Laplacian Research Articles

Efficient modeling of fractional Laplacian viscoacoustic wave equation with fractional finite-difference method

The fractional viscoacoustic/viscoelastic wave equation, which accurately quantifies the frequency-independent anelastic effects, has been the focus of seismic industry in recent years. The pseudo-spectral (PS) method stands as one of the most widely used numerical methods for solving the fractional wave equation. However, the PS method often suffers from low accuracy and efficiency, particularly when modeling wave propagation in heterogeneous media. To address these issues, we propose a novel and efficient fractional finite-difference (FD) method for solving the wave equation with fractional Laplacian operators. This method develops an arbitrary high-order FD operator via the generating function of our fractional FD (F-FD) scheme, enhancing accuracy with L2-optimal FD coefficients. Similar to classic FD methods, our F-FD method is characterized by straightforward programming and excellent 3D extensibility. It surpasses the PS method by eliminating the need for Fast Fourier Transform (FFT) and inverse-FFT (IFFT) operations at each time step, offering significant benefits for 3D applications. Consequently, the F-FD method proves more adept for wave-equation-based seismic data processes like imaging and inversion. Compared with existing F-FD methods, our approach uniquely approximates the entire fractional Laplacian operator and stands as a local numerical algorithm, with an adjustable F-FD operator order based on model parameters for enhanced practicality. Accuracy analyses confirm that our method matches the precision of the PS method with a correctly ordered F-FD operator. Numerical examples show that the proposed method has good applicability for complex models. Finally, we have carried out reverse time migration on the Marmousi-2 model, and the imaging profiles indicate that the proposed method can be effectively applied to seismic imaging, demonstrating good practicability.

Laplace neural operator for solving differential equations

Laplace neural operator for solving differential equations

A novel physics-aware graph network using high-order numerical methods in weather forecasting model

In this paper, we present a novel architecture for accurate weather forecasting within the framework of Physics-aware Graph Networks (PaGN), which aims to improve the prediction of weather time series by integrating both data and physical equations defined on sparsely distributed spatial domains. This approach employs geometric deep learning by accounting for both supervised loss and physics loss. Solely in supervised learning, data-driven training demands a large amount of data, whereas integrating physics information can reduce this requirement, resulting in faster training. The physics awareness also provides a stronger inductive bias and better generalization, directing the machine learning process by clarifying the complexity of weather data. Thus, a more accurate discretization of physics equations can enhance the physics awareness of the complex weather dynamics. Meanwhile, weather data is essentially gathered at observatories that are not located densely, since covering the Earth with numerous observatories is prohibitively expensive. Given its reliance on graph structure, the proposed PaGN is well-suited for handling sparsely distributed data particularly defined on arbitrary geometric domain. One of the main contributions of this study is the use of high-order numerical methods approximating physics equations defined on graph neural networks for achieving accuracy improvements. The fractional graph Laplacian operator is further integrated with the physics equations to account for the non-local characteristics of complex weather data. For computational efficiency, the accuracy improved PaGN architecture is smoothly implemented in the embedding space. A series of numerical tests on weather datasets is performed to verify the improved accuracy, robustness, and applicability of the proposed methodology. We first carry out the numerical study on the synthetic datasets obtained from the physics equations with an extra forcing term. We then extensively investigate the accuracies of weather forecasting for various configurations in terms of the geophysical regions and the accuracy-improved PaGN approaches for diffusion and wave equations. Moreover, we emphasize that the fractional graph Laplacian operator incorporated into our PaGN model can further improve the accuracy for weather forecasting.

Isotropic optimizations of finite difference discretization

The isotropy is a fundamental requirement for solving partial differential equations by the finite difference scheme. In this work, we first introduce the concept of the virtual nodes to construct the finite difference scheme, and propose several stencils of finite difference discretization to optimize the isotropy of the gradient and Laplacian operators. The isotropic error of the optimized gradient stencil is reduced to 4.1% and 7.5% of the conventional scheme and the typical 4th-order isotropic stencil that is widely used in the multiphase lattice Boltzmann method, while the optimized Laplacian stencil displays the best performance in the Fourier analysis. Then, the optimized stencils are applied to suppress the spurious currents in the multiphase lattice Boltzmann method. The spurious current is reduced to only 7.2% of that calculated by the typical 4th-order isotropic stencil at the reduced temperature 0.6, at which the liquid/gas density ratio is near to 1000. Furthermore, they are adopted to rectify the dendrite directions in the alloy solidification simulations by the phase field method. The angle deviation of dendrite growth is reduced to 15.7% of the conventional scheme.

