross section of material “i.”

(v) The macroscopic total cross section is denoted as Σt and is defined as follows for the homogeneous mixture:

ΣtΣa+Σs.(17)

(vi) The source Su is considered to be a simplified “spontaneous fission” source stemming from fissionable actinides, such as 239Pu and 240Pu, emitting monoenergetic neutrons at the highest energy (i.e., zero lethargy). Such a source is comprised within the OECD/NEA polyethylene-reflected plutonium (PERP) OECD/NEA reactor physics benchmark (Valentine, 2006; Cacuci and Fang, 2023), which can be modeled using the following simplified expression:

Su=S0δu;S0k=12λkSNkSFkSνkSWkS,(18)

where the superscript “S” indicates the “source; ” the subscript index k = 1 indicates material properties pertaining to the isotope 239Pu; the subscript index k = 2 indicates material properties pertaining to the isotope 240Pu; λkS denotes the decay constant; NkS denotes the atomic density of the respective actinide; FkS denotes the spontaneous fission branching ratio; νkS denotes the average number of neutrons per spontaneous fission; and WkS denotes a function of parameters used in Watt’s fission spectrum to approximate the spontaneous fission neutron spectrum of the respective actinide. The detailed forms of the parameters WkS are unimportant for illustrating the application of the nth-FASAM-L. The nominal values for these imprecisely known parameters are available from a library file contained in SOURCES 4C (Wilson et al., 2002).

Mirroring the considerations for the Boltzmann transport equation presented in Equations 16, the “adjoint slowing-down model” is constructed in the Hilbert space HB of square-integrable functions φuHB and ψuHB endowed with the following inner product, denoted as φu,ψuB:

φu,ψuB0uthφuψudu.(19)

Using the inner product φu,ψuB defined in Equation 19, the adjoint slowing-down model is constructed by the usual procedure, i.e., by (i) constructing the inner product of Equation 12 with a function ψuHB; (ii) integrating by parts the resulting relation so as to transfer the differential operation from the forward function φu onto the adjoint function ψu; (iii) using the initial condition provided in Equation 13 and eliminating the unknown function φuth by choosing the final-value condition ψuth=0; and (iv) choosing the source for the resulting adjoint slowing-down model so as to satisfy the generalized reciprocity relation shown in Equation 6. The result of these operations is the following adjoint slowing-down model for the adjoint slowing-down function ψu:

dψudu+f1αψu=δuud,(20)
ψuth=0,atu=uth.(21)

The “contributon-flux response density” Rcφ,ψ, as generally defined in Equation 8, specialized for the neutron slowing-down model, coincides with the inner product used in this context, i.e.,

Rcφ,ψ0uthφuψuduφu,ψuB.(22)

It is important to note that Rcφ,ψ does not depend explicitly on either the feature function fα or any primary model parameter. Therefore, the G-differential of Rcφ,ψ will not comprise a direct-effect term but will consist entirely of the indirect-effect term.

For this “contributon-flux response density” model, the following primary model parameters are subject to experimental uncertainties.

(i) For each material “i,i=1,...,M, included in the homogeneous mixture, the following are primary model parameters: the atomic number densities Nmi; the microscopic radiative-capture cross section σγi; and the scattering cross section σsi;

(ii) The source parameters λkS, NkS, FkS, νkS, and WkS, for k = 1,2.

The above primary parameters are considered to constitute the components of a “vector of primary model parameters” defined as follows:

αNm1,σγ1,σs1,...,NmM,σγM,σsM,λ1S,λ2S,N1S,N2S,F1S,F2S,ν1S,ν2S,W1S,W2Sα1,,αTP;TP3M+10.(23)

The first-level forward/adjoint system (1st-LFAS) for the “first-level forward/adjoint function” u12;uφu,ψu comprises Equations 12, 13, 20, and 21. The structure of the 1st-LFAS suggests that the components fiα of the feature function fα can be defined as follows:

fαf1α,f2α;f1αΣaαξ¯αΣtα;f2αS0αξ¯αΣtα.(24)

Solving Equations 12, 13 while using the definitions introduced in Equation 24 yields the following expression for the flux φu in terms of the components fiα of the feature function fα:

φu=Huf2αexpuf1α;H0=0;Hu=1,ifu>0.(25)

Solving the above adjoint slowing-down model yields the following closed-form expression for the adjoint slowing-down function ψu:

ψu=Huduexpuudf1α.(26)

In terms of the components fiα of the feature function fα, the closed-form expression of the “contributon response density” is obtained by substituting the expressions provided in Equations 25, 26 into Equation 22 and performing the integration over lethargy, which yields

Rcφ,ψ=0uthHuf2αexpuf1αHuduexpuudf1αdu=udf2αexpudf1α.(27)

In terms of the primary model parameters, the closed-form expression of the “contributon response density” is

Rcφ,ψ=udS0αξ¯αΣtαexpudΣaαξ¯αΣtα.(28)

As Equation 28 indicates, the model response can be considered to depend directly on TP3M+10 primary model parameters. In view of Equation 27, however, the model response can alternatively be considered to depend directly on two feature functions and only indirectly (through the two feature functions) on the primary model parameters. In the former consideration/interpretation, the response sensitivities to the primary model parameters will be obtained by applying the nth-CASAM-L. In the later consideration/interpretation, the response sensitivities to the primary model parameters will be obtained by applying the nth-FASAM-L, which will involve two stages: (a) the response sensitivities with respect to the feature functions will be obtained in the first stage; (b) the subsequent computation of the response sensitivities to the primary model parameters will be performed in the second stage by using the response sensitivities with respect to the feature functions obtained in the first stage. The computational distinctions that stem from these differing considerations/interpretations underlying the nth-CASAM-L versus the nth-FASAM-L will become evident in the next section by using a paradigm neutron slowing-down model, which is representative of the general situation for any linear system.

3 First-order adjoint sensitivity analysis of the contributon flux to the slowing-down model’s features and parameters

The first-order sensitivities of the response Rcu12;u, where u12;uφu,ψu, are obtained by determining the first-order Gateaux (G-)- differential, denoted as δRcu12;u,v12;uα0, of this response for variations v12;uδφu,δψu around the phase-space point φ0,ψ0. By definition, the first-order G-differential δRcu12;u,v12;uα0 is obtained as follows:

δRcu12;u,v12;uα0ddε0uthφ0u+εv1uψ0u+εδψuduε=0=0uthv1uψu+φuδψuduα0.(29)

The sensitivities of Rcu12;u with respect to the feature functions (and subsequently to the primary model parameters) will be determined in Section 3.1 by applying the 1st-FASAM-L. Alternatively, the sensitivities of Rcu12;u directly with respect to the primary model parameters will be determined in Section 3.2 by applying the 1st-CASAM-L.

