A Particle-Based Algorithm for Distributional Optimization on \textit{Constrained Domains} via Variational Transport and Mirror Descent

We consider the optimization problem of minimizing an objective functional, which admits a variational form and is defined over probability distributions on the constrained domain, which poses challenges to both theoretical analysis and algorithmic design. Inspired by the mirror descent algorithm for constrained optimization, we propose an iterative particle-based algorithm, named Mirrored Variational Transport (mirrorVT), extended from the Variational Transport framework [7] for dealing with the constrained domain. In particular, for each iteration, mirrorVT maps particles to an unconstrained dual domain induced by a mirror map and then approximately perform Wasserstein gradient descent on the manifold of distributions defined over the dual space by pushing particles. At the end of iteration, particles are mapped back to the original constrained domain. Through simulated experiments, we demonstrate the effectiveness of mirrorVT for minimizing the functionals over probability distributions on the simplex- and Euclidean ball-constrained domains. We also analyze its theoretical properties and characterize its convergence to the global minimum of the objective functional

Stochastic Estimation of Nuclear Level Density in the Nuclear Shell Model: An Application to Parity-Dependent Level Density in 58^{58}Ni

We introduce a novel method to obtain level densities in large-scale shell-model calculations. Our method is a stochastic estimation of eigenvalue count based on a shifted Krylov-subspace method, which enables us to obtain level densities of huge Hamiltonian matrices. This framework leads to a successful description of both low-lying spectroscopy and the experimentally observed equilibration of $J^\pi=2^+$ and $2^-$ states in $^{58}$Ni in a unified manner.Comment: 13 pages, 4 figure