Newton-Based Alternating Methods for the Ground State of a Class of Multicomponent Bose–Einstein Condensates

SIAM Journal on Optimization, Volume 34, Issue 3, Page 3136-3162, September 2024.
Abstract. The computation of the ground state of special multicomponent Bose–Einstein condensates (BECs) can be formulated as an energy functional minimization problem with spherical constraints. It leads to a nonconvex quartic-quadratic optimization problem after suitable discretizations. First, we generalize the Newton-based methods for single-component BECs to the alternating minimization scheme for multicomponent BECs. Second, the global convergent alternating Newton-Noda iteration (ANNI) is proposed. In particular, we prove the positivity preserving property of ANNI under mild conditions. Finally, our analysis is applied to a class of more general “multiblock” optimization problems with spherical constraints. Numerical experiments are performed to evaluate the performance of proposed methods for different multicomponent BECs, including pseudo spin-1/2, antiferromagnetic spin-1 and spin-2 BECs. These results support our theory and demonstrate the efficiency of our algorithms.