3.1 Application of the 1st-FASAM-L

The first-level variational sensitivity function v12;uv1u,δψu is the solution of the first-level variational sensitivity system (1st-LVSS) obtained by differentiating the 1st-LFAS. The function v1u is obtained by taking the first-order G-differentials of Equations 12, 13 to obtain

ddεdφ0+εv1du+f10+εδf1φ0+εv1ε=0=δuddεf20+εδf2ε=0,(30)
ddεφ0u+εv1uε=0=0;atu=0.(31)

Carrying out the differentiations with respect to ε in the above equations and setting ε=0 in the resulting expressions yields the following relations:

dv1udu+f1α0v1u=δf2δuδf1φ0u,(32)
v1u=0;atu=0.(33)

The equations satisfied by the variational function δψu are obtained by G-differentiating Equations 20, 21 to obtain the equations below:

dduδψu+f1α0δψu=δf1ψu,(34)
δψuth=0,atu=uth.(35)

Concatenating Equations 3235 yields the following 1st-LVSS for the first-level variational sensitivity function v12;uδφu,δψu:

V12×2;u;fv12;uα0=qV12;u12;u;f;δfα0,(36)
bv1v1;f;δfα0=0,(37)

where

V12×2;u;fd/du+f100d/du+f1,bv1v1;f;δfv10δψuth,(38)
qV12;u1;f;δfδf2δuδf1φuδf1ψu.(39)

Rather than repeatedly solving the 1st-LVSS for every possible variations δfi, i=1,2, the appearance of the first-level variational sensitivity function v12;uδφu,δψu will be eliminated from the expression of the G-differential of the response δRcu12;u,v12;uα0, defined in Equation 29), by applying the principles of the 1st-FASAM-L outlined in the accompanying “Part I” by Cacuci (2024c). The specific steps are as follows:

1. A Hilbert space, denoted as H1, is introduced endowed with the following inner product denoted as χ12;u,θ12;u1, between two elements, χ12;uχ11u,χ21uH1 and θ12;uθ11u,θ21uH1:

χ12;u,θ12;u1i=120uthχi1uθi1udu.(40)

2. In the Hilbert H1, the inner product of Equation 36 is formed with a yet undefined vector-valued function a12;ua11u,a21uH1 to obtain the following relation:

a12;u,V12×2;u;f0v12;u1α0=a12;u,qV12;u12;u;f;δf1α0.(41)

3. The left-side of Equation 41 is integrated by parts to obtain the following relation, where the specification α0 is omitted to simplify the notation:

0utha11udv1du+f1v1du+0utha21udduδψ+f1δψdu=0uthv1ddua11u+f1a11udu+0uthδψuddua21u+f1a21udu+a11uthv1utha110v10a21uthδψuth+a210δψ0.(42)

4. The first two terms on the right side of Equation 42 are required to represent the G-differentiated response defined in Equation 29, and the unknown boundary values of the function v12;u are eliminated from the bilinear concomitant on the right side of Equation 42 to obtain the following 1st-LASS for the first-level adjoint sensitivity function a12;ua11u,a21u:

A12×2;x;fa12;x=qA12;u12;x;f,(43)
bA1u12;u;a12;u;fα0a11utha210=0,(44)

where

A12×2;u;fd/du+f100d/du+f1=V12×2;u;f*,(45)
qA12;u12;x;fψuφu.(46)

5. It follows from Equations 29, 4144 that G-differentiated response defined in Equation 29 takes the following expression in terms of the first-level adjoint sensitivity function a12;ua11u,a21u:

δRcu12;u,a12;uα0=0utha11uδf2δuδf1φuduα0+0utha21uδf1ψuduα0,(47)

The expressions of the sensitivities of the response Rcφ,ψ with respect to the components of the feature function fα are given by the expressions that multiply the respective components of fα in Equation 47, i.e.,

Rcφ,ψf1=0utha11uφu+a21uψudu,(48)
Rcφ,ψf2=0utha11uδudu.(49)

The above expressions are to be evaluated at the nominal parameter values α0, but the indication α0 has been omitted for simplicity.

The first-order sensitivities of the response Rcφ,ψ with respect to the primary model parameters are obtained by using the results obtained in Equations 48, 49, respectively, in conjunction with the “chain rule” of differentiating the components of the feature function fα with respect to the primary model parameters defined in Equation 29 to obtain the following expressions:

Rcφ,ψαi=Rcφ,ψf1f1αi+Rcφ,ψf2f2αi=f1αi0utha11uφu+a21uψudu+f2αi0utha11uδudu.(50)

Solving the 1st-LASS defined by Equations 43, 44 yields the following closed-form expressions for the components of the first-level adjoint sensitivity function a12;ua11u,a21u:

a11u=uduHuduexpuudf1α,(51)
a21u=uf2αexpuf1α.(52)

Using the above expressions in Equations 48, 49 yields the following closed-form expressions for the respective sensitivities:

Rcφ,ψf1=ud2f2αexpudf1α,(53)
Rcφ,ψf2=udexpudf1α.(54)

The correctness of the expressions obtained in Equations 53, 54 can be verified by differentiating accordingly the closed-form expression given in Equation 27.

3.2 Application of the 1st-CASAM-L

The 1st-CASAM-L delivers the first-order sensitivities of the response directly with respect to the primary model parameters. The expression of the G-differentiated response is as shown in Equation 29, but the source term on the right side of the 1st-LVSS takes the following form:

qV12;u1;f;δfδui=1TPf2αiδαiφui=1TPf1αiδαiψui=1TPf1αiδαi.(55)

If one were to actually solve the 1st-LVSS to obtain the first-level variational function and subsequently use the respective variational function to compute each sensitivity, one would need to solve the 1st-LVSS TP-times, using each time a source that would correspond to the ith-primary parameter, of the form qV1i;2;u1;f;δfqδuf2/αiφuf1/αi,ψuf1/αi, for each primary parameter i=1,...,TP.

Since the left side of the 1st-LVSS remains the same as in Equation 36 and the boundary conditions also remain the same as obtained in Equation 37, it follows that the 1st-LASS and its solution a12;ua11u,a21u remain unchanged. It therefore follows that the counterpart of the expression of the G-differential obtained in Equation 47 takes the following form:

δRcu12;u,a12;uα0=i=1TPf1αiδαi0utha21uψuduα0+0utha11uδui=1TPf2αiδαiφui=1TPf1αiδαiduα0.(56)

The first-order sensitivities of the response Rcφ,ψ with respect to the primary model parameters αi,i=1,...,TP are obtained by identifying the expressions that multiply the respective variations δαi in Equation 47, which yields the following result:

Rcφ,ψαi=f1αi0utha11uφu+a21uψudu+f2αi0utha11uδudu.(57)

As expected, the result obtained from Equation 57 is identical to the result produced from Equation 50 by using the 1st-FASAM-L. Both the 1st-FASAM-L and 1st-CASAM-L require “one large-scale computation” for solving the 1st-LASS represented by Equations 43, 44.

4 Second-order adjoint sensitivity analysis of the contributon flux to the slowing-down model’s features and parameters

In practice, closed-form expressions such as those shown in Equations 53, 54 are unavailable. The 1st-FASAM-L yields the expressions provided in Equations 48, 49, while the 1st-CASAM-L yields the expressions provided in Equation 57. Hence, these expressions will provide the starting points for obtaining the second-order sensitivities that stem from the respective first-order sensitivities. As outlined within the general frameworks of both the nth-FASAM-L and nth-CASAM-L methodologies, the second-order sensitivities are obtained by conceptually considering them to arise as the “first-order sensitivities of the first-order sensitivities.”

