Spectral analysis by direct and adjoint Monte Carlo methods

International audienceTime or $\alpha$ 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 $\alpha$ eigenvalue problem, which paves the way towards the spectral analysis of time-dependent systems (Betzler, 2014; Betzler et al., 2014, 2015, 2018). In this work, we improve the existing methods in two directions. We first show that the $\alpha$-k modified power iteration scheme can be successfully applied to the estimation of the matrix elements in the direct formulation of the $\alpha$ eigenvalue problem, which removes the bias on the fundamental eigenvalue and eigenvector of the discretized matrix, similarly to what happens for the fission matrix in the k-eigenvalue problems. Then, we show that the matrix elements for the adjoint formulation of the eigenvalue problem can be estimated by using the Generalized Iterated Fission Probability method, which we have introduced in order to compute the fundamental adjoint $\alpha$ eigenfunction. We will verify the proposed algorithms and probe their convergence as a function of the size of the discretized matrices on some simplified benchmark configurations where exact reference solutions can be obtained