Integral Fractional Laplacian Research Articles

Weighted Analytic Regularity for the Integral Fractional Laplacian in Polyhedra

Weighted Analytic Regularity for the Integral Fractional Laplacian in Polyhedra

Fast Implementation of FEM for Integral Fractional Laplacian on Rectangular Meshes

Fast Implementation of FEM for Integral Fractional Laplacian on Rectangular Meshes

Exponential Convergence of \(hp\)-FEM for the Integral Fractional Laplacian in Polygons

Exponential Convergence of \(hp\)-FEM for the Integral Fractional Laplacian in Polygons

Finite difference discretization for one-dimensional higher-order integral fractional Laplacian and its application

A simple and easy-to-implement discrete approximation is proposed for one-dimensional higher-order integral fractional Laplacian (IFL), and our method is applied to discrete the fractional biharmonic equation, multi-term fractional differential model and fractal KdV equation. Based on the generating function, a fractional analogue of the central difference scheme to higher-order IFL is provided, the convergence of the discrete approximation is proved. Extensive numerical experiments are provided to confirm our analytical results. Moreover, some new observations are discovered from our numerical results.

A Monotone Discretization for the Fractional Obstacle Problem and Its Improved Policy Iteration

In recent years, the fractional Laplacian has attracted the attention of many researchers, the corresponding fractional obstacle problems have been widely applied in mathematical finance, particle systems, and elastic theory. Furthermore, the monotonicity of the numerical scheme is beneficial for numerical stability. The purpose of this work is to introduce a monotone discretization method for addressing obstacle problems involving the integral fractional Laplacian with homogeneous Dirichlet boundary conditions over bounded Lipschitz domains. Through successful monotone discretization of the fractional Laplacian, the monotonicity is preserved for the fractional obstacle problem and the uniform boundedness, existence, and uniqueness of the numerical solutions of the fractional obstacle problems are proved. A policy iteration is adopted to solve the discrete nonlinear problems, and the convergence after finite iterations can be proved through the monotonicity of the scheme. Our improved policy iteration, adapted to solution regularity, demonstrates superior performance by modifying discretization across different regions. Numerical examples underscore the efficacy of the proposed method.

Correction to: Adaptive finite element approximation of optimal control problems with the integral fractional Laplacian

Correction to: Adaptive finite element approximation of optimal control problems with the integral fractional Laplacian

Adaptive finite element approximation of optimal control problems with the integral fractional Laplacian

Adaptive finite element approximation of optimal control problems with the integral fractional Laplacian

Finite element approximation of optimal control problem with integral fractional Laplacian and state constraint

Finite element approximation of optimal control problem with integral fractional Laplacian and state constraint

Robust BPX preconditioner for fractional Laplacians on bounded Lipschitz domains

We propose and analyze a robust Bramble-Pasciak-Xu (BPX) preconditioner for the integral fractional Laplacian of order s ∈ ( 0 , 1 ) s\in (0,1) on bounded Lipschitz domains. Compared with the standard BPX preconditioner, an additional scaling factor 1 − γ ~ s 1-\widetilde {\gamma }^s , for some fixed γ ~ ∈ ( 0 , 1 ) \widetilde {\gamma } \in (0,1) , is incorporated to the coarse levels. For either quasi-uniform grids or graded bisection grids, we show that the condition numbers of the resulting systems remain uniformly bounded with respect to both the number of levels and the fractional power.

A new ‘walk on spheres’ type method for fractional diffusion equation in high dimensions based on the Feynman–Kac formulas

In this paper, we introduce an efficient stochastic method for solving space-fractional diffusion equations in high dimensions based on the Feynman–Kac formula. The key idea is to approximate the trajectory of the process by using a series of balls. As an extension of our first work Sheng et al. (2022) for the Poisson equation, the new algorithm finds remarkably efficient in solving time-dependent linear problems with integral fractional Laplacian on the bounded and unbounded domains. We present some numerical examples to validate the robustness and efficiency of our methods.

Constructive Approximation on Graded Meshes for the Integral Fractional Laplacian

We consider the homogeneous Dirichlet problem for the integral fractional Laplacian $$(-\Delta )^s$$ . We prove optimal Sobolev regularity estimates in Lipschitz domains provided the solution is $$C^s$$ up to the boundary. We present the construction of graded bisection meshes by a greedy algorithm and derive quasi-optimal convergence rates for approximations to the solution of such a problem by continuous piecewise linear functions. The nonlinear Sobolev scale dictates the relation between regularity and approximability.