4.1 Application of the 2nd-FASAM-L

The 2nd-FASAM-L uses the first-order sensitivities obtained from the 1st-CASAM-L, as provided in Equations 48, 49, to obtain the respective second-order sensitivities, as presented in Sections 4.1.1 and 4.1.2.

4.1.1 Second-order sensitivities stemming from the first-order sensitivity Rc/f1

The second-order sensitivities that stem from the first-order sensitivity Rc/f1 are obtained by determining the G-differential of Rc/f1. For subsequent “bookkeeping” purposes, this first-order sensitivity will be denoted as R11;u222;u;fαRc/f1, where the superscript “(1)” denotes “first-order” (sensitivity) and the argument “1” indicates that this sensitivity is with respect to the first component, i.e., f1α, of the feature function fα. This sensitivity also depends on the function u222;uu12;u,a12;u, which is the solution of the “second-level forward/adjoint system (2nd-LFAS)” obtained by concatenating the 1st-LFAS with the 1st-LASS, comprising Equations 12, 13, 20, 21, 43, and 44.

Applying the definition of the G-differential to Equation 48 yields the following expression for the G-differential δR11;u222;u;v222;u;fαα0:

δR11;u222;u;v222;u;fαα0ddε0utha11u+εδa11uφu+εv1uduα0,ε=0ddε0utha21u+εδa21uψu+εδψuduα0,ε=0=0uthφuδa11udu0utha11uv1udu0uthψuδa21udu0utha21uδψuduj=122Rφ;ffjf1δfj.(58)

The components v1u, δψu, δa11u, and δa21u of the second-level variational sensitivity function v222;uv1u,δψu,δa11u,δa21u are the solutions of the 2nd-LVSS, which is obtained by G-differentiating the 2nd-LFAS. Thus, performing the G-differentiation of Equations 12, 13, 20, 21, 43, and 44 yields the following 2nd-LVSS for the second-level variational sensitivity function v222;uv1u,δψu,δa11u,δa21u:

V222×22;u;fv222;uα0=qV222;u;f;δfα0,(59)
bv2u;f;δfα0=0,(60)

where

V222×22;u;fd\/du+f10000ddu+f0001ddu+f10100ddu+f1;(61)
qV222;u;f;δfδf2δuδf1φuδf1ψuδf1a11uδf1a21u;bv2u;f;δfv10δψuthδa11uthδa210.(62)

The second-level variational sensitivity function v222;u will be eliminated from the expression of δR11;u222;u;v222;u;fαα0 by constructing the 2nd-LASS corresponding to the above 2nd-LVSS. The solution of the 2nd-LASS will be used in Equation 58 to construct δR11;u222;u;v222;u;fαα0, an alternative expression that will not depend on v222;u. This 2nd-LASS will be constructed in a Hilbert space denoted as H2, comprising four-component vector-valued functions of the form χ222;1;uχ121;u,χ221;u,χ321;u,χ421;uH2 as elements, and is endowed with the following inner product between two vectors χ222;1;u and θ222;1;u:

χ222;u,θ222;u2i=1220uthχi21;uθi21;udu.(63)

The inner product defined in Equation 63 will be used to construct the inner product of Equation 59 with a function denoted as a222;1;ua121;u,a221;u,a321;u,a421;uH2, where the argument “1” of the function a222;1;u indicates that this (adjoint) function corresponds to the first-order sensitivity of the response with respect to the “first” component, f1α, of the feature function fα. Constructing this inner product yields the following relation, where the specification α0 has been omitted to simplify the notation:

a222;1;x,V222×22;u;fv222;u22=0utha121;udv1/du+f1v1du+0utha221;udδψ/du+f1δψdu+0utha321;uδψdδa11/du+f1δa11du+0utha421;uv1u+dδa21/du+f1δa21du=0utha121;uδf2δuδf1φudu+0utha221;uδf1ψudu+0utha321;uδf1a11udu+0utha421;uδf1a21udu.(64)

Integrating by parts the left side of Equation 64 yields the following relation:

0utha121;udv1/du+f1v1du+0utha221;udδψ/du+f1δψdu+0utha321;uδψdδa11/du+f1δa11du+0utha421;uv1u+dδa21/du+f1δa21du=a121;uthv1utha121;0v10+0uthv1uda121;u/du+f1a121;udua221;uthδψuth+a221;0δψ0+0uthδψda221;u/du+f1a221;udua321;uthδa11uth+a321;0δa1100uthδψa321;udu+0uthδa11uda321;u/du+f1a321;udu0uthv1ua421;udu+a421;uthδa21utha421;0δa210+0uthδa21uda421;u/du+f1a421;udu.(65)

The right side of Equation 65 is now tailored to represent the G-differential δR11;u222;u;v222;u;fαα0 expressed by Equation 58 by requiring the second-level adjoint sensitivity function a222;1;u to be the solution of the following 2nd-LASS:

jA222×22;u;fa222;1;uα0=s222;1;u;fα0,(66)
bA2u;fα0=0,(67)

where

A222×22;u;fd/du+f10010d/du+f11000d/du+f10000d/du+f1;(68)
s222;1;u;fa11ua21uφuψu;bA2u;fa121;utha221;0a321;0a421;uth.(69)

Implementing the equations underlying the 2nd-LVSS and the 2nd-LASS and substituting Equation 58 into Equation 64 provide the following alternative expression for the G-differential δR11;u222;u;v222;u;fαα0:

δR11;u222;u;v222;u;fαα0=0utha121;uδf2δuδf1φuduα0+0utha221;uδf1ψuduα0+0utha321;uδf1a11uduα0+0utha421;uδf1a21uduα0.(70)

The expressions that multiply the respective components of fα in Equation 70 are the expressions of the second-order sensitivities 2Rcφ,ψ/f1fj (stemming from the first-order sensitivity Rc/f1) of the response Rcφ,ψ, with respect to the components of the feature function fα. Thus, identifying in Equation 70 the expressions that multiply the respective variations in the components of the feature function fα yields the following relations:

2Rcφ,ψf1f1=0utha121;uφudu0utha221;uψudu0utha321;ua11udu0utha421;ua21udu;(71)
Rcφ,ψf2f1=0utha121;uδudu.(72)

Solving the 2nd-LASS represented by Equations 66, 67 yields the following closed-form expressions for the components of the second-level adjoint sensitivity function a222;1;u:

a121;u=udu2Huduexpuudf1α,(73)
a221;u=f2αu2expuf1α,(74)
a321;u=f2αuexpuf1α,(75)
a421;u=uduHuduexpuudf1α.(76)

Using the explicit closed-form expressions obtained in Equations 7376 and substituting them in Equations 71, 72 yield the following closed-form explicit expressions for the respective second-order sensitivities:

2Rcφ,ψf1f1=ud3f2αexpudf1α,(77)
Rcφ,ψf2f1=ud2expudf1α.(78)

The correctness of the expressions obtained in Equations 77, 78 can be verified by differentiating accordingly the closed-form expression given in Equation 53.

