Application of the Second-Order Comprehensive Adjoint Sensitivity Analysis Methodology to Compute First- and Second-Order Sensitivities of Flux Functionals in a Multiplying System With Source

This work presents an application of the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) to the neutron transport Boltzmann equation that models a multiplying subcritical system comprising a nonfission neutron source to compute efficiently and exactly all of the first- and second-order functional derivatives (sensitivities) of a detector’s response to all of the model’s parameters, including isotopic number densities, microscopic cross sections, fission spectrum, sources, and detector response function. As indicated by the general theoretical considerations underlying the 2nd-ASAM, the number of computations required to obtain the first and second orders increases linearly in augmented Hilbert spaces as opposed to increasing exponentially in the original Hilbert space. The results presented in this work are currently being implemented in several production-oriented three-dimensional neutron transport code systems for analyzing specific subcritical systems

Towards Overcoming the Curse of Dimensionality: The Third-Order Adjoint Method for Sensitivity Analysis of Response-Coupled Linear Forward/Adjoint Systems, Uncertainty Quantification and Predictive Modeling with Applications to Nuclear Energy Systems

This work presents the Third-Order Adjoint Sensitivity Analysis Methodology (3rd-ASAM) for response-coupled forward and adjoint linear systems. The 3rd-ASAM enables the efficient computation of the exact expressions of the 3rd-order functional derivatives (“sensitivities”) of a general system response, which depends on both the forward and adjoint state functions, with respect to all of the parameters underlying the respective forward and adjoint systems. Such responses are often encountered when representing mathematically detector responses and reaction rates in reactor physics problems. The 3rd-ASAM extends the 2nd-ASAM in the quest to overcome the “curse of dimensionality” in sensitivity analysis, uncertainty quantification and predictive modeling. This work also presents new formulas that incorporate the contributions of the 3rd-order sensitivities into the expressions of the first four cumulants of the response distribution in the phase-space of model parameters. Using these newly developed formulas, this work also presents a new mathematical formalism, called the 2nd/3rd-BERRU-PM “Second/Third-Order Best-Estimated Results with Reduced Uncertainties Predictive Modeling”) formalism, which combines experimental and computational information in the joint phase-space of responses and model parameters, including not only the 1st-order response sensitivities, but also the complete hessian matrix of 2nd-order second-sensitivities and also the 3rd-order sensitivities, all computed using the 3rd-ASAM. The 2nd/3rd-BERRU-PM uses the maximum entropy principle to eliminate the need for introducing and “minimizing” a user-chosen “cost functional quantifying the discrepancies between measurements and computations,” thus yielding results that are free of subjective user-interferences while generalizing and significantly extending the 4D-VAR data assimilation procedures. Incorporating correlations, including those between the imprecisely known model parameters and computed model responses, the 2nd/3rd-BERRU-PM also provides a quantitative metric, constructed from sensitivity and covariance matrices, for determining the degree of agreement among the various computational and experimental data while eliminating discrepant information. The mathematical framework of the 2nd/3rd-BERRU-PM formalism requires the inversion of a single matrix of size Nr Nr, where Nr denotes the number of considered responses. In the overwhelming majority of practical situations, the number of responses is much less than the number of model parameters. Thus, the 2nd-BERRU-PM methodology overcomes the curse of dimensionality which affects the inversion of hessian matrices in the parameter space

Introducing the Journal of Nuclear Engineering: An Interdisciplinary Open Access Journal Dedicated to Publishing Research in Nuclear and Radiation Sciences and Applications

Note: In lieu of an abstract, this is an excerpt from the first page. In 1938, Strassmann, Hahn and Meitner discovered neutron-induced nuclear fission in uranium, forever changing our world and opening multiple paths to developing nuclear energy, nuclear medicine, instrumentation, space propulsion, environmental monitoring, remediation and nuclear security [...

First-Order Comprehensive Adjoint Method for Computing Operator-Valued Response Sensitivities to Imprecisely Known Parameters, Internal Interfaces and Boundaries of Coupled Nonlinear Systems: II. Application to a Nuclear Reactor Heat Removal Benchmark

This work illustrates the application of a comprehensive first-order adjoint sensitivity analysis methodology (1st-CASAM) to a heat conduction and convection analytical benchmark problem which simulates heat removal from a nuclear reactor fuel rod. This analytical benchmark problem can be used to verify the accuracy of numerical solutions provided by software modeling heat transport and fluid flow systems. This illustrative heat transport benchmark shows that collocation methods require one adjoint computation for every collocation point while spectral expansion methods require one adjoint computation for each cardinal function appearing in the respective expansion when recursion relations cannot be developed between the corresponding adjoint functions. However, it is also shown that spectral methods are much more efficient when recursion relations provided by orthogonal polynomials make it possible to develop recursion relations for computing the corresponding adjoint functions. When recursion relations cannot be developed for the adjoint functions, the collocation method is probably more efficient than the spectral expansion method, since the sources for the corresponding adjoint systems are just Dirac delta functions (which makes the respective computation equivalent to the computation of a Green’s function), rather than the more elaborated sources involving high-order Fourier basis functions or orthogonal polynomials. For systems involving many independent variables, it is likely that a hybrid combination of spectral expansions in some independent variables and collocation in the remaining independent variables would provide the most efficient computational outcome

Illustrative Application of the nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems to the Nordheim–Fuchs Reactor Dynamics/Safety Model

The application of the recently developed “nth-order comprehensive sensitivity analysis methodology for nonlinear systems” (abbreviated as “nth-CASAM-N”) has been previously illustrated on paradigm nonlinear space-dependent problems. To complement these illustrative applications, this work illustrates the application of the nth-CASAM-N to a paradigm nonlinear time-dependent model chosen from the field of reactor dynamics/safety, namely the well-known Nordheim–Fuchs model. This phenomenological model describes a short-time self-limiting power transient in a nuclear reactor system having a negative temperature coefficient in which a large amount of reactivity is suddenly inserted, either intentionally or by accident. This model is sufficiently complex to demonstrate all the important features of applying the nth-CASAM-N methodology yet admits exact closed-form solutions for the energy released in the transient, which is the most important system response. All of the expressions of the first- and second-level adjoint functions and, subsequently, the first- and second-order sensitivities of the released energy to the model’s parameters are obtained analytically in closed form. The principles underlying the application of the 3rd-CASAM-N methodology for the computation of the third-order sensitivities are demonstrated for both mixed and unmixed second-order sensitivities. For the Nordheim–Fuchs model, a single adjoint computation suffices to obtain the six 1st-order sensitivities, while two adjoint computations suffice to obtain all of the 36 second-order sensitivities (of which 21 are distinct). This illustrative example demonstrates that the number of (large-scale) adjoint computations increases at most linearly within the nth-CASAM-N methodology, as opposed to the exponential increase in the parameter-dimensional space which occurs when applying conventional statistical and/or finite difference schemes to compute higher-order sensitivities. For very large and complex models, the nth-CASAM-N is the only practical methodology for computing response sensitivities comprehensively and accurately, overcoming the curse of dimensionality in sensitivity analysis of nonlinear systems