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Variance reduction techniques

Publié le 30 novembre 2018

By default, Tripoli-4® uses common variance reduction techniques such as implicit capture, particle splitting and Russian roulette, so that particle transport is non-analog.

Moreover, Tripoli-4® has a special built-in variance reduction module called INIPOND. The key algorithm of INIPOND is based on the Exponential Transform method [1], with an automatic pre-calculation of the importance map. For shielding configurations associated to deep penetration problems, Tripoli-4® automatically pre-computes an importance map. This importance function is assumed to be factorized in space, energy, angle and time variables, with a coupling between space and energy variables. A space and energy grid has to be defined by the user for the initialization of the importance map by the code. An arbitrary of fictitious detectors (typically, one or a few) are actually placed by the user in the geometry and act as 'attractors' for simulated particles. These attractors are especially useful for dealing with passing-round configurations.

At the beginning of the simulation, adjustments of the importance map are performed from group to group by taking advantage of statistics on discrepancies between particle weights and reference weights given by the importance map, when particles slow down from group to group. If the Exponential Transform technique is used to perform non-analog flights, the sampling from the free flight kernel is performed by replacing the total cross-section by the biased macroscopic total cross-section, and statistical weights are adjusted accordingly.


Left to right: Reactor Pressure Vessel flux assessment, display of collision sites with optimised variance reduction

In addition to producing an internally-generated importance map, Tripoli-4® is also capable of reading an external importance function produced by a deterministic code by solving the adjoint Boltzmann equation on a space and energy mesh. Successful tests have been performed by coupling the importance map generated by the deterministic code APOLLO3 to Tripoli-4® for shielding problems [2].

For coupled neutron-photon simulations, two importance maps may be required. For such configurations, a diagnosis tool is now available in Tripoli-4® so as to facilitate the adjustment of the neutron importance map [3].

During the simulation, the Exponential Transform method is also combined with a quota sampling algorithm and splitting and Russian roulette are finally performed before each track-length and collision sampling, according to the importance map.


Ex-core chamber : example of source contribution calculation using Green’s functions.

Tripoli-4® allows for Green's functions calculations for parametric source analyses [4]. The algorithm is based on the pre-calculation of Green's functions between source and tally areas. The storage is performed during a first standard run generally based on the efficient variance reduction techniques of the code. Then, for any new source configuration, the requested tallies are computed by recombining the Green's function values with the new source distribution, without simulating any particle history. Fine source contribution analysis can also be achieved through this feature. Analyses can thus be performed in much shorter calculation times, when compared with full Monte Carlo simulations.

A new variance reduction method has been recently proposed in the literature of applied mathematics: the Adaptive Multilevel Splitting (AMS) [5]. Originally designed to help the simulation of rare events associated to continuous Markov chains, it has been extended to discrete Markov chains [6]. The AMS algorithm has been successfully transposed to the field of particle transport and implemented in Tripoli-4® as an effective variance reduction technique for neutron and photon transport [7], including coupled neutron-photon transport [8].


[1] F.H. Clark, The Exponential Transform as an importance sampling device. A review. Oak Ridge National Laboratory report ORNL-RSIC-14 (1966).

[2] D. Mancusi et al., in Proceedings of RPSD2018, Santa Fe, USA (2018).

[3] O. Petit, Y.K. Lee, C.M. Diop, Progr. Nucl. Sci. Tech. 4, 408-412 (2014).

[4] S. Bourganel, O. Petit, C.M. Diop, Nuclear Technology 184, 29-41 (2013).

[5] F. Cérou and A. Guyader, Stoch. Anal. Appl. 25, 417 (2007).

[6] C.-E. Bréhier et al., Ann. App. Prob. 26, 3559 (2016).

[7] H. Louvin et al., EPJ Nuclear Sci. Technol. 3, 29 (2017).

[8] H. Louvin, O. Petit, in Proceedings of RPSD2018, Santa Fe, USA (2018).