4.1.2 Second-order sensitivities stemming from the first-order sensitivity Rc/f2

The second-order sensitivities that stem from the first-order sensitivity Rc/f2 are obtained by determining the G-differential of Rc/f2. For subsequent “bookkeeping” purposes, this first-order sensitivity will be denoted as R12;u222;u;fαRc/f2, where the superscript “(1)” denotes “first-order” (sensitivity) and the argument “2” indicates that this sensitivity is with respect to the second component, i.e., f2α, of the feature function fα. This sensitivity also depends on the function u222;uu12;u,a12;u. Applying the definition of the G-differential to the expression provided in Equation 49 yields the result below for the G-differential δR12;u222;u;v222;u;fαα0:

δR12;u222;u;v222;u;fαα0=0uthδa11uδuduj=122Rφ;ffjf2δfj.(79)

The function δa11u, as shown in Equation 79, is the component of the second-level variational sensitivity function v222;uv1u,δψu,δa11u,δa21u, which is the solution of the 2nd-LVSS comprising Equations 59, 60. The component δa11u will be eliminated from the expression of δR12;u222;u;v222;u;fαα0 by following the same procedure as described in Section 4.1.1 to construct a 2nd-LASS, the solution of which will be denoted as a222;2;ua122;u,a222;u,a322;u,a422;uH2 and will be used in Equation 79 to eliminate δa11u. The argument “2” in a222;2;u indicates that this second-level adjoint sensitivity function corresponds to the first-order sensitivity of the response with respect to the “second” component, f2α, of the feature function fα. The 2nd-LASS for the function a222;2;u will have the same left side and boundary conditions as obtained in Equations 66, 67, but the right-side of this 2nd-LASS will correspond to the G-differential obtained in Equation 79, which leads to the following 2nd-LASS:

A222×22;u;fa222;2;uα0=s222;2;u;fα0,(80)
bA2u;fα0=0,(81)

where

s222;1;u;f0,0,δu,0.(82)

The alternative expression for the G-differential δR12;u222;u;v222;u;fαα0 in terms of the components of a222;2;u has the same formal expression as shown in Equation 70 but with the components of the function a222;1;u being replaced by the components of a222;2;u, i.e.,:

δRc/f2=0utha122;uδf2δuδf1φudu+0utha222;uδf1ψudu+0utha322;uδf1a11udu+0utha422;uδf1a21udu.(83)

Solving the 2nd-LASS represented by Equations 80, 81 yields the following expressions:

a122;u=0,(84)
a222;u=uexpuf1α,(85)
a322;u=Huexpuf1α,(86)
a422;u=0.(87)

Identifying in Equation 83 the expressions that multiply the respective variations δfi, i=1,2, in the components of the feature function fα and using the closed-form expressions obtained in Equations 8487, 26, 51 yield the following closed-form explicit expressions for the respective second-order sensitivities:

2Rcφ,ψf1f2=0utha222;uψudu0utha322;ua11udu=ud2expudf1α,(88)
Rcφ,ψf2f2=0.(89)

The correctness of the expressions obtained in Equations 88, 89 can be verified by differentiating accordingly the closed-form expression given in Equation 54.

Notably, due to the symmetry of the mixed second-order sensitivities, the expressions obtained in Equations 88, 72 provide an intrinsic mutual verification mechanism of the accuracy of the computations of the second-level adjoint sensitivity functions a222;1;u and a222;2;u.

4.2 Application of the 2nd-CASAM-L

The starting point for the application of the 2nd-CASAM-L is to determine the G-differential of the TP first-order sensitivities represented by Equation 57. For “bookkeeping” purposes, it is convenient to designate these TP first-order sensitivities as follows:

Rc1i;u222;u;αRcφ,ψ/αi=g1i;α0utha11uφu+a21uψudu+g2i;α0utha11uδudu,(90)

where

g1i;αf1/αi;g2i;αf2/αi;i=1,...,TP.(91)

The G-differential of the expression in Equation 90 is obtained, by definition, as follows:

δRc1i;u222;u;v222;u;α;δαα00utha11uφu+a21uψududdεg1i;α+εδαα0,ε=0g1i;αddε0utha11u+εδa11uφu+εv1uduα0,ε=0g1i;αddε0utha21u+εδa21uψu+εδψuduα0,ε=0+0utha11uδududdεg2i;α+εδαα0,ε=0+g2i;αddε0utha11u+εδa11uδuduα0,ε=0δRc1i;u222;u;v222;u;αind+δRc1i;u222;u;α;δαdir,(92)

where the direct-effect and indirect-effect terms are defined, respectively, as follows:

δRc1i;u222;u;α;δαdirj=1TPg2i;ααjδαj0utha11uδuduα0j=1TPg1i;ααjδαj0utha11uφu+a21uψuduα0,(93)
δRc1i;u222;u;v222;u;αindg2i;α0uthδa11uδuduα0g1i;α0utha11uv1u+φuδa11uduα0g1i;α0utha21uδψu+ψuδa21uduα0.(94)

The direct-effect term can be evaluated/computed already at this stage. On the other hand, the indirect-effect depends on the second-level variational function v222;uv1u,δψu,δa11u,δa21u, which is the solution of the counterpart of 2nd-LVSS defined by Equations 59, 60, with the same boundary conditions and right-side but with distinct source terms, each source term involving the quantities g1i;α/αj and g2i;α/αj for i,j=1,...,TP. If this path were chosen to compute the second-order sensitivities, the 2nd-LVSS would need to be solved TP2 times, with TP2 different sources on the respective right sides, albeit with the same left side and boundary conditions.

The components v1u,δψu,δa11u,δa21u are eliminated from the expression of the indirect-effect term δRc1i;u222;u;v222;u;αind defined in Equation 94 by constructing a corresponding 2nd-LASS in the Hilbert space H2 by following the same sequence of steps as described in Section 4.1. The formal expression of the 2nd-LASS thus obtained will have the same left side and boundary conditions as those described in Section 4.1, but the right side of this formal 2nd-LASS will have a source term that will correspond to the indirect-effect term defined in Equation 94 and, hence, will be different for each i=1,...,TP, i.e.,

A222×22;u;αa222;i;2;uα0=s222;i;u;αα0,i=1,...,TP;(95)
bA2u;αα0=0;i=1,...,TP;(96)

where

s222;i;u;α[g1i;αa11,g1i;αa21,g2i;αδug1i;αφ,g1i;αψ].(97)

In terms of the solution a222;i;2;u of the 2nd-LASS represented by Equations 95, 96, the indirect-effect term δRc1i;u222;u;v222;u;αind defined in Equation 94 will have a representation that will formally resemble the expressions provided in Section 4.1, e.g., Equation 83, but with the second-level adjoint function(s) from Section 4.1 being replaced by the second-level adjoint sensitivity function a222;i;2;u. Finally, the total G-differential δRc1i;u222;u;v222;u;α;δαα0 will be obtained, as shown in Equation 92, by adding the expression of the indirect-effect term obtained in terms of the second-level adjoint sensitivity function a222;i;2;u and the expression of the direct-effect term provided in Equation 93. The expression of the individual second-order sensitivities 2Rcφ,ψ/αiαj, i,j=1,...,TP will subsequently be obtained by identifying in the final expression of the total G-differential δRc1i;u222;u;v222;u;α;δαα0 those terms that multiply the parameter variations αj, j=1,...,TP.