On asymptotics of solutions for superdiffusion and subdiffusion equations with the Riemann-Liouville fractional derivative

<abstract><p>In the present paper, we focus on the study of the asymptotic behaviors of solutions for the Cauchy problem of time-space fractional superdiffusion and subdiffusion equations with integral initial conditions, where the Riemann-Liouville derivative is used in the temporal direction and the integral fractional Laplacian is applied in the spatial variables. The fundamental solutions of the considered equations, which can be represented in terms of the Fox $ H $-function, are constructed and investigated by using asymptotic expansions of the Fox $ H $-function. Then, we obtain the asymptotic behaviors of solutions in the sense of $ L^{p}(\mathbb{R}^{d}) $ and $ L^{p, \infty}(\mathbb{R}^{d}) $ norms, where Young's inequality for convolution plays a very important role. Finally, gradient estimates and large time behaviors of solutions are also provided. In particular, we derive the optimal $ L^{2} $- decay estimate for the subdiffusion equation.</p></abstract>

Besov regularity for the Dirichlet integral fractional Laplacian in Lipschitz domains

We prove Besov regularity estimates for the solution of the Dirichlet problem involving the integral fractional Laplacian of order s in bounded Lipschitz domains Ω:‖u‖B˙2,∞s+r(Ω)≤C‖f‖L2(Ω),r=min⁡{s,1/2},if s≠1/2,‖u‖B˙2,∞1−ϵ(Ω)≤C‖f‖L2(Ω),ϵ∈(0,1),if s=1/2, with explicit dependence of C on s and ϵ. These estimates are consistent with the regularity on smooth domains and show that there is no loss of regularity due to Lipschitz boundaries. The proof uses elementary ingredients, such as the variational structure of the problem and the difference quotient technique.

Fast Q1 finite element for two-dimensional integral fractional Laplacian

To investigate 2-dimensional integral fractional Laplacian, a fast Q1 finite element method is proposed based on a weighted trapezoidal rule. Different from the very limited existing finite element approximations, the singular integrals are handled numerically, but the rest of integrals can be exactly evaluated. In addition, the entries of stiffness matrix with Toeplitz structure can be efficiently expressed as c2,sh2−γ144w·e with known vectors w and e, where γ∈(2s,2] and 0<s<1. We also prove the upper and lower bounds of the maximum and minimum eigenvalues of stiffness matrix and its condition number, and then study the error analysis of the developed scheme in discrete energy norm. Finally, a preconditioned conjugate gradient method coupled with fast Fourier transform (FFT)-based fast algorithm for matrix-vector product is provided to solve linear systems, and then some numerical experiments are carried out to verify the effectiveness of our scheme.

Weighted Analytic Regularity for the Integral Fractional Laplacian in Polygons

Weighted Analytic Regularity for the Integral Fractional Laplacian in Polygons

Monte Carlo fPINNs: Deep learning method for forward and inverse problems involving high dimensional fractional partial differential equations

We introduce a sampling-based machine learning approach, Monte Carlo fractional physics-informed neural networks (MC-fPINNs), for solving forward and inverse fractional partial differential equations (FPDEs). As a generalization of the physics-informed neural networks (PINNs), MC-fPINNs utilize a Monte Carlo approximation strategy to compute the fractional derivatives of the DNN outputs, and construct an unbiased estimation of the physical soft constraints in the loss function. Our sampling approach can yield lower overall computational cost compared to fPINNs proposed in Pang et al.(2019), hence it can solve high dimensional FPDEs at reasonable cost. We validate the performance of MC-fPINNs via several examples, including high dimensional integral fractional Laplacian equations, parametric identification of time–space fractional PDEs, and fractional diffusion equation with random inputs. The results show that MC-fPINNs are flexible and quite effective in tackling high dimensional FPDEs.

Finite Difference Method for Inhomogeneous Fractional Dirichlet Problem

We make the split of the integral fractional Laplacian as $(-\Delta)^s u=(-\Delta)(-\Delta)^{s-1}u$, where $s\in(0,\frac{1}{2})\cup(\frac{1}{2},1)$. Based on this splitting, we respectively discretize the one- and two-dimensional integral fractional Laplacian with the inhomogeneous Dirichlet boundary condition and give the corresponding truncation errors with the help of the interpolation estimate. Moreover, the suitable corrections are proposed to guarantee the convergence in solving the inhomogeneous fractional Dirichlet problem and an $\mathcal{O}(h^{1+\alpha-2s})$ convergence rate is obtained when the solution $u\in C^{1,\alpha}(\bar{\Omega}^{\delta}_{n})$, where $n$ is the dimension of the space, $\alpha\in(\max(0,2s-1),1]$, $\delta$ is a fixed positive constant, and $h$ denotes mesh size. Finally, the performed numerical experiments confirm the theoretical results.

Local Convergence of the FEM for the Integral Fractional Laplacian

We provide for first order discretizations of the integral fractional Laplacian sharp local error estimates on proper subdomains in both the local $H^1$-norm and the localized energy norm. Our estimates have the form of a local best approximation error plus a global error measured in a weaker norm.

Fractional Semilinear Optimal Control: Optimality Conditions, Convergence, and Error Analysis

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 30 July 2020Accepted: 10 August 2021Published online: 05 January 2022Keywordsoptimal control problem, fractional diffusion, integral fractional Laplacian, regularity estimates, finite elements, convergence, a priori error estimatesAMS Subject Headings35R11, 49J20, 49M25, 65K10, 65N15, 65N30Publication DataISSN (print): 0036-1429ISSN (online): 1095-7170Publisher: Society for Industrial and Applied MathematicsCODEN: sjnaam

Local discontinuous Galerkin method for the fractional diffusion equation with integral fractional Laplacian

In this paper, we provide a framework of designing the local discontinuous Galerkin scheme for integral fractional Laplacian (−Δ)s with s∈(0,1) in two dimensions. We theoretically prove and numerically verify the numerical stability and convergence of the scheme with the convergence rate no worse than O(hk+12) with k≥1.