arXiv subjects

Abhishek Chaudhary

Publications and source records attributed to Abhishek Chaudhary.

13 recordsLinked to original sources

arXiv2026

Convergence of a fully discrete approximation for the stochastic 2D Euler equations with Kraichnan transport noise

We study a fully discrete approximation theory for the two-dimensional incompressible Euler equations on $\mathbb{T}^{2}$, in vorticity form, driven by Kraichnan-type transport noise and with $\mathbb{H}^{-1}(\mathbb{T}^2)$-valued initial vorticity. The main new ingredient is a finite-dimensional coercivity estimate for the torus Kraichnan model, which captures at the discrete level the regularizing mechanism induced by the rough transport noise. Together with a new compactness strategy based on continuous equation-based interpolants and localized discrete estimates, this coercivity estimate gives tightness of the approximations and shows that subsequential limit is a weak martingale solution of the Euler equations.

math.PR↗

arXiv2026

Error analysis for 3D Navier--Stokes equations with additive noise

We consider the three-dimensional stochastic Navier--Stokes equations with additive stochastic forcing. We study temporal discretizations based on a semi-implicit Euler as well as a Crank-Nicolson scheme. We prove that locally in time the former converges with rate 1 while the latter converges with rate 3/2. These theoretical results are confirmed by extensive numerical simulations. To the best of our knowledge this is the first time that numerical simulations for the three-dimensional stochastic Navier--Stokes equations have been performed.

math.NA↗

arXiv2025

Convergence analysis for an implementable scheme to solve the linear-quadratic stochastic optimal control problem with stochastic wave equation

We study an optimal control problem for the stochastic wave equation driven by affine multiplicative noise, formulated as a stochastic linear-quadratic (SLQ) problem. By applying a stochastic Pontryagin's maximum principle, we characterize the optimal state-control pair via a coupled forward-backward SPDE system. We propose an implementable discretization using conforming finite elements in space and an implicit midpoint rule in time. By a new technical approach we obtain strong convergence rates for the discrete state-control pair without relying on Malliavin calculus. For the practical computation we develop a gradient-descent algorithm based on artificial iterates that employs an exact computation for the arising conditional expectations, thereby eliminating costly Monte Carlo sampling. Consequently, each iteration has a computational cost that is proportional to the number of spatial degrees of freedom, producing a scalable method that preserves the established strong convergence rates. Numerical results validate its efficiency.

math.OC↗

arXiv2025

Higher order numerical schemes for SPDEs with additive Noise

We present high-order numerical schemes for linear stochastic heat and wave equations with Dirichlet boundary conditions, driven by additive noise. Standard Euler schemes for SPDEs are limited to an order convergence between 1/2 and 1 due to the low temporal regularity of noise. For the stochastic heat equation, a modified Crank-Nicolson scheme with proper numerical quadrature rule for the noise term in its reformulation as random PDE achieves a strong convergence rate of 3/2. For the stochastic wave equation with additive noise a corresponding approach leads to a scheme which is of order 2.

math.NA↗

arXiv2024

Dissipative measure-valued solutions and weak-strong uniqueness for the Euler alignment system

We introduce the concept of a dissipative measure-valued solution to the Euler alignment system. This approach incorporates a modified total energy balance, utilizing a binary tensor Young measure. The central finding is a weak (measure-valued)--strong uniqueness principle: if both a dissipative measure-valued solution and a classical smooth solution originate from the same initial data, they will be identical as long as the classical solution exists.

math.AP↗

arXiv2024

A numerical method to simulate the stochastic linear-quadratic optimal control problem with control constraint in higher dimensions

We propose an {\em implementable} numerical scheme for the discretization of linear-quadratic optimal control problems involving SDEs in higher dimensions with {\em control constraint}. For time discretization, we employ the implicit Euler scheme, deriving discrete optimality conditions that involve time discretization of a backward stochastic differential equations. We develop a recursive formula to compute conditional expectations in the time discretization of the BSDE whose computation otherwise is the most computationally demanding step. Additionally, we present the error analysis for the rate of convergence. We provide numerical examples to demonstrate the efficiency of our scheme in higher dimensions.

