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Ananta K. Majee

Publications and source records attributed to Ananta K. Majee.

16 recordsLinked to original sources
arXiv2023

On rate of convergence of finite difference scheme for degenerate parabolic-hyperbolic PDE with Levy noise

In this article, we consider a semi discrete finite difference scheme for a degenerate parabolic-hyperbolic PDE driven by Lévy noise in one space dimension. Using bounded variation estimations and a variant of classical Kružkov's doubling of variable approach, we prove that expected value of the $L^1$-difference between the unique entropy solution and approximate solution converges at a rate of $(Δx)^\frac{1}{7}$, where $Δx$ is the spatial mesh size.

math.NA↗
arXiv2023

Stochastic Fractional Conservation Laws: Large deviation principle, Central limit theorem and Moderate deviation principle

In this article, we establish the Freidlin-Wentzell type large deviation principle and central limit theorem for stochastic fractional conservation laws with small multiplicative noise in kinetic formulation framework. The weak convergence method and doubling variables method play a crucial role. As a consequence, we also establish moderate deviation principle for the underlying problem.

math.PR↗
arXiv2023

Nonlinear SPDE driven by Levy noise: Well-posedness, optimal control and invariant measure

In this article, we study a nonlinear stochastic control problem perturbed by multiplicative Levy noise, where the nonlinear operator in divergence form satisfies p type growth with coercivity assumptions. By using Aldous tightness criteria and Jakubowski version of the Skorokhod theorem on nonmetric spaces along with standard contraction method, we establish existence of pathwise unique strong solution. Formulating the associated control problem, and using variational approach together with the convexity property of cost functional in control variable, we establish existence of a weak optimal solution of the underlying problem. We use the technique of Maslowski and Seidler to prove existence of an invariant measure for uncontrolled SPDE driven with multiplicative Levy noise.

math.PR↗
arXiv2023

Convergence of an operator splitting scheme for fractional conservation laws with Levy noise

In this paper, we are concerned with a operator splitting scheme for linear fractional and fractional degenerate stochastic conservation laws driven by multiplicative Levy noise. More specifically, using a variant of classical Kruzkov's doubling of variable approach, we show that the approximate solutions generated by the splitting scheme converges to the unique stochastic entropy solution of the underlying problems.Finally, the convergence analysis is illustrated by several numerical examples.

math.NA↗
arXiv2019

On the Development of a Coupled Non-linear Telegraph-Diffusion Model for Image Restoration

In this work, we propose a telegraph coupled partial differential equation (TCPDE) based model for image restoration. New framework interpolates between a couple of non-linear telegraph equation and a parabolic equation. Proposed strategy can be applied to significantly preserve the oscillatory and texture pattern in an image, even in low signal-to-noise ratio. First, we prove that the present model has a unique global weak solution using Banach's fixed point theorem. Then apply our model over a set of gray-level images to illustrate the superiority of the proposed model over the recently developed hyperbolic-parabolic PDE based models as well as coupled diffusion-based model.

math.AP↗
arXiv2019

Well-posedness study of a non-linear hyperbolic-parabolic coupled system applied to image speckle reduction

In this article, we consider a non-linear hyperbolic-parabolic coupled system based on telegraph diffusion framework applied to image despeckling. A separate equation is used to calculate the edge variable, which improves the quality of the despeckled images. A well-posedness result of the proposed coupled system is settled via Schauder's fixed point theorem. Numerical experiments are reported to illustrate the effectiveness of the proposed model, with recently developed models, over a set of gray level test images contaminated by speckle noise.

math.AP↗
arXiv2019

Analysis and Simulation of a Coupled Diffusion based Image Denoising Model

In this study, a new coupled Partial Differential Equation (CPDE) based image denoising model incorporating space-time regularization into non-linear diffusion is proposed. This proposed model is fitted with additive Gaussian noise which performs efficient image smoothing along with the preservation of edges and fine structures. For this purpose, we propose a new functional minimization framework to remove the image noise, which results in solving a system of three partial differential equations (PDEs). Our proposed model is dissimilar from the existing CPDE models as it includes two additional evolution equations to handle edge strength function and data fidelity term. These two evolution equations control the smoothing process and force the resultant denoised solution to be close to the initial solution. To the best of our knowledge, the proposed model is the only work, which deciphers the combined effect of both the terms using separate PDEs. Furthermore, we establish the existence and uniqueness of a weak solution of the proposed system using the time discretization method with $H^1$ initial data. Finally, we used a generalized weighted average finite difference scheme to efficiently solve the coupled system and experiment results show the effectiveness of the proposed CPDE model.

math.NA↗
arXiv2019

Stochastic optimal control of a evolutionary $p$-Laplace equation with multiplicative Lévy noise

