Related papers: Data Assimilation with Higher Order Finite Element…
Nudging is a data assimilation method amenable to both analysis and implementation. It also has the (reported) advantage of being insensitive to model errors compared to other assimilation methods. However, nudging behavior in the presence…
Continuous data assimilation (CDA) methods, such as the nudging algorithm introduced by Azouani, Olson, and Titi (AOT), have proven to be highly effective in deterministic settings for asymptotically synchronizing approximate solutions with…
This paper presents a robust, adaptive numerical scheme for simulating high density ratio and high shear multiphase flows on locally refined Cartesian grids that adapt to the evolving interfaces and track regions of high vorticity. The…
We investigate novel fitted finite element approximations for two-phase Navier--Stokes flow. In particular, we consider both Eulerian and Arbitrary Lagrangian--Eulerian (ALE) finite element formulations. The moving interface is approximated…
In this paper we propose the use of a continuous data assimilation algorithm for miscible flow models in a porous medium. In the absence of initial conditions for the model, observed sparse measurements are used to generate an approximation…
We study a finite element approximation of a coupled fluid-structure interaction consisting of a three-dimensional incompressible viscous fluid governed by the unsteady Stokes equations and a two-dimensional elastic plate. To avoid the use…
We study a continuous data assimilation algorithm proposed by Azouani, Olson, and Titi (AOT) in the context of an unknown Reynolds number. We determine the large-time error between the true solution of the 2D Navier-Stokes equations and the…
A two-grid scheme based on mixed finite-element approximations to the incompressible Navier-Stokes equations is introduced and analyzed. In the first level the standard mixed finite-element approximation over a coarse mesh is computed. In…
In this paper, we try to uncover the second-order essence of several first-order optimization methods. For Nesterov Accelerated Gradient, we rigorously prove that the algorithm makes use of the difference between past and current gradients,…
Data assimilation combines (imperfect) knowledge of a flow's physical laws with (noisy, time-lagged, and otherwise imperfect) observations to produce a more accurate prediction of flow statistics. Assimilation by nudging (from 1964), while…
In contrast with the diffusion equation which smoothens the initial data to $C^\infty$ for $t>0$ (away from the corners/edges of the domain), the subdiffusion equation only exhibits limited spatial regularity. As a result, one generally…
The ultra-long relaxation time of glass transition makes it difficult to construct atomic models of amorphous materials by conventional methods. We propose a novel method for building such atomic models using data assimilation method by…
This paper contains the latest installment of the authors' project on developing ensemble based data assimilation methodology for high dimensional fluid dynamics models. The algorithm presented here is a particle filter that combines model…
We consider a non-linear Bayesian data assimilation model for the periodic two-dimensional Navier-Stokes equations with initial condition modelled by a Gaussian process prior. We show that if the system is updated with sufficiently many…
We analyze the performance of a data-assimilation algorithm based on a linear feedback control when used with observational data that contains measurement errors. Our model problem consists of dynamics governed by the two-dimension…
An immersed-boundary method for the incompressible Navier--Stokes equations is presented. It employs discrete forcing for a sharp discrimination of the solid-fluid interface, and achieves second-order accuracy, demonstrated in examples with…
In this paper, we introduce a new, fast data assimilation algorithm for a 2D linear advection equation with divergence-free coefficients. We first apply the nodal discontinuous Galerkin (DG) method to discretize the advection equation, and…
Second-order pooling, a.k.a.~bilinear pooling, has proven effective for deep learning based visual recognition. However, the resulting second-order networks yield a final representation that is orders of magnitude larger than that of…
This paper considers improving the Picard and Newton iterative solvers for the Navier-Stokes equations in the setting where data measurements or solution observations are available. We construct adapted iterations that use continuous data…
In this article we consider the 2D Navier-Stokes equations with variable viscosity depending on the vertical position. As our main result we establish linear enhanced dissipation near the non-affine stationary states replacing Couette flow.…