Banach空间中演化一阶方程的两点问题

利用演化深度神经网络求解偏微分方程的位置嵌入方法

Positional Embeddings for Solving PDEs with Evolutional Deep Neural Networks

本文提出基于Laplace-Beltrami算子特征函数的位置嵌入，将演化深度神经网络扩展到复杂几何区域上的参数化时变PDE求解，并通过Krylov求解、主动采样和改进Rosenbrock方法提升性能，数值实验验证了其有效性。

Approximation of the fixed point of the product of two operators in Banach algebras with applications to some functional equations

In this paper, the existence and uniqueness of the fixed point for the product of two nonlinear operator in Banach algebra is discussed. In addition, an approximation method of the fixed point of hybrid nonlinear equations in Banach algebras is established. This method is applied to two interesting different types of functional equations. In addition, to illustrate the applicability of our results we give some numerical examples.

Krasnoselskij-type algorithms for variational inequality problems and fixed point problems in Banach spaces

Existence and uniqueness as well as the iterative approximation of fixed points of enriched almost contractions in Banach spaces are studied. The obtained results are generalizations of the great majority of metric fixed point theorems, in the setting of a Banach space. The main tool used in the investigations is to work with the averaged operator $T_λ$ instead of the original operator $T$. The effectiveness of the new results thus derived is illustrated by appropriate examples. An application of the strong convergence theorems to solving a variational inequality is also presented.

Banach 空间中 $m$ 点问题的快速收敛方法

Fast convergent method for the $m$-point problem in Banach space

针对 Banach 空间中一阶微分方程的非局部 $m$ 点问题，提出并验证了一种基于 Dunford-Cauchy 积分和短求和求积公式的指数收敛算法。

A novel two-point gradient method for Regularization of inverse problems in Banach spaces

In this paper, we introduce a novel two-point gradient method for solving the ill-posed problems in Banach spaces and study its convergence analysis. The method is based on the well known iteratively regularized Landweber iteration method together with an extrapolation strategy. The general formulation of iteratively regularized Landweber iteration method in Banach spaces excludes the use of certain functions such as total variation like penalty functionals, $L^1$ functions etc. The novel scheme presented in this paper allows to use such non-smooth penalty terms that can be helpful in practica…

Banach空间中Hölder梯度下降与自适应正则化方法用于一阶点优化

Hölder Gradient Descent and Adaptive Regularization Methods in Banach Spaces for First-Order Points

本文研究了在光滑无限维空间中优化光滑非凸函数的算法，提出Hölder梯度下降法用于寻找正则化多项式函数的一阶点，并分析了自适应正则化方法的评估复杂度。