Eigenvalue Estimates on Weighted Manifolds

We derive various eigenvalue estimates for the Hodge Laplacian acting on differential forms on weighted Riemannian manifolds. Our estimates unify and extend various results from the literature and provide a number of geometric applications. In particular, we derive an inequality which relates the eigenvalues of the Jacobi operator for f-minimal hypersurfaces and the spectrum of the Hodge Laplacian.

A novel Laplacian vector directional filter designed for real-time removal of impulsive noise from color images

Eliminating noise from a color image represents an essential task in image processing, given that noise can markedly diminish visual quality and impact the precision of subsequent analyses or applications (such as visual quality inspection, medical image analysis, etc.). In this study, we introduce an innovative approach designed to remove noise while preserving the edges and fine details of a color image. This method is based on Laplacian operators, threshold values for identifying potentially corrupted pixels due to noise, and the Vector Directional Filter (VDF) for substituting noisy pixels. Consequently, this technique is termed Laplacian VDF (LVDF). Additionally, a Hardware/Software (HW/SW) design is formulated for the real-time implementation of the LVDF filter. The hardware architecture is designed through the High-Level Synthesis (HLS) flow, while the software component is executed using the ARM Cortex-A53 hardcore processor. Furthermore, five Direct Memory Access (DMAs) are employed to enhance data throughput between the hardware coprocessor and Double Data Rate (DDR) memory. Performance evaluations conducted on the ZCU102 kit reveal that the LVDF HW/SW design facilitates the restoration of 74 images per second, resulting in a 94 % reduction in execution time compared to the software implementation, all while maintaining consistent objective and subjective image quality.

A Visual Measurement Method for Deep Holes in Composite Material Aerospace Components.

The visual measurement of deep holes in composite material workpieces constitutes a critical step in the robotic assembly of aerospace components. The positioning accuracy of assembly holes significantly impacts the assembly quality of components. However, the complex texture of the composite material surface and mutual interference between the imaging of the inlet and outlet edges of deep holes significantly challenge hole detection. A visual measurement method for deep holes in composite materials based on the radial penalty Laplacian operator is proposed to address the issues by suppressing visual noise and enhancing the features of hole edges. Coupled with a novel inflection-point-removal algorithm, this approach enables the accurate detection of holes with a diameter of 10 mm and a depth of 50 mm in composite material components, achieving a measurement precision of 0.03 mm.

Positive solutions for a coupled systems involving the fractional p(x)- Laplacian operator in unbounded domains

In this paper, we study the existence of at least one weak solution for a class of nonlocal elliptic systems involving the fractional p(x)-Laplacian. These operators are utilized to solve a system defined within unbounded domain. The right sides of systems treated are closely related to a type of gradient C1-functional which satisfy a sublinear growth conditions. Under some additional assumptions on the nonlinearities and leveraging the theory of Lebesgue and fractional Sobolev spaces with variable exponents, we can use a variational approach to establish the existence result.

The Existence and Uniqueness of Radial Solutions for Biharmonic Elliptic Equations in an Annulus

This paper concerns with the existence of radial solutions of the biharmonic elliptic equation ▵2u=f(|x|,u,|∇u|,▵u) in an annular domain Ω={x∈RN:r1<|x|<r2}(N≥2) with the boundary conditions u|∂Ω=0 and ▵u|∂Ω=0, where f:[r1,r2]×R×R+×R→R is continuous. Under certain inequality conditions on f involving the principal eigenvalue λ1 of the Laplace operator −▵ with boundary condition u|∂Ω=0, an existence result and a uniqueness result are obtained. The inequality conditions allow for f(r,ξ,ζ,η) to be a superlinear growth on ξ,ζ,η as |(ξ,ζ,η)|→∞. Our discussion is based on the Leray–Schauder fixed point theorem, spectral theory of linear operators and technique of prior estimates.