4.2.1 Comparing the 2nd-FASAM-L versus the 2nd-CASAM-L

The computational savings provided by using, whenever possible, the 2nd-FASAM-L rather than the 2nd-CASAM-L are evident by comparing the results obtained in Section 4.1 versus the results obtained in Section 4.2. The feature function fα comprises two components fiα, i=1,2; consequently, the 2nd-FASAM-L requires two large-scale computations (to solve the corresponding 2nd-LASS) to obtain the second-order response sensitivities with respect to the components of the feature function. Subsequently, the second-order response sensitivities with respect to the primary model parameters are obtained analytically using the chain-rule of differentiation.

In contradistinction, there is TP3M+10, where the number (M) of materials in the medium can easily exceed two dozen primary model parameters. Consequently, the 2nd-CASAM-L requires TP large-scale computations (to solve the corresponding 2nd-LASS) to obtain the second-order response sensitivities with respect to the primary model parameters. The boundary conditions and the operators on the left sides for all of the 2nd-LASS, for both the 2nd-FASAM-L and 2nd-CASAM-L, are the same; only the source terms on the left sides of these 2nd-LASS differ from each other. It is therefore computationally advantageous if the inverse operators of the left sides of these 2nd-LASS could be computed just once and stored for subsequent use, in which case the computational advantage of using the 2nd-FASAM-L would not be massive. Such a procedure could be feasible for relatively small models but would be impractical for large-scale problems, for which the advantage of using the 2nd-FASAM-L rather than the 2nd-CASAM-L increases as the number of model parameters increases.

5 Third-order adjoint sensitivity analysis of the contributon flux to the slowing-down model’s features and parameters

The 3rd-FASAM-L determines the third-order sensitivities by applying the principles of the 1st-FASAM to the second-order sensitivities, i.e., considering that the third-order sensitivities are “the first-order sensitivities of the second-order sensitivities.” The unmixed second-order sensitivity 2Rcφ,ψ/f2f2 is identically zero. The two non-zero second-order sensitivities of the model response with respect to the components of the feature function fα are as follows: (i) the unmixed second-order sensitivity 2Rcφ,ψ/f1f1, expressed in Equation 71, and (ii) the mixed second-order sensitivity 2Rcφ,ψ/f1f2=2Rcφ,ψ/f2f1, expressed in either Equation 72 or Equation 88, which are equivalent, in view of the symmetry property of the mixed second-order sensitivities. Therefore, either the expression obtained in Equation 88 or Equation 72 can be used as the starting point for obtaining the third-order sensitivities stemming from this mixed second-order sensitivity. It appears that the expression provided in Equation 72 is the simpler of the two, so it will be used as the starting point for obtaining the corresponding third-order sensitivities.

The second-order sensitivity 2Rcφ,ψ/f1f1 expressed in Equation 71 depends on the components of the third-level forward/adjoint function, denoted as u323;1;1;u=u222;u,a222;1;u, which is the solution of the third-level forward/adjoint system (3rd-LFAS) obtained by concatenating the 2nd-LFAS with the 2nd-LASS, thus comprising Equations 12, 13, 20, 21, 43, 44, 66 and 67. The argument “1;1” of u323;1;1;u indicates that this third-level function corresponds to the (unmixed) second-order sensitivity 2Rcφ,ψ/f1f1 of the response with respect to the “first” feature function, f1. Therefore, the second-order sensitivity 2Rcφ,ψ/f1f1 is denoted as follows: R21;1;u3;fα2Rcφ,ψ/f1f1, where the argument “1;1” indicates that this third-level function corresponds to the (unmixed) second-order sensitivity 2Rcφ,ψ/f1f1 and the arguments of the function u323;1;1;u were omitted, for simplicity. Similarly, the mixed second-order sensitivity 2Rcφ,ψ/f1f2 depends on the components of the same function u323;1;1;u and will, therefore, be denoted as R22;1;u3;fα2Rcφ,ψ/f2f1, where the argument “2;1” indicates that this second-order sensitivity is with respect to the components f2,f1 of fα.

5.1 Application of the 3rd-FASAM-L to compute the third-order sensitivities stemming from 2Rcφ,ψ/f1f1

The third-order sensitivities stemming from R21;1;u3;fα2Rcφ,ψ/f1f1 are obtained from the G-differential of Equation 71, which will be denoted as δR21;1;u3;v3;fαα0δ2Rcφ,ψ/f1f1α0, and they are, by definition, determined as follows:

δR21;1;u3;v3;fαα0ddε0utha121;u+εδa121;uφu+εv1uduα0,ε=0ddε0utha221;u+εδa221;uψu+εδψuduα0,ε=0ddε0utha321;u+εδa321;ua11u+εδa11uduα0,ε=0ddε0utha421;u+εδa421;ua21u+εδa21uduα0,ε=0.(98)

Performing the differentiation with respect to ε in Equation 98 and setting ε=0 in the resulting expression yield

δR21;1;u3;v3;fαα0=0utha121;uv1u+φuδa121;uduα00utha221;uδψu+ψuδa221;uduα00utha321;uδa11u+a11uδa321;uduα00utha421;uδa21u+a21uδa421;uduα0.(99)

The third-level variational function v3v323;1;1;uv222;u,δa222;1;u, where δa222;1;uδa121;u,δa221;u,δa321;u,δa421;u, is the solution of the 3rd-LVSS obtained by concatenating the 2nd-LVSS (i.e., Equations 59, 60), with the equations obtained by G-differentiating the 2nd-LASS, represented by Equations 66, 67, for the function a222;1;u. The resulting 3rd-LVSS for the third-level variational function v323;1;1;u comprises the following matrix equation, where the dots are used to denote zero-elements for better visibility of the structure:

L········M·······1M·····1··L······1·M··1···1·L1·1·····L··1·····Mv1uδψuδa11uδa21uδa121;uδa221;uδa321;uδa421;u=δf2δuδf1φuδf1ψuδf1a11uδf1a21uδf1a121;uδf1a221;uδf1a321;uδf1a421;u,Luddu+f1α;Muddu+f1α;Mu=L*u;(100)
v10=0;δψuth=0;δa11uth=0;δa210=0;δa121;uth=0;δa221;0=0;δa321;0=0;δa421;uth=0.(101)

The 3rd-LVSS comprising Equations 100, 101 can be formally expressed in the following 23×23-matrix form:

V323×23;u;fv323;1;1;u=qV323;u323;u;f;δf,(102)
bv3v323;1;1;u=0.(103)

The above matrix form of the 3rd-LVSS will be used as a “condensed notation” to construct the 3rd-LASS, the solution of which will be used to derive the alternative expression for the G-differential δR21;1;u323;1;1;u;v323;1;1;u;fαα0. This 3rd-LASS will be constructed in a Hilbert space denoted as H3, comprising as elements eight-component vector-valued functions of the form χ323;1;1;uχ121;1;u,...,χ821;1;uH3, and endowed with the following inner product between two vectors χ323;1;1;u and θ323;1;1;u:

χ323;1;1;u,θ323;1;1;u3i=1230uthχi31;1;uθi31;1;udu.(104)

