分数拉普拉斯方程：解的正则性及有限元近似

Affine approximation of parametrized kernels and model order reduction for nonlocal and fractional Laplace models

We consider parametrized problems driven by spatially nonlocal integral operators with parameter-dependent kernels. In particular, kernels with varying nonlocal interaction radius $δ> 0$ and fractional Laplace kernels, parametrized by the fractional power $s\in(0,1)$, are studied. In order to provide an efficient and reliable approximation of the solution for different values of the parameters, we develop the reduced basis method as a parametric model order reduction approach. Major difficulties arise since the kernels are not affine in the parameters, singular, and discontinuous. Moreover, th…

An inverse problem of determining fractional orders in a fractal solute transport model

A fractal mobile-immobile (MIM in short) solute transport model in porous media is set forth, and an inverse problem of determining the fractional orders by the additional measurements at one interior point is investigated by Laplace transform. The unique existence of the solution to the forward problem is obtained based on the inverse Laplace transform, and the uniqueness of the inverse problem is proved in the real-space of Laplace transform by the maximum principle, and numerical inversions with noisy data are presented to demonstrate a numerical stability of the inverse problem.

用分数阶算子正则化拉普拉斯方程的柯西问题

Regularising the Cauchy problem for Laplace's equation by fractional operators

本文提出用分数阶导数替代整数阶导数以正则化拉普拉斯方程柯西问题，在矩形/圆柱几何中利用Abel型分数阶导数的稳定化效应，并用牛顿法处理自由边界恢复及阻抗函数的同时恢复，给出数值重构与收敛性结果。

Fractional Elliptic Problems on Lipschitz Domains: Regularity and Approximation

This survey hinges on the interplay between regularity and approximation for linear and quasi-linear fractional elliptic problems on Lipschitz domains. For the linear Dirichlet integral Laplacian, after briefly recalling Hölder regularity and applications, we discuss novel optimal shift theorems in Besov spaces and their Sobolev counterparts. These results extend to problems with finite horizon and are instrumental for the subsequent error analysis. Moreover, we dwell on extensions of Besov regularity to the fractional $p$-Laplacian, and review the regularity of fractional minimal graphs and s…

用于非各向同性分数阶拉普拉斯、亥姆霍兹和汤川方程的立方游动蒙特卡罗模拟

Walk-on-Cubes Monte Carlo Simulation for nonisotropic fractional Laplace, Helmholtz, and Yukawa equations

研究拉普拉斯、亥姆霍兹和汤川方程的非各向同性分数阶类似物，基于非各向同性分数阶拉普拉斯方程与对称α - 稳定列维过程的关系，给出立方游动算法模拟亥姆霍兹和汤川方程的解，还提供了相关对应关系和重构。

On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: regularity and error estimates

We are interested in the classical ill-posed Cauchy problem for the Laplace equation. One method to approximate the solution associated with compatible data consists in considering a family of regularized well-posed problems depending on a small parameter $\varepsilon>0$. In this context, in order to prove convergence of finite elements methods, it is necessary to get regularity results of the solutions to these regularized problems which hold uniformly in $\varepsilon$. In the present work, we obtain these results in smooth domains and in 2D polygonal geometries. In presence of corners, due t…

一个用于最优控制问题的有限元方法：受分数拉普拉斯方程约束

A FEM for an optimal control problem subject to the fractional Laplace equation

本文研究了受分数拉普拉斯方程约束的线性二次最优控制问题的数值近似，提出了一种基于平衡公式离散化的有限元方法，并通过数值实验验证了两种不同离散化方案的有限元误差估计。

Weighted Sobolev regularity and rate of approximation of the obstacle problem for the integral fractional Laplacian

We obtain regularity results in weighted Sobolev spaces for the solution of the obstacle problem for the integral fractional Laplacian. The weight is a power of the distance to the boundary. These bounds then serve us as a guide in the design and analysis of an optimal finite element scheme over graded meshes.