通过磨光化对逆拉普拉斯变换的正则化

A variational technique of mollification applied to backward heat conduction problems

This paper addresses a backward heat conduction problem with fractional Laplacian and time-dependent coefficient in an unbounded domain. The problem models generalized diffusion processes and is well-known to be severely ill-posed. We investigate a simple and powerful variational regularization technique based on mollification. Under classical Sobolev smoothness conditions, we derive order-optimal convergence rates between the exact solution and regularized approximation in the practical case where both the data and the operator are noisy. Moreover, we propose an order-optimal a-posteriori par…

一种 mollification 方法用于分形传输及树形优化

A Mollification Approach to Ramified Transport and Tree Shape Optimization

本文提出一种 mollification 方法用于计算最优灌溉模式，通过 Lagrangian 框架正则化灌溉成本函数，并应用于树根和枝干最优形状优化问题。

演化$p$-拉普拉斯方程的路径式噪声正则化现象

A pathwise regularization by noise phenomenon for the evolutionary $p$-Laplace equation

该研究针对势函数随时间平移的演化$p$-拉普拉斯问题，在轨迹局部时足够正则的条件下，证明了经典方法失效的奇异势函数下解的存在性，确立了该非线性问题的路径式噪声正则化现象。

Regularization of radial solutions of $p$-Laplace equations, and computations using infinite series

We consider radial solutions of equations with the $p$-Laplace operator in $R^n$. We introduce a change of variables, which in effect removes the singularity at $r=0$. While solutions are not of class $C^2$, in general, we show that solutions are $C^2$ functions of $\displaystyle r^{\frac{p}{2(p-1)}}$. Then we express the solution as an infinite series in powers of $\displaystyle r^{\frac{p}{p-1}}$, and give explicit formulas for its coefficients. We implement this algorithm, using Mathematica software. Mathematica's ability to perform the exact computations turns out to be crucial.