The inner product defined in Equation 104 will be used to construct the inner product of Equation 102 with a function denoted as a323;1;1;ua131;1;u,...,a831;1;uH3, where the argument “1,1” of the function indicates that this (third-level adjoint) function corresponds to the unmixed second-order sensitivity of the response with respect to the “first” component, f1α, of the feature function fα. Constructing this inner product yields the following relation, where the specification α0 has been omitted to simplify the notation:

0utha131;1;udduv1u+f1v1udu+0utha231;1;udduδψu+f1δψudu+0utha331;1;uδψudduδa11u+f1δa11udu+0utha431;1;uv1u+dduδa21u+f1δa21udu+0utha531;1;udduδa121;u+f1δa121;uδa42u+δa11udu+0utha631;1;udduδa221;u+f1δa221;uδa32u+δa21udu+0utha731;1;udduδa321;u+f1δa321;u+v1udu+0utha831;1;udduδa421;u+f1δa421;u+δψudu=0utha131;1;uδf2δuδf1φudu+0utha231;1;uδf1ψudu+0utha331;1;uδf1a11udu+0utha431;1;uδf1a21udu+0utha531;1;uδf1a121;udu+0utha631;1;uδf1a221;udu+0utha731;1;uδf1a321;udu+0utha831;1;uδf1a421;udu.(105)

The component for Equation 105 can be written as follows:

a323;1;1;u,V323×23;u;fv323;1;1;u3=a323;1;1;u,qV323;u323;u;f;δf3.(106)

The left side of Equation 106 is integrated by parts to obtain the relation given below, in which the argument “1;1” has been omitted when writing the components ai31;1;u,i=1,...,8 to simplify the notation:

0utha13udduv1u+f1v1udu+0utha23udduδψu+f1δψudu+0utha33uδψudduδa11u+f1δa11udu+0utha43uv1u+dduδa21u+f1δa21udu+0utha53udduδa121;u+f1δa121;uδa42u+δa11udu+0utha63udduδa221;u+f1δa221;uδa32u+δa21udu+0utha73udduδa321;u+f1δa321;u+v1udu+0utha83udduδa421;u+f1δa421;u+δψudu=a13uthv1utha130v10+0uthv1uddua13u+f1a13udua23uthδψuth+a230δψ0+0uthδψuddua23u+f1a23udua33uthδa11uth+a330δa110+0uthδa11uddua33u+f1a33udu0utha33uδψudu0utha43uv1udu+a43uthδa21utha430δa210+0uthδa21uddua43u+f1a43udua53uthδa121;uth+a530δa121;0+0uthδa121;uddua53u+f1a53udu+0utha53uδa42u+δa11udu+a63uthδa221;utha630δa221;0+0uthδa221;uddua63u+f1a63udu+0utha63uδa32u+δa21udu+a73uthδa321;utha730δa321;0+0uthδa321;uddua73u+f1a73udu+0utha73uv1udua83uthδa421;uth+a830δa421;0+0uthδa421;uddua83u+f1a83udu+0utha831;1;uδψudu.(107)

The boundary terms that appear in Equation 107 will vanish by using Equation 101 and imposing the following boundary conditions on the components ai31;1;u,i=1,...,8 of the third-level adjoint sensitivity function a323;1;1;u:

a131;1;uth=0;a231;1;0=0;a331;1;0=0;a431;1;uth=0;a531;1;0=0;a631;1;uth=0;a731;1;uth=0;a831;1;0=0.(108)

Equation 107 can be written in matrix form as follows:

a323;1;1;u,V323×23;u;fv323;1;1;u3=v323;1;1;u,A323×23;u;fa323;1;1;u3,(109)

where A323×23;u;fV323×23;u;f* denotes the formal adjoint of V323×23;u;f. The right side of Equation 109 is now required to represent the G-differential δR21;1;u323;1;1;u;v323;1;1;u;fαα0 by imposing the following relation:

A323×23;u;fa323;1;1;u=sA323;1;1;f,(110)

where

sA323;1;1;fa121;u,a221;u,a321;u,a421;u;φu,ψu,a11u,a21u.(111)

The relations provided in Equations 108, 110 constitute the 3rd-LASS for the third-level adjoint sensitivity function a323;1;1;u. In component form, Equation 110 has the following expression, where the dots are used to denote zero-elements for better visibility of the structure:

M··1··1··L1····1··L·1······M·1······L········M·······1M·····1··La131;1;ua231;1;ua331;1;ua431;1;ua531;1;ua631;1;ua731;1;ua831;1;u=a121;ua221;ua321;ua421;uφuψua11ua21u.(112)

Using the relations in Equations 99, 102, 103, 108, 110 yields the following alternative expression for δR21;1;u323;1;1;u;v323;1;1;u;fαα0:

δR21;1;u323;1;1;u;v323;1;1;u;fαα0=0utha131;1;uδf2δuδf1φuduα00utha231;1;uδf1ψuduα00utha331;1;uδf1a11uduα00utha431;1;uδf1a21uduα00utha531;1;uδf1a121;uduα00utha631;1;uδf1a221;uduα00utha731;1;uδf1a321;uduα00utha831;1;uδf1a421;uduα0.(113)

The third-order sensitivities stemming from the relation obtained in Equation 113 are the expressions that multiply the respective variations δf1 and δf2 and are as follows:

3Rcφ,ψ/f1f1f1=0utha131;1;uφudu0utha231;1;uψudu0utha331;1;ua11udu0utha431;1;ua21udu0utha531;1;ua121;udu0utha631;1;ua221;udu0utha731;1;ua321;udu0utha831;1;ua421;udu;(114)
3Rcφ,ψ/f1f1f2=0utha131;1;uδudu.(115)

The expressions obtained in Equations 114, 115 are to be evaluated at the nominal values of parameters and state functions, but the notation α0 has been omitted for simplicity.

Solving Equations 112, 108 yields the following expressions for the components of the third-level adjoint sensitivity function a323;1;1;u:

a131;1;u=udu3Huduexpuudf1α,(116)
a231;1;u=f2αu3expuf1α,(117)
a331;1;u=f2αu2expuf1α,(118)
a431;1;u=udu2Huduexpuudf1α,(119)
a531;1;u=f2αuexpuf1α,(120)
a631;1;u=uduHuduexpuudf1α,(121)
a731;1;u=udu2Huduexpuudf1α,(122)
a831;1;u=f2αu2expuf1α.(123)

Using the expressions obtained in in Equations 132, 133 and performing the respective operations yield the following results:

3Rcφ,ψ/f1f1f1=ud4f2αexpudf1α,(124)
3Rcφ,ψ/f1f1f2=ud3expudf1α.(125)

5.2 Application of the 3rd-FASAM-L to compute the 3rd-order sensitivities stemming from 2Rcφ,ψ/f1f2=2Rcφ,ψ/f2f1

The third-order sensitivities stemming from R22;1;u3;fα2Rcφ,ψ/f2f1 will be obtained from the G-differential of (Equation 72), which will be denoted as δR22;1;u3;v3;fαα0δ2Rcφ,ψ/f2f1α0, and which is by definition determined as follows:

δR22;1;u3;v3;fαα0δ2Rcφ,ψ/f2f1α0ddε0utha121;u+εδa121;uδuduα0,ε=0=0uthδa121;uδudu.(126)

The function δa121;u is one of the components of the third-level variational function v323;1;1;u, which is the solution of the 3rd-LVSS represented by Equations 101, 102. To avoid the need for solving the 3rd-LVSS, the appearance of this function will be eliminated from Equation 126 by deriving an alternative expression for the G-differential δR22;1;u3;v3;fαα0 in terms of a third-level adjoint sensitivity function, denoted as a323;2;1;ua132;1;u,...,a832;1;uH3. The argument “2,1” of the function a323;2;1;u indicates that this (third-level adjoint) function corresponds to the mixed second-order sensitivity of the response with respect to the “second and first” components, f2,f1, of the feature function fα.

