  We study different approaches to implementing sparse-in-time observations
into the the Azouani-Olson-Titi data assimilation algorithm. We propose a new
method which introduces a "data assimilation window" separate from the
observational time interval. We show that by making this window as small as
possible, we can drastically increase the strength of the nudging parameter
without losing stability. Previous methods used old data to nudge the solution
until a new observation was made. In contrast, our method stops nudging the
system almost immediately after an observation is made, allowing the system
relax to the correct physics. We show that this leads to an order-of-magnitude
improvement in the time to convergence in our 3D Navier-Stokes simulations.
Moreover, our simulations indicate that our approach converges at nearly the
same rate as the idealized method of direct replacement of low Fourier modes
proposed by Hayden, Olson, and Titi (HOT). However, our approach can be readily
adapted to non-idealized settings, such as finite element methods, finite
difference methods, etc., since there is no need to access Fourier modes as our
method works for general interpolants. It is in this sense that we think of our
approach as ``second best;'' that is, the ``best'' method would be the direct
replacement of Fourier modes as in HOT, but this idealized approach is
typically not feasible in physically realistic settings. While our method has a
convergence rate that is slightly sub-optimal compared to the idealized method,
it is directly compatible with real-world applications. Moreover, we prove
analytically that these new algorithms are globally well-posed, and converge to
the true solution exponentially fast in time. In addition, we provide the first
3D computational validation of HOT algorithm.
