A generalized strong convergence algorithm in the presence of the errors for the variational inequality problems in Hilbert spaces

A Priori Error Analysis for an Optimal Control Problem Governed by a Variational Inequality of the Second Kind

We consider an optimal control problem governed by an elliptic variational inequality of the second kind. The problem is discretized by linear finite elements for the state and a variational discrete approach for the control. Based on a quadratic growth condition we derive nearly optimal a priori error estimates. Moreover, we establish second order sufficient optimality conditions that ensure a quadratic growth condition. These conditions are rather restrictive, but allow us to construct a one-dimensional locally optimal solution with reduced regularity, which serves as an exact solution for n…

非线性不适定问题的变分不等式方法的扩展

An extension of the variational inequality approach for nonlinear ill-posed problems

本文在Banach空间框架下，将变分不等式方法从Hölder速率扩展到更一般的速率，包括对数收敛速率，用于非线性不适定算子方程的Tikhonov正则化。

Convergence Analysis of Penalty Based Numerical Methods for Constrained Inequality Problems

This paper presents a general convergence theory of penalty based numerical methods for elliptic constrained inequality problems, including variational inequalities, hemivariational inequalities, and variational-hemivariational inequalities. The constraint is relaxed by a penalty formulation and is re-stