Second-order adjoint sensitivity analysis procedure (SO-ASAP) for computing exactly and efficiently first- and second-order sensitivities in large-scale linear systems:II. Illustrative application to a paradigm particle diffusion problem

This work presents an illustrative application of the second-order adjoint sensitivity analysis procedure (SO-ASAP) to a paradigm neutron diffusion problem, which is sufficiently simple to admit an exact solution, thereby making transparent the mathematical derivations underlying the SO-ASAP. The illustrative application presented in this work shows that the actual number of adjoint computations needed for computing all of the first- and second-order response sensitivities may significantly less than 2*N+1 per response. For this illustrative problem, four (4) large-scale adjoint computations sufficed for the complete and exact computations of all 4 first- and 10 distinct second-order derivatives. Furthermore, the construction and solution of the SASS requires very little additional effort beyond the construction of the adjoint sensitivity system needed for computing the first-order sensitivities. Only the sources on the right-sides of the diffusion operator needed to be modified; the left-side of the differential equations remained unchanged. Most of the second-order relative sensitivities are just as large as or larger than the first-order ones. We show that the second-order sensitivities cause the expected value of the response to differ from the computed nominal value of the response; and they contribute decisively to causing asymmetries in the response distribution. Neglecting the second-order sensitivities would nullify the third-order response correlations, and hence would nullify the skewness of the response; consequently, any events occurring in a response's long and/or short tails, which are characteristic of rare but decisive events would likely be missed. We expect the SO-ASAP to affect significantly other fields that need efficiently computed second-order response sensitivities, e.g., optimization, data assimilation/adjustment, model calibration, and predictive modeling

Fluctuation-Response Relations for Multi-Time Correlations

We show that time-correlation functions of arbitrary order for any random variable in a statistical dynamical system can be calculated as higher-order response functions of the mean history of the variable. The response is to a ``control term'' added as a modification to the master equation for statistical distributions. The proof of the relations is based upon a variational characterization of the generating functional of the time-correlations. The same fluctuation-response relations are preserved within moment-closures for the statistical dynamical system, when these are constructed via the variational Rayleigh-Ritz procedure. For the 2-time correlations of the moment-variables themselves, the fluctuation-response relation is equivalent to an ``Onsager regression hypothesis'' for the small fluctuations. For correlations of higher-order, there is a new effect in addition to such linear propagation of fluctuations present instantaneously: the dynamical generation of correlations by nonlinear interaction of fluctuations. In general, we discuss some physical and mathematical aspects of the {\it Ans\"{a}tze} required for an accurate calculation of the time correlations. We also comment briefly upon the computational use of these relations, which is well-suited for automatic differentiation tools. An example will be given of a simple closure for turbulent energy decay, which illustrates the numerical application of the relations.Comment: 28 pages, 1 figure, submitted to Phys. Rev.

The european project NURISP for nuclear reactor simulation

The NURISP project aims at developing the European NURESIM reference simulation platform [1] for nuclear reactor. A first version of NURESIM was delivered in 2008. 22 organizations from 14 European countries contribute to the further development of this platform. NURISP also includes a User’s Group (UG) whose members are not NURISP partners and come from the industrial nuclear sector or European and non-European R&D labs. Users can benefit from the use of the NURESIM platform, methods, results and modules and they provide concrete input and feedback on the use of these elements

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

Monte Carlo uncertainty propagation approaches in ADS burn-up calculations

In activation calculations, there are several approaches to quantify uncertainties: deterministic by means of sensitivity analysis, and stochastic by means of Monte Carlo. Here, two different Monte Carlo approaches for nuclear data uncertainty are presented: the first one is the Total Monte Carlo (TMC). The second one is by means of a Monte Carlo sampling of the covariance information included in the nuclear data libraries to propagate these uncertainties throughout the activation calculations. This last approach is what we named Covariance Uncertainty Propagation, CUP. This work presents both approaches and their differences. Also, they are compared by means of an activation calculation, where the cross-section uncertainties of 239Pu and 241Pu are propagated in an ADS activation calculation

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