math.AP↗

arXiv2022

Stochastic fractional conservation laws

In this paper, we consider the Cauchy problem for the nonlinear fractional conservation laws driven by a multiplicative noise. In particular, we are concerned with the well-posedness theory and the study of the long-time behavior of solutions for such equations. We show the existence of desired kinetic solution by using the vanishing viscosity method. In fact, we establish strong convergence of the approximate viscous solutions to a kinetic solution. Moreover, under a nonlinearity-diffusivity condition, we prove the existence of an invariant measure using the well-known Krylov-Bogoliubov theorem. Finally, we show the uniqueness and ergodicity of the invariant measure.

math.AP↗

arXiv2022

Semantically Distributed Robust Optimization for Vision-and-Language Inference

Analysis of vision-and-language models has revealed their brittleness under linguistic phenomena such as paraphrasing, negation, textual entailment, and word substitutions with synonyms or antonyms. While data augmentation techniques have been designed to mitigate against these failure modes, methods that can integrate this knowledge into the training pipeline remain under-explored. In this paper, we present \textbf{SDRO}, a model-agnostic method that utilizes a set linguistic transformations in a distributed robust optimization setting, along with an ensembling technique to leverage these transformations during inference. Experiments on benchmark datasets with images (NLVR$^2$) and video (VIOLIN) demonstrate performance improvements as well as robustness to adversarial attacks. Experiments on binary VQA explore the generalizability of this method to other V\&L tasks.

cs.CV↗

arXiv2021

Stochastic Degenerate Fractional Conservation Laws

We consider the Cauchy problem for a degenerate fractional conservation laws driven by a noise. In particular, making use of an adapted kinetic formulation, a result of existence and uniqueness of solution is established. Moreover, a unified framework is also established to develop the continuous dependence theory. More precisely, we demonstrate L^1-continuous dependence estimates on the initial data, the order of fractional Laplacian, the diffusion matrix, the flux function, and the multiplicative noise function present in the equation.

math.AP↗

arXiv2021

A convergent finite volume scheme for the stochastic barotropic compressible Euler equations

In this paper, we analyze a semi-discrete finite volume scheme for the three-dimensional barotropic compressible Euler equations driven by a multiplicative Brownian noise. We derive necessary a priori estimates for numerical approximations, and show that the Young measure generated by the numerical approximations converge to a dissipative measure--valued martingale solution to the stochastic compressible Euler system. These solutions are probabilistically weak in the sense that the driving noise and associated filtration are integral part of the solution. Moreover, we demonstrate strong convergence of numerical solutions to the regular solution of the limit systems at least on the lifespan of the latter, thanks to the weak (measure-valued)--strong uniqueness principle for the underlying system. To the best of our knowledge, this is the first attempt to prove the convergence of numerical approximations for the underlying system.

math.AP↗

arXiv2021

Convergence of a spectral method for the stochastic incompressible Euler equations

We propose a spectral viscosity method (SVM) to approximate the incompressible Euler equations driven by a multiplicative noise. We show that SVM solution converges to a dissipative measure-valued martingale solution. These solutions are weak in the probabilistic sense i.e. the probability space and the driving Wiener process are an integral part of the solution. We also exhibit weak (measure-valued)-strong uniqueness principle. Moreover, we establish strong convergence of approximate solutions to the regular solution of the limit system at least on the lifespan of the latter, thanks to the weak (measure-valued)--strong uniqueness principle for the underlying system.

math.AP↗

arXiv2021

Creating and Implementing a Smart Speaker

We have seen significant advancements in Artificial Intelligence and Machine Learning in the 21st century. It has enabled a new technology where we can have a human-like conversation with the machines. The most significant use of this speech recognition and contextual understanding technology exists in the form of a Smart Speaker. We have a wide variety of Smart Speaker products available to us. This paper aims to decode its creation and explain the technology that makes these Speakers, "Smart."

cs.HC↗

arXiv2020

On weak-strong uniqueness for stochastic equations of incompressible fluid flow

We introduce a novel concept of dissipative measure-valued martingale solution to the stochastic Euler equations describing the motion of an inviscid incompressible fluid. These solutions are characterized by a parametrized Young measure and a concentration defect measure in the total energy balance. Moreover, they are weak in the probablistic sense i.e., the underlying probablity space and the driving Wiener process are intrinsic part of the solution. In a significant departure from the existing literature, we first exhibit the relative energy inequality for the incompressible Euler equations driven by a multiplicative noise, and then demonstrate pathwise weak-strong uniqueness principle. Finally, we also provide a sufficient condition, a la Prodi and Serrin, for the uniqueness of weak martingale solutions to stochastic Naiver-Stokes system in the class of finite energy weak martingale solutions.

math.AP↗