The third-level adjoint sensitivity function a323;2;1;u will be the solution of 3rd-LASS to be constructed in the Hilbert space H3 using Equation 104 to construct the inner product of a323;2;1;u with Equation 102. Constructing this inner product yields the following relation, where the specification α0 has been omitted to simplify the notation:

a323;2;1;u,V323×23;u;fv323;1;1;u3=a323;2;1;u,qV323;u323;u;f;δf3.(127)

The left side of Equation 127 is integrated by parts to obtain the following relation:

a323;2;1;u,V323×23;u;fv323;1;1;u3=v323;1;1;u,A323×23;u;fa323;2;1;u3,(128)

where the following boundary conditions were imposed on the components ai32;1;u, i=1,...,8, of the third-level adjoint sensitivity function a323;2;1;u:

a132;1;uth=0;a232;1;0=0;a332;1;0=0;a432;1;uth=0;a532;1;0=0;a632;1;uth=0;a732;1;uth=0;a832;1;0=0.(129)

The right side of Equation 109 is now required to represent the G-differential δR22;1;u323;1;1;u;v323;1;1;u;fαα0 by imposing the following relation:

A323×23;u;fa323;2;1;u=sA323;2;1;f0,0,0,0,0,δu,0,0.(130)

The relations provided in Equations 108, 110 constitute the 3rd-LASS for the third-level adjoint sensitivity function a323;1;1;u. Using the relations in Equations 99, 102, 103, 108, and 110 yields the following alternative expression for δR22;1;u323;1;1;u;a323;2;1;u;fαα0, in which the function v323;1;1;u has been replaced by the function a323;2;1;u:

δR22;1;u323;1;1;u;a323;2;1;u;fαα0=0utha132;1;uδf2δuδf1φuduα00utha232;1;uδf1ψuduα00utha332;1;uδf1a11uduα00utha432;1;uδf1a21uduα00utha532;1;uδf1a121;uduα00utha632;1;uδf1a221;uduα00utha732;1;uδf1a321;uduα00utha832;1;uδf1a421;uduα0.(131)

The third-order sensitivities stemming from the relation obtained in Equation 131 are the expressions that multiply the respective variations δf1 and δf2 and are as follows:

3Rcφ,ψ/f1f2f1=0utha132;1;uφudu0utha232;1;uψudu0utha332;1;ua11udu0utha432;1;ua21udu0utha532;1;ua121;udu0utha632;1;ua221;udu0utha732;1;ua321;udu0utha832;1;ua421;udu;(132)
3Rcφ,ψ/f2f2f1=0utha132;1;uδudu.(133)

The expressions obtained in Equations 132, 133 are to be evaluated at the nominal values of parameters and state functions, but the notation α0 has been omitted for simplicity.

In component form, the 3rd-LASS for the third-level adjoint sensitivity function a323;2;1;u has the following expression, where dots are used to denote zero-elements for better visibility of the structure:

M··1··1··L1····1··L·1······M·1······L········M·······1M·····1··La132;1;ua232;1;ua332;1;ua432;1;ua532;1;ua632;1;ua732;1;ua832;1;u=0000δu000.(134)

Solving Equation 134 yields the following expressions for the components of the third-level adjoint sensitivity function a323;2;1;u:

a132;1;u=a432;1;u=a632;1;u=a732;1;u=0;a532;1;u=Huexpuf1α;a232;1;u=u2expuf1α;a332;1;u=uexpuf1α=a832;1;u.(135)

Using the expressions obtained in Equation 135, substituting them into Equations 132, 133 and performing the respective operations yield the following results:

3Rcφ,ψ/f1f2f1=0utha232;1;uψudu0utha332;1;ua11udu0utha532;1;ua121;udu0utha832;1;ua421;udu=ud3expudf1α(136)
3Rcφ,ψ/f2f2f1=0.(137)

6 Concluding discussion

This work has presented illustrative applications of the “nth-FASAM-L,” which has been specifically developed to be the most efficient methodology for computing exact expressions of sensitivities of responses (of such unique linear models) to features of model parameters and, subsequently, to the model parameters themselves. The efficiency of the nth-FASAM-L stems from the maximal reduction of the number of adjoint computations (which are “large-scale” computations) compared to the extant conventional high-order adjoint sensitivity analysis methodology nth-CASAM-L (Cacuci, 2022). The unique characteristics of the nth-FASAM-L have been illustrated in this work using a paradigm model of a “contributon-flux density response” that occurs in the energy distribution of neutrons stemming from a fission source in a homogeneous mixture of materials. This analytically solvable illustrative paradigm model has been used to demonstrate the following general conclusions regarding the characteristics and applicability of the nth-FASAM-L.

(i) Comparing the mathematical framework of the nth-FASAM-L to that of the nth-CASAM-L indicates that the components fiα,i=1,...,TF of the “feature function” fαf1α,...,fTFα play within the nth-FASAM-L the same role as played by the components αj,j=1,...,TP of the “vector of primary model parameters” αα1,,αTP within the framework of the nth-CASAM-L. It is paramount to underscore, at the outset, that the total number of model parameters is always larger (usually by a wide margin) than the total number of components of the feature function fα, i.e., TPTF. The illustrative paradigm model of “neutron slowing down in a homogeneous mixture of materials” presented in this work comprised a feature function with two components (i.e., TF=2) denoted as f1α and f2α, which were, in turn, functions of TP3M+10 imprecisely known model parameters (where M denotes the number of materials and/or isotopes in the mixture, which is of the order of 20–50 in a nuclear reactor, depending on its service in operation).

(ii) For computing the exact expressions of the first-order sensitivities of a model response to the uncertain parameters, boundaries, and internal interfaces of the model, both the 1st-FASAM-L and 1st-CASAM-L require a single large-scale “adjoint” computation. This “large-scale” computation using either the 1st-FASAM-L or 1st-CASAM-L involves solving the same operator equations and boundary conditions within the respective 1st-LASS; only the sources for the respective 1st-LASS differ from each other. The 1st-FASAM-L enjoys a slight computational advantage since it requires only TF quadratures (one quadrature per component of the feature function), while the 1st-CASAM-L requires TP quadratures (one quadrature per model parameter). For the illustrative “contributon response of the neutron slowing-down” paradigm model, the computation of the first-order response sensitivities with respect to the model parameters required two quadratures using the 1st-FASAM-L, while the 1st-CASAM-L required TP-quadratures. Within the 1st-FASAM-L, the sensitivities with respect to the primary model parameters are obtained by using the first-order sensitivities Rc/f1 and Rc/f2 (with respect to the components of the feature function) in conjunction with the chain rule of differentiation of the exactly known expressions of the components f1α and f2α in terms of the primary model parameters.