Generic spectrum of the weighted Laplacian operator on Cayley graphs

Generic spectrum of the weighted Laplacian operator on Cayley graphs

Robin problem involving the $p(x)$-Laplacian operator without Ambrosetti-Rabinowizt condition

The paper deals with the following Robin problem$$\left\lbrace\begin{aligned}- \mathcal{M} \left( \int _{\Omega} \frac{1}{p(x)} \vert \nabla u \vert ^{p(x)} dx + \int _{\partial \Omega } \frac{a(x)}{p(x)} \vert \nabla u \vert ^{p(x)} d \sigma \right) \mathop{\rm div} (\vert \nabla u \vert ^{p(x)-2} \nabla u) = \lambda h(x,u) \ \ \text{ in } \Omega,\\\vert \nabla u \vert ^{p(x)-2} \frac{\partial u}{\partial \nu} + a(x) \vert u \vert ^{p(x)-2} u &=0 \quad \quad \quad \ \text{ on } \partial \Omega .\end{aligned}\right.$$The goal is to determine the precise positive interval of $\lambda $ for which the problem admits at least two nontrivial solutions via variational approach for the above problem without assuming the Ambrosetti-Rabinowitz condition. Next, we give a result on the existence of an unbounded sequence of nontrivial weak solutions by employing the fountain theoreom with Cerami condition.

On a class of generalized Choquard system in fractional Orlicz-Sobolev spaces

We study a category of non-homogeneous Choquard systems involving the fractional ϕi(⋅)-Laplacian operator. By establishing suitable hypotheses on the Choquard nonlinearities and the continuous function ϕi, we confirm the existence of nontrivial solutions to the problem at hand. This achievement occurs within the structure of the fractional Orlicz-Sobolev space. Our research makes use of the mountain pass theorem, which circumvents the necessity for the Palais-Smale condition, and strategically leverages the Hardy-Littlewood-Sobolev inequality to enhance our analysis.

Direct and inverse problems for the Hilfer fractional differential equation

The article studies direct and inverse problems for subdiffusion equations involving a Hilfer fractional derivative. An arbitrary positive self-adjoint operator $A$ is taken as the elliptic part of the equation. In particular, as the operator $A$ we can take the Laplace operator with the Dirichlet condition. First, the existence and uniqueness of a solution to the direct problem is proven. Then, using the representation of the solution to the direct problem, the existence and uniqueness of the inverse problem of finding the right-hand side of the equation, which depends only on the spatial variable, is proved.

Phase-space entropy cascade and irreversibility of stochastic heating in nearly collisionless plasma turbulence.

We consider a nearly collisionless plasma consisting of a species of "test particles" in one spatial and one velocity dimension, stirred by an externally imposed stochastic electric field-a kinetic analog of the Kraichnan model of passive advection. The mean effect on the particle distribution function is turbulent diffusion in velocity space-known as stochastic heating. Accompanying this heating is the generation of fine-scale structure in the distribution function, which we characterize with the collisionless (Casimir) invariant C_{2}∝∫∫dxdv〈f^{2}〉-a quantity that here plays the role of (negative) entropy of the distribution function. We find that C_{2} is transferred from large scales to small scales in both position and velocity space via a phase-space cascade enabled by both particle streaming and nonlinear interactions between particles and the stochastic electric field. We compute the steady-state fluxes and spectrum of C_{2} in Fourier space, with k and s denoting spatial and velocity wave numbers, respectively. In our model, the nonlinearity in the evolution equationfor the spectrum turns into a fractional Laplacian operator in k space, leading to anomalous diffusion. Whereas even the linear phase mixing alone would lead to a constant flux of C_{2} to high s (towards the collisional dissipation range) at every k, the nonlinearity accelerates this cascade by intertwining velocity and position space so that the flux of C_{2} is to both high k and high s simultaneously. Integrating over velocity (spatial) wave numbers, the k-space (s-space) flux of C_{2} is constant down to a dissipation length (velocity) scale that tends to zero as the collision frequency does, even though the rate of collisional dissipation remains finite. The resulting spectrum in the inertial range is a self-similar function in the (k,s) plane, with power-law asymptotics at large k and s. Our model is fully analytically solvable, but the asymptotic scalings of the spectrum can also be found via a simple phenomenological theory whose key assumption is that the cascade is governed by a "critical balance" in phase space between the linear and nonlinear timescales. We argue that stochastic heating is made irreversible by this entropy cascade and that, while collisional dissipation accessed via phase mixing occurs only at small spatial scales rather than at every scale as it would in a linear system, the cascade makes phase mixing even more effective overall in the nonlinear regime than in the linear one.

