Related papers: Continuous Data Assimilation for Semilinear Parabo…
We address an open problem posed by H. Brezis, M. Marcus and A.C. Ponce in: Nonlinear elliptic equations with measures revisited. In: Mathematical Aspects of Nonlinear Dispersive Equations (J. Bourgain, C. Kenig, S. Klainerman, eds.),…
The accuracy of simulation-based forecasting in chaotic systems is heavily dependent on high-quality estimates of the system state at the time the forecast is initialized. Data assimilation methods are used to infer these initial conditions…
In this paper, a higher-order time-discretization scheme is proposed, where the iterates approximate the solution of the stochastic semilinear wave equation driven by multiplicative noise with general drift and diffusion. We employ a…
We study the effect of Gaussian perturbations on a class of model hyperbolic partial differential equations with double symplectic characteristics in low spatial dimensions, extending some recent work in [5]. The coefficients of our partial…
We present new convergence analyses for parallel subspace correction methods for unconstrained semicoercive and nearly semicoercive convex optimization problems, generalizing the theory of singular and nearly singular linear problems to a…
We consider backward problems for semilinear coupled parabolic systems in bounded domains. We prove conditional stability estimates for linear and semilinear systems of strongly coupled parabolic equations involving general semilinearities.…
Multiplicative noise (also known as speckle noise) models are central to the study of coherent imaging systems, such as synthetic aperture radar and sonar, and ultrasound and laser imaging. These models introduce two additional layers of…
Nudging is an important data assimilation technique where partial field measurements are used to control the evolution of a dynamical system and/or to reconstruct the entire phase-space configuration of the supplied flow. Here, we apply it…
Data assimilation (DA) estimates the state of an evolving dynamical system from noisy, partial observations, and is widely used in scientific simulation as well as weather and climate science. In practice, filtering methods rely on…
This paper is a contribution in the context of variational data assimilation combined with statistical learning. The framework of data assimilation traditionally uses data collected at sensor locations in order to bring corrections to a…
This paper is concerned with developing and analyzing two novel implicit temporal discretization methods for the stochastic semilinear wave equations with multiplicative noise. The proposed methods are natural extensions of well-known…
Data assimilation (DA) aims at forecasting the state of a dynamical system by combining a mathematical representation of the system with noisy observations taking into account their uncertainties. State of the art methods are based on the…
We consider a recently introduced continuous data assimilation (CDA) approach for downscaling a coarse resolution configuration of the 2D Benard convection equations into a finer grid. In this CDA, a nudging term, estimated as the misfit…
A continuing mystery in understanding the empirical success of deep neural networks is their ability to achieve zero training error and generalize well, even when the training data is noisy and there are more parameters than data points. We…
This paper focuses on continuous data assimilation (CDA) for the Navier-Stokes equations with nonlinear slip boundary conditions. CDA methods are typically employed to recover the original system when initial data or viscosity coefficients…
In this paper, we study the inverse problem for a class of abstract ultraparabolic equations which is well-known to be ill-posed. We employ some elementary results of semi-group theory to present the formula of solution, then show the…
A semi-Lagrangian method for parabolic problems is proposed, that extends previous work by the authors to achieve a fully conservative, flux-form discretization of linear and nonlinear diffusion equations. A basic consistency and…
Data assimilation combines information from physical observations and numerical simulation results to obtain better estimates of the state and parameters of a physical system. A wide class of physical systems of interest have solutions that…
Consider a continuous dynamical system for which partial information about its current state is observed at a sequence of discrete times. Discrete data assimilation inserts these observational measurements of the reference dynamical system…
This work focuses on the regularization by nonlinear noise for a class of partial differential equations that may only have local solutions. In particular, we obtain the global existence, uniqueness and the Feller property for stochastic 3D…