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Runaway electrons (RE) generated during magnetic disruptions present a majorthreat to the safe operation of plasma nuclear fusion reactors. A criticalaspect of understanding RE dynamics is to calculate the runaway probability,i.e., the probability that an electron in the phase space will runaway on, orbefore, a prescribed time. Such probability can be obtained by solving theadjoint equation of the underlying Fokker-Planck equation that controls theelectron dynamics. In this effort, we present a sparse-grid probabilisticscheme for computing the runaway probability. The key ingredient of ourapproach is to represent the solution of the adjoint equation as a conditionalexpectation, such that discretizing the differential operator reduces to theapproximation of a set of integrals. Adaptive sparse grid interpolation isutilized to approximate the map from the phase space to the runawayprobability. The main novelties of this effort are the integration of thesparse-grid method into the probabilistic numerical scheme for computing escapeprobability, as well as the demonstration in computing RE probabilities. Twonumerical examples are given to illustrate that the proposed method can achieve$\mathcal{O}(\Delta t)$ convergence, as well as the adaptive refinementstrategy can effectively handle the sharp transition layer between the runawayand non-runaway regions.

A sparse-grid probabilistic scheme for approximation of the runaway probability of electrons in fusion tokamak simulation