In this article, we are interested in an initial value optimal control problem for a evolutionary $p$-Laplace equation driven by multiplicative Lévy noise. We first present wellposedness of a weak solution by using an implicit time discretization of the problem, along with the Jakubowski version of the Skorokhod theorem for a non-metric space. We then formulate associated control problem, and establish existence of an optimal solution by using variational method and exploiting the convexity property of the cost functional.

math.AP↗
arXiv2017

Optimal Strong Rates of Convergence for a Space-Time Discretization of the Stochastic Allen-Cahn Equation with multiplicative noise

The stochastic Allen-Cahn equation with multiplicative noise involves the nonlinear drift operator ${\mathscr A}(x) = Δx - \bigl(\vert x\vert^2 -1\bigr)x$. We use the fact that ${\mathscr A}(x) = -{\mathcal J}^{\prime}(x)$ satisfies a weak monotonicity property to deduce uniform bounds in strong norms for solutions of the temporal, as well as of the spatio-temporal discretization of the problem. This weak monotonicity property then allows for the estimate $ \underset{1 \leq j \leq J}\sup {\mathbb E}\bigl[ \Vert X_{t_j} - Y^j\Vert_{{\mathbb L}^2}^2\bigr] \leq C_δ(k^{1-δ} + h^2)$ for all small $δ>0$, where $X$ is the strong variational solution of the stochastic Allen-Cahn equation, while $\big\{Y^j:0\le j\le J\big\}$ solves a structure preserving finite element based space-time discretization of the problem on a temporal mesh $\{ t_j;\, 1 \leq j \leq J\}$ of size $k>0$ which covers $[0,T]$.

math.AP↗
arXiv2016

Continuous dependence estimate for a degenerate parabolic-hyperbolic equation with Levy noise

In this article, we are concerned with a multidimensional degenerate parabolic-hyperbolic equation driven by Levy processes. Using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities of the entropy solutions under the assumption that Levy noise depends only on the solution. This result is used to show the error estimate for the stochastic vanishing viscosity method. In addition, we establish fractional BV estimate for vanishing viscosity approximations in case the noise coefficients depend on both the solution and spatial variable.

math.AP↗
arXiv2016

A finite difference scheme for conservation laws driven by Levy noise

In this paper, we analyze a semi-discrete finite difference scheme for a conservation laws driven by a homogeneous multiplicative Levy noise. Thanks to BV estimates, we show a compact sequence of approximate solutions, generated by the finite difference scheme, converges to the unique entropy solution of the underlying problem, as the spatial mesh size \Dx-->0. Moreover, we show that the expected value of the L^1-difference between the approximate solution and the unique entropy solution converges at a rate O(\sqrt{\Dx}).

math.AP↗
arXiv2016

On the Cauchy problem of a degenerate parabolic-hyperbolic PDE with Lévy noise

In this article we deal with stochastic perturbation of degenerate parabolic partial differential equations (PDEs). The particular emphasise is on analysing the effect of multiplicative Lévy noise to such problems and establishing wellposedness by developing a suitable weak entropy solution framework. The proof of existence is based on the vanishing viscosity technique. The uniqueness is settled by interpreting Kruzkov's doubling technique in the presence noise.

math.AP↗
arXiv2015

Continuous dependence estimate for conservation laws with Lévy noise

We are concerned with multidimensional stochastic balance laws driven by Lévy processes. Using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities of the entropy solutions under the assumption that Lévy noise only depends on the solution. This result is used to show the error estimate for the stochastic vanishing viscosity method. In addition, we establish fractional $BV$ estimate for vanishing viscosity approximations in case the noise coefficient depends on both the solution and spatial variable.

math.AP↗
arXiv2014

Conservation laws driven by Lévy white noise

We consider multidimensional conservation laws perturbed by multiplicative Lévy noise. We establish existence and uniqueness results for entropy solutions. The entropy inequalities are formally obtained by the Itô-Lévy chain rule. The multidimensionality requires a generalized interpretation of the entropy inequalities to accommodate Young measure-valued solutions. We first establish the existence of entropy solutions in the generalized sense via the vanishing viscosity method, and then establish the $L^1$-contraction principle. Finally, the $L^1$ contraction principle is used to argue that the generalized entropy solution is indeed the classical entropy solution.

math.AP↗
arXiv2014

Stochastic conservation laws: weak-in-time formulation and strong entropy condition

This article is an attempt to complement some recent developments on conservation laws with stochastic forcing. In a pioneering development, Feng $&$ Nualarthave developed the entropy solution theory for such problems and the presence of stochastic forcing necessitates introduction of {\it strong entropy condition}. However, the authors' formulation of entropy inequalities are weak-in-space but strong-in-time. In the absence of a-priori path continuity for the solutions, we take a critical outlook towards this formulation and offer an entropy formulation which is weak-in-time and weak-in-space.

math.AP↗