"Fast numerical methods in continuous and discrete times"Nguyen, Dang-KhoaWe study continuous-time dynamics and discrete algorithms in real Hilbert spaces for finding zeros of a single-valued, monotone, continuous operator. We begin with a second-order system featuring vanishing damping, for which we establish fast convergence rates for both the operator residual and the gap function, and we prove weak convergence of the trajectory to an operator zero. Time discretizations yield accelerated variants of Optimistic Gradient Descent Ascent (OGDA) that inherit these asymptotic properties from the continuous system. By adding a Tikhonov regularization term, we further guarantee strong convergence to the minimal-norm solution, with explicit decay rates for both velocity and operator norm. Additionally, we explore flows incorporating time-rescaling and an anchor point, showing how they complement recent Fast OGDA results. Finally, we present some connections between the Heavy-ball method and Nesterov acceleration in both optimization and monotone operator frameworks. Joint works with Hedy Attouch, Radu Ioan Bot, Ernö Robert Csetnek, and David Alexander Hulett. |
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