Calculating Alpha Eigenvalues and Eigenfunctions with a Markov Transition Rate Matrix Monte Carlo Method

Spectral analysis by direct and adjoint Monte Carlo methods

“…Despite some successful attempts, direct Monte Carlo simulation of higher α eigenvalues and eigenmodes has received only limited attention (Yamamoto, 2011). In this context, matrix-filling Monte Carlo methods have recently drawn much in-terest Betzler et al, , 2015Betzler et al, , 2018: the underlying idea is to estimate by Monte Carlo simulation the elements of a matrix whose eigenvalues and eigenvectors converge to the true α eigenvalues and eigenmodes in the limit of a sufficiently fine discretization of the phase space (Betzler et al, 2018) 1 . This approach is very similar in spirit to the better-known fission matrix method for k-eigenvalues (Dufek and Gudowski, 2009;Carney et al, 2014).…”

“…Although the α eigenvalues and eigenmodes thus estimated are generally biased because of the finite size of the matrix, this method allows obtaining a fairly accurate picture of the entire spectrum and thus grasping the time evolution of the system (Betzler et al, , 2015, even for complex three-dimensional configurations Betzler et al, 2018). Moreover, once the α spectrum and the associated eigenvectors have been determined from the matrix, the full time-dependent evolution of the neutron and precursor populations can also be reconstructed, at least in principle, by using the direct and adjoint matrices (Betzler et al, 2018).…”

“…The proposed matrix-filling Monte Carlo method based on a transition rate method related to the adjoint formulation of the α eigenvalue equation 2 suffers however from two approximations: the first is due to the fact that the exact adjoint formulation is in practice replaced by a forward formulation, in order to avoid the explicit simulation of backward random walks Betzler et al, 2018). The second is due to the fact that the matrix elements are estimated and filled in the course of a k-eigenvalue or c-eigenvalue Monte Carlo calculation, which induces a systematic bias even on the fundamental eigenvalue and eigenvector Betzler et al, 2018).…”

“…The proposed matrix-filling Monte Carlo method based on a transition rate method related to the adjoint formulation of the α eigenvalue equation 2 suffers however from two approximations: the first is due to the fact that the exact adjoint formulation is in practice replaced by a forward formulation, in order to avoid the explicit simulation of backward random walks Betzler et al, 2018). The second is due to the fact that the matrix elements are estimated and filled in the course of a k-eigenvalue or c-eigenvalue Monte Carlo calculation, which induces a systematic bias even on the fundamental eigenvalue and eigenvector Betzler et al, 2018). Although both approximations vanish in the limit of a sufficiently fine discretization of the phase space, for realistic systems this might require very large matrix sizes, entailing severe memory footprint issues: contrary to the fission matrix, where only a spatial discretization is required, the matrix associated to α-eigenvalue problems demands a full discretization of the phase space, including space, direction and energy .…”

“…

Time or α eigenvalues are key to several applications in reactor physics, encompassing start-up analysis and reactivity measurements. In a series of recent works, a Monte Carlo method has been proposed in order to estimate the elements of the matrices that represent the discretized formulation of the operators involved in the α-eigenvalue problem, which paves the way towards the spectral analysis of time-dependent systems Betzler et al, , 2015Betzler et al, , 2018. In this work, we improve the existing methods in two directions.

…”