Related papers: Continuing dynamic assimilation of the inner regio…

We use a diffuse interface method for solving Poisson's equation with a Dirichlet condition on an embedded curved interface. The resulting diffuse interface problem is identified as a standard Dirichlet problem on approximating regular…

Data assimilation is uniquely challenging in weather forecasting due to the high dimensionality of the employed models and the nonlinearity of the governing equations. Although current operational schemes are used successfully, our…

This work applies a continuous data assimilation scheme---a particular framework for reconciling sparse and potentially noisy observations to a mathematical model---to Rayleigh-B\'enard convection at infinite or large Prandtl numbers using…

Obtaining accurate high-resolution representations of model outputs is essential to describe the system dynamics. In general, however, only spatially- and temporally-coarse observations of the system states are available. These observations…

This paper improves the spectrally-filtered direct-insertion downscaling method for discrete-in-time data assimilation by introducing a relaxation parameter that overcomes a constraint on the observation frequency. Numerical simulations…

In this paper we prove optimal error estimates for {solutions with natural regularity} of the equations describing the unsteady motion of incompressible shear-thinning fluids. We consider a full space-time semi-implicit scheme for the…

Spatially referenced data are increasingly available thanks to the development of modern GPS technology. They also provide rich opportunities for spatial analytics in the field of marketing science. Our main interest is to propose a new…

Experimental design is central to science and engineering. A ubiquitous challenge is how to maximize the value of information obtained from expensive or constrained experimental settings. Bayesian optimal experimental design (OED) provides…

Data assimilation (DA) enables hydrologic models to update their internal states using near-real-time observations for more accurate forecasts. With deep neural networks like long short-term memory (LSTM), using either lagged observations…

A boundary integral based method for the stable reconstruction of missing boundary data is presented for the governing hyperbolic equation of elastodynamics in annular planar domains. Cauchy data in the form of the solution and traction is…

For scalar fully nonlinear partial differential equations depending on the Hessian andspatial coordinates, we present a general theory for obtaining comparison principles and well posedness for the associated Dirichlet problem with…

We discuss the Cauchy problem for anisotropic wave equations. Precisely, we address the question to know which kind of Cauchy data on the lateral boundary are necessary to guarantee uniqueness of solutions of an anisotropic wave equation.…

This paper is concerned with the recovery of (approximate) solutions to parabolic problems from incomplete and possibly inconsistent observational data, given on a time-space cylinder that is a strict subset of the computational domain…

In this paper we analyze the finite element approximation of the Stokes equations with non-smooth Dirichlet boundary data. To define the discrete solution, we first approximate the boundary datum by a smooth one and then apply a standard…

We introduce the Optimizing a Discrete Loss (ODIL) framework for the numerical solution of Partial Differential Equations (PDE) using machine learning tools. The framework formulates numerical methods as a minimization of discrete residuals…

To improve the predictability of complex computational models in the experimentally-unknown domains, we propose a Bayesian statistical machine learning framework utilizing the Dirichlet distribution that combines results of several…

Preconditioning has long been a staple technique in optimization, often applied to reduce the condition number of a matrix and speed up the convergence of algorithms. Although there are many popular preconditioning techniques in practice,…

Micrometeorology, city comfort, land use management and air quality monitoring increasingly become important environmental issues. To serve the needs, meteorology needs to achieve a serious advance in representation and forecast on…

In this paper we propose a numerical method to solve the Cauchy problem based on the viscous shallow water equations in an horizontally moving domain. More precisely, we are interested in a flooding and drying model, used to modelize the…

We consider the Cauchy problem for the Helmholtz equation with a domain in R^d, d>2 with N cylindrical outlets to infinity with bounded inclusions in R^{d-1}. Cauchy data are prescribed on the boundary of the bounded domains and the aim is…