Related papers: Continuous Data Assimilation for Semilinear Parabo…

In this paper we consider the solution of monotone inverse problems using the particular example of a parameter identification problem for a semilinear parabolic PDE. For the regularized solution of this problem, we introduce a total…

The present article is devoted to well-posedness by noise for the continuity equation. Namely, we consider the continuity equation with non-linear and partially degenerate stochastic perturbations in divergence form. We prove the existence…

We develop an algorithm based on the nudging data assimilation scheme for the concurrent (on-the-fly) estimation of scalar parameters for a system of evolutionary dissipative partial differential equations in which the state is partially…

Sparse sensor networks in weather and ocean modeling observe only a small fraction of the system state, which destabilizes standard nudging-based data assimilation. We introduce Interpolated Discrepancy Data Assimilation (IDDA), which…

The recent statistical theory of neural networks focuses on nonparametric denoising problems that treat randomness as additive noise. Variability in image classification datasets does, however, not originate from additive noise but from…

We study a nonlinear-nudging modification of the Azouani-Olson-Titi continuous data assimilation (downscaling) algorithm for the 2D incompressible Navier-Stokes equations. We give a rigorous proof that the nonlinear-nudging system is…

The existence of random attractors for singular stochastic partial differential equations (SPDE) perturbed by general additive noise is proven. The drift is assumed only to satisfy the standard assumptions of the variational approach to…

The understanding of nonlinear, high dimensional flows, e.g, atmospheric and ocean flows, is critical to address the impacts of global climate change. Data Assimilation techniques combine physical models and observational data, often in a…

We interpret steady linear statistical inverse problems as artificial dynamic systems with white noise and introduce a stochastic differential equation (SDE) system where the inverse of the ending time $T$ naturally plays the role of the…

Many ecological studies and conservation policies are based on field observations of species, which can be affected by systematic variability introduced by the observation process. A recently introduced causal modeling technique called…

We consider the convected Helmholtz equation with a generalized Myers boundary condition (a boundary condition of the second-order) and characterize the set of physical parameters for which the problem is weakly well-posed. The model comes…

Data assimilation (DA) is widely used to combine physical knowledge and observations. It is nowadays commonly used in geosciences to perform parametric calibration. In a context of climate change, old calibrations can not necessarily be…

In this paper we propose a continuous data assimilation (downscaling) algorithm for the B\'enard convection in porous media using only coarse mesh measurements of the temperature. In this algorithm, we incorporate the observables as a…

A hybrid data assimilation algorithm is developed for complex dynamical systems with partial observations. The method starts with applying a spectral decomposition to the entire spatiotemporal fields, followed by creating a machine learning…

In using data assimilation to import information from observations to estimate parameters and state variables of a model, one must assume a distribution for the noise in the measurements and in the model errors. Using the path integral…

The aim of this paper is to extend the global error estimation and control addressed in Lang and Verwer [SIAM J. Sci. Comput. 29, 2007] for initial value problems to finite difference solutions of semilinear parabolic partial differential…

It is well understood that dynamic instability is among the primary drivers of forecast uncertainty in chaotic, physical systems. Data assimilation techniques have been designed to exploit this phenomena, reducing the effective dimension of…

These notes present an alternative approach to the asymptotic stability of stochastic partial differential equations driven by multiplicative noise, applicable to a wide range of dissipative systems. The method builds on general criteria…

We consider the numerical reconstruction of the spatially dependent conductivity coefficient and the source term in elliptic partial differential equations in a two-dimensional convex polygonal domain, with the homogeneous Dirichlet…

In this paper, we study a mixed variational problem subject to perturbations, where the noise term is modelled by means of a bilinear form that has to be understood to be "small" in some sense. Indeed, we consider a family of such problems…