On the global and local convergence of damped inverse iterations for the Gross-Pitaevskii eigenvalue problem

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In this talk we discuss the computation of ground states of Bose-Einstein condensates by solving the stationary Gross-Pitaevskii equation (GPE). The stationary GPE is an eigenvalue problem with a nonlinearity in the eigenfunction. For solving it we propose an energy-adaptive Riemannian gradient method with provable global convergence. Furthermore, we show how the method can be interpreted as a generalized inverse iteration with damping and how we can recover explicit convergence rates depending on spectral gaps of a linearized Gross-Pitaevskii operator. We also discuss the implications of our findings on the usage of spectral shifting strategies in such nonlinear settings.

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