arXiv subjects

Akash Ashirbad Panda

Publications and source records attributed to Akash Ashirbad Panda.

5 recordsLinked to original sources

arXiv2026

Stability, Convergence, and Error Analysis of Finite Element Methods for 3D Magnetohydrodynamics with $p$-Laplacian Viscosity

This paper develops a fully discrete finite element method for three-dimensional incompressible magnetohydrodynamic (MHD) flows with nonlinear $p$-Laplace viscosity. The scheme combines spatial finite elements with a semi-implicit Euler time discretisation. Convergence to a weak solution is proved using a time-translation compactness argument together with Minty's monotonicity method. Under additional regularity assumptions, we derive unconditional error estimates for both the velocity and magnetic field, with no coupling restriction between the time step and mesh size. The framework extends finite element analysis of incompressible MHD systems to non-Newtonian shear-thickening fluids ($p>2$), while recovering the classical Newtonian case ($p=2$). Finally, numerical simulations are provided to validate the theoretical convergence rates and demonstrate the robustness of the proposed method.

math.NA↗

arXiv2026

Nonlocal Tikhonov Regularization: Hilbert Scales, Explicit Rates, and the Classical Limit

We study fractional-Sobolev Tikhonov regularization for linear inverse problems on a bounded Lipschitz domain. The regularization penalty is generated by the restricted Dirichlet fractional Laplacian, and the associated variational problem is shown to admit a unique minimizer that depends Lipschitz continuously on the data. Identifying the positive self-adjoint operator $$A_s=I+(-Δ)^s,\, D(A_s^{1/2})=H_0^s(Ω),$$ we transform the problem isometrically into a classical Hilbert-space Tikhonov problem with observation operator $B=KA_s^{-1/2}$. This yields explicit mean-square error bounds and an order-optimal \emph{a priori} and \emph{a posteriori} parameter rules under Hölder-type source conditions. The framework is illustrated by partial observations and by the backward fractional heat equation. In the latter case, $$ B^*B=A_s^{-1}e^{-2tA_s}, $$ which permits a mode-wise description of the source condition, the singular-value decay, and the effective reconstruction bandwidth. We also study the local limit $s\to1^-$: after Bourgain--Brezis--Mironescu normalization, the fractional functionals $Γ$-converge in $L^2(Ω)$ to the classical $H_0^1$-Tikhonov functional, and the corresponding minimizers converge strongly in $L^2(Ω)$. Numerical experiments for the backward fractional heat problem illustrate the reconstruction procedure and the influence of the penalty order, and confirm the predicted mean-square convergence rate to within a few percent via Monte Carlo simulation, with Morozov's discrepancy principle attaining the same order-optimal rate a posteriori.

math.AP↗

arXiv2025

Least Square Estimation: SDEs Perturbed by Lévy Noise with Sparse Sample Paths

This article investigates the least squares estimators (LSE) for the unknown parameters in stochastic differential equations (SDEs) that are affected by Lévy noise, particularly when the sample paths are sparse. Specifically, given $n$ sparsely observed curves related to this model, we derive the least squares estimators for the unknown parameters: the drift coefficient, the diffusion coefficient, and the jump-diffusion coefficient. We also establish the asymptotic rate of convergence for the proposed LSE estimators. Additionally, in the supplementary materials, the proposed methodology is applied to a benchmark dataset of functional data/curves, and a small simulation study is conducted to illustrate the findings.

stat.ME↗

arXiv2022

Higher order time discretization for the stochastic semilinear wave equation with multiplicative noise

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 variational method for its error analysis and prove an improved convergence order of 3/2 for the approximates of the solution. The core of the analysis is Holder continuity in time and moment bounds for the solutions of the continuous and the discrete problem. Computational experiments are also presented.

math.NA↗

arXiv2021

The stochastic Gierer-Meinhardt system

The Gierer-Meinhardt system occurs in morphogenesis, where the development of an organism from a single cell is modelled. One of the steps in the development, is the formation of spatial patterns of the cell structure, starting from an almost homogeneous cell distribution. Turing proposed in his pioneering work different activator-inhibitor systems with different diffusion rates, which could trigger the emergence of such cell structures. Mathematically, one describes these activator-inhibitor systems as a coupled systems of reaction-diffusion equations with hugely different diffusion coefficients and highly nonlinear interaction. One famous example of these systems is the Gierer-Meinhardt system. These systems usually are not of monotone type, such that one has to apply other techniques. The purpose of this article is to study the stochastic reaction-diffusion Gierer-Meinhardt system with homogeneous Neumann boundary condition on a one or two-dimensional bounded spatial domain. To be more precise, we perturb the original Gierer-Meinhardt system by an infinite-dimensional Wiener process and show under which conditions on the Wiener process and the system, a solution exists. In dimension one, we even show the pathwise uniqueness. In dimension two, uniqueness is still an open question.

math.AP↗