(iii) Both the 2nd-FASAM-L and 2nd-CASAM-L conceptually determine the second-order sensitivities by using the fundamental concept that “the second-order sensitivities are the first-order sensitivities of the first-order sensitivities.” For computing the exact expressions of the second-order response sensitivities with respect to the primary model’s parameters, the fundamental difference between the 2nd-FASAM-L and 2nd-CASAM-L is obtained as follows: the 2nd-FASAM-L requires as many large-scale “adjoint” computations as there are “feature functions of parameters” fiα,i=1,...,TF (where TF denotes the total number of feature functions) for solving the left side of the 2nd-LASS with TF distinct sources on its right side. In contradistinction, the 2nd-CASAM-L requires TP (where TP denotes the total number of model parameters or non-zero first-order sensitivities) large-scale computations for solving the same left side of the 2nd-LASS but with TP distinct sources. Remarkably, the types of “large-scale” computations are the same in both the 2nd-FASAM-L and 2nd-CASAM-L since they both solve the same operator equations and boundary conditions within the respective 2nd-LASS systems; only the sources for these adjoint systems differ from each other. Since TFTP, the 2nd-FASAM-L is considerably more efficient than the 2nd-CASAM-L for computing the exact expressions of the second-order sensitivities of a model response to the uncertain parameters, boundaries, and internal interfaces of the model. For the illustrative contributon-response paradigm model, the computation of the second-order response sensitivities with respect to the model parameters using the 2nd-FASAM-L requires just two large-scale computations, for solving the two 2nd-LASS that correspond to the first-order sensitivities, Rc/f1 and Rc/f2, of the contributon response with respect to the respective components, f1α and f2α, of the model’s “feature function” fα. In contradistinction, computing the second-order sensitivities to the model parameters using the 2nd-CASAM-L requires TP large-scale computations, one for solving each of the 2nd-LASS that corresponds to each one of the distinct first-order sensitivities Rc/αi, i=1,...,TP, of the response with respect to the TP model parameters. Remarkably, only the unmixed second-order sensitivity 2Rcφ,ψ/f1f1 and the mixed second-order sensitivity 2Rcφ,ψ/f1f2=2Rcφ,ψ/f2f1 are non-zero. The unmixed second-order sensitivity is identically zero, i.e., 2Rcφ,ψ/f2f20. In contradistinction, computing the second-order sensitivities to the model parameters using the 2nd-CASAM-L requires TP large-scale computations, one for solving each of the 2nd-LASS that corresponds to one of the distinct TP model parameters. None of the second-order sensitivities with respect to the primary model parameters vanish.

(iv) For computing the exact expressions of the third-order response sensitivities with respect to the primary model’s parameters, the 3rd-FASAM-L requires at most TFTF+1/2 large-scale “adjoint” computations for solving the 3rd-LASS with TFTF+1/2 distinct sources, while the 3rd-CASAM-L requires at most TPTP+1/2 large-scale computations for solving the 3rd-LASS with TPTP+1/2 distinct sources. For the illustrative “contributon response of the neutron slowing-down” paradigm model, the computation of the third-order response sensitivities with respect to the model parameters using the 3rd-FASAM-L requires only two large-scale computations for solving the two 3rd-LASS that correspond to the respective non-zero second-order sensitivities 2Rcφ,ψ/f1f1 and 2Rcφ,ψ/f1f2=2Rcφ,ψ/f2f1. Only the unmixed third-order sensitivity 3Rcφ,ψ/f1f1f1 and the mixed third-order sensitivity 3Rcφ,ψ/f1f1f2 are non-zero; all other third-order sensitivities vanish identically. In contradistinction, the 3rd-CASAM-L requires all TPTP+1/2 large-scale computations for solving the 3rd-LASS since all of the second-order sensitivities with respect to the primary model parameters are non-zero. Furthermore, all of the third-order response sensitivities with respect to the primary model parameters are non-zero.

(v) The same computational count of “large-scale computations” caries over when computing the fourth- and higher-order sensitivities, i.e., the formula for calculating the “number of large-scale adjoint computations” is formally the same for both the nth-FASAM-N (Cacuci, 2024a, 2024b) and nth-CASAM-N (Cacuci, 2023a), but the “variable” in the formula for determining the number of adjoint computations for the nth-FASAM-N is TF (i.e., total number of feature functions), while the counterpart for the formula for determining the number of adjoint computations for the nth-CASAM-N is TP (i.e., total number of model parameters). Since TFTP, it follows that the higher the order of computed sensitivities, the more efficient the nth-FASAM-N (Cacuci, 2024a, 2024b) becomes compared to the nth-CASAM-N (Cacuci, 2023a).

(vi) The probability of encountering vanishing sensitivities is much higher when using the nth-FASAM-L than when using the nth-CASAM-L. For the illustrative “contributon response of the neutron slowing-down” paradigm model, it is evident that the only a few of the response sensitivities of fourth order (and higher order) with respect to the components of the feature function fα will not vanish, and the non-vanishing sensitivities will all involve the component f1α of the feature function since this component appears in an exponential, whereas the other component appears just as a multiplicative factor. In contradistinction, none of the higher-order response sensitivities with respect to the primary model parameters will vanish using the 2nd-CASAM-L.

(vii) When a model has no “feature” functions of parameters, but only comprises primary parameters, the nth-FASAM-L becomes identical to the nth-CASAM-L.

(viii) Both the nth-FASAM-L and nth-CASAM-L are formulated in linearly increasing higher-dimensional Hilbert spaces—as opposed to exponentially increasing parameter-dimensional spaces—thus overcoming the limitation of dimensionality in the sensitivity analysis of linear systems. Both the nth-FASAM-L and nth-CASAM-L are incomparably more efficient and more accurate than any other method (statistical, finite differences, etc.) for computing the exact expressions of response sensitivities (of any order) with respect to the uncertain parameters, boundaries, and internal interfaces of the model.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.

Author contributions

DC: conceptualization, data curation, formal analysis, investigation, methodology, project administration, resources, software, supervision, validation, visualization, writing–original draft, and writing–review and editing.

Funding

The author(s) declare that no financial support was received for the research, authorship, and/or publication of this article.

Conflict of interest

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Keywords: arbitrarily high-order adjoint sensitivity analysis, nth-order feature adjoint sensitivity analysis methodology for response-coupled forward/adjoint linear systems, response-coupled forward/adjoint systems, neutron-slowing down, sensitivity of responses to model features

Citation: Cacuci DG (2024) nth-order feature adjoint sensitivity analysis methodology for response-coupled forward/adjoint linear systems: II. Illustrative application to a paradigm energy system. Front. Energy Res. 12:1421519. doi: 10.3389/fenrg.2024.1421519

Received: 22 April 2024; Accepted: 14 August 2024;
Published: 05 September 2024.

Edited by:

Shripad T. Revankar, Purdue University, United States

Reviewed by:

Shichang Liu, North China Electric Power University, China
Jian Guo, Chinese Academy of Sciences (CAS), China

Copyright © 2024 Cacuci. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Dan Gabriel Cacuci, Y2FjdWNpQGNlYy5zYy5lZHU=

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.