Related papers: On Bayesian Consistency for Flows Observed Through…
We consider a multi-dimensional model of a compressible fluid in a bounded domain. We want to estimate the density and velocity of the fluid, based on the observations for only velocity. We build an observer exploiting the symmetries of the…
We investigate the shallow flow of viscous fluid into and out of a channel whose gap width increases as a power-law ($x^n$), where $x$ is the downstream axis. The fluid flows slowly, while injected at a rate in the form of $t^\alpha$, where…
The advection of a passive scalar by a quenched (frozen) incompressible velocity field is studied by extensive high precision numerical simulation and various approximation schemes. We show that second order self consistent perturbation…
We show how a complete mathematical description of a complicated physical phenomenon can be learned from observational data via a hybrid approach combining three simple and general ingredients: physical assumptions of smoothness, locality,…
Laplacian flows model the rate of change of each node's state as being proportional to the difference between its value and that of its neighbors. Typically, these flows capture diffusion or synchronization dynamics and are well-studied.…
Flow-based latent generative models such as Stable Diffusion 3 are able to generate images with remarkable quality, even enabling photorealistic text-to-image generation. Their impressive performance suggests that these models should also…
In this paper, we propose and study several inverse problems of identifying/determining unknown coefficients for a class of coupled PDE systems by measuring the average flux data on part of the underlying boundary. In these coupled systems,…
We present a new application of Lagrangian Perturbation Theory (LPT): the stability analysis of fluid flows. As a test case that demonstrates the framework we focus on the plane Couette flow. The incompressible Navier-Stokes equation is…
The need to blend observational data and mathematical models arises in many applications and leads naturally to inverse problems. Parameters appearing in the model, such as constitutive tensors, initial conditions, boundary conditions, and…
In this dissertation two-dimensional buoyancy-driven flows are investigated. While usually the Navier-Stokes equations are equipped with no-slip boundary conditions here we focus on the Navier-slip conditions that, depending on the system…
In a Bayesian inverse problem setting, the solution consists of a posterior measure obtained by combining prior belief, information about the forward operator, and noisy observational data. This measure is most often given in terms of a…
For $D$ a bounded domain in $\mathbb R^d, d \ge 2,$ with smooth boundary $\partial D$, the non-linear inverse problem of recovering the unknown conductivity $\gamma$ determining solutions $u=u_{\gamma, f}$ of the partial differential…
Many modern applications of Bayesian inference, such as in cosmology, are based on complicated forward models with high-dimensional parameter spaces. This considerably limits the sampling of posterior distributions conditioned on observed…
We develop and implement a Bayesian approach for the estimation of the shape of a two dimensional annular domain enclosing a Stokes flow from sparse and noisy observations of the enclosed fluid. Our setup includes the case of direct…
Inverse problems involve making inference about unknown parameters of a physical process using observational data. This paper investigates an important class of inverse problems -- the estimation of the initial condition of a…
Multiplicative noise arises in inverse problems when, for example, uncertainty on measurements is proportional to the size of the measurement itself. The likelihood that arises is hence more complicated than that from additive noise. We…
Bayesian approaches are one of the primary methodologies to tackle an inverse problem in high dimensions. Such an inverse problem arises in hydrology to infer the permeability field given flow data in a porous media. It is common practice…
Inverse problems governed by partial differential equations (PDEs) play a crucial role in various fields, including computational science, image processing, and engineering. Particularly, Darcy flow equation is a fundamental equation in…
We present a continuation method that entails generating a sequence of transition probability density functions from the prior to the posterior in the context of Bayesian inference for parameter estimation problems. The characterization of…
This work proposes a statistically enhanced framework to address the instability and limited predictive capability of conventional Galerkin-Proper Orthogonal Decomposition (Galerkin-POD) models. The method reformulates the correction of the…