The Nehari Manifold for a Class of Singular $\psi$-Riemann-Liouville Fractional with $p$-Laplacian Operator Differential Equations

The Nehari Manifold for a Class of Singular $\psi$-Riemann-Liouville Fractional with $p$-Laplacian Operator Differential Equations

The effects of nonlinear perturbation terms on comparison principles for the p-Laplacian

In this paper, we investigate various comparison principles for quasilinear elliptic equations of [Formula: see text]-Laplace type with lower-order terms that depend on the solution and its gradient. More specifically, we study comparison principles for equations of the following form: [Formula: see text] where [Formula: see text] is the [Formula: see text]-Laplace operator with [Formula: see text], and [Formula: see text] is a continuous function that satisfies a structure condition. Many of these results lead to comparison principles for the model equations [Formula: see text] where [Formula: see text] are non-decreasing and [Formula: see text]. Our results either improve or complement those that appear in the literature.

Nonlinear Asymptotic Stability and Transition Threshold for 2D Taylor–Couette Flows in Sobolev Spaces

In this paper, we investigate the stability of the 2-dimensional (2D) Taylor–Couette (TC) flow for the incompressible Navier–Stokes equations. The explicit form of velocity for 2D TC flow is given by u=(Ar+Br)(-sinθ,cosθ)T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u=(Ar+\\frac{B}{r})(-\\sin \ heta , \\cos \ heta )^T$$\\end{document} with (r,θ)∈[1,R]×S1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(r, \ heta )\\in [1, R]\ imes \\mathbb {S}^1$$\\end{document} being an annulus and A, B being constants. Here, A, B encode the rotational effect and R is the ratio of the outer and inner radii of the annular region. Our focus is the long-term behavior of solutions around the steady 2D TC flow. While the laminar solution is known to be a global attractor for 2D channel flows and plane flows, it is unclear whether this is still true for rotating flows with curved geometries. In this article, we prove that the 2D Taylor–Couette flow is asymptotically stable, even at high Reynolds number (Re∼ν-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Re\\sim \ u ^{-1}$$\\end{document}), with a sharp exponential decay rate of exp(-ν13|B|23R-2t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\exp (-\ u ^{\\frac{1}{3}}|B|^{\\frac{2}{3}}R^{-2}t)$$\\end{document} as long as the initial perturbation is less than or equal to ν12|B|12R-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u ^\\frac{1}{2} |B|^{\\frac{1}{2}}R^{-2}$$\\end{document} in Sobolev space. The powers of ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document} and B in this decay estimate are optimal. It is derived using the method of resolvent estimates and is commonly recognized as the enhanced dissipative effect. Compared to the Couette flow, the enhanced dissipation of the rotating Taylor–Couette flow not only depends on the Reynolds number but also reflects the rotational aspect via the rotational coefficient B. The larger the |B|, the faster the long-time dissipation takes effect. We also conduct space-time estimates describing inviscid-damping mechanism in our proof. To obtain these inviscid-damping estimates, we find and construct a new set of explicit orthonormal basis of the weighted eigenfunctions for the Laplace operators corresponding to the circular flows. These provide new insights into the mathematical understanding of the 2D Taylor–Couette flows.

Second order discrete boundary value problem with the $(p_1(k); p_2(k))$-Laplacian

In this paper we investigate existence and non-existence of solutions for a Dirichlet boundary value problem involving the $(p_1(k), p_2(k))$-Laplacian operator when variational methods are applied to obtain the results.

The hyperellipsoidal system

One hundred ninety years ago, Lamé introduced the celebrated ellipsoidal system, which is an orthogonal curvilinear system that incorporates any anisotropic characteristics of three‐dimensional space. Since then, the theory of ellipsoidal harmonics has been successfully developed, but almost nothing has been achieved for ellipsoidal waves. This is due to the fact that, in order to deal with the ellipsoidal harmonics, Lamé introduced some ingenious arguments that led to a complete spectral theory for the Kernel space of the Laplacian, but no such arguments have been proposed for the Kernel space of the d'Alembert operator. So, one possible approach to develop a method that will give us ellipsoidal wave functions is to try to generalize the Lamé arguments (to the extent that this is possible) to four‐dimensional ellipsoidal harmonics. This procedure will give us the hyperellipsoidal harmonics. Then we can replace the fourth coordinate by in order for the four‐dimensional Laplace operator to become the three‐dimensional wave operator of d'Alembert and the four‐dimensional harmonics to become three dimensional waves. The present work is the first stage of this project. It develops the geometry of a four‐dimensional ellipsoidal system and calculates the Laplace operator in this hyperellipsoidal system.

Normalized Hodge Laplacian Matrix and Application to Random Walk on Simplicial Complexes

Normalized Hodge Laplacian Matrix and Application to Random Walk on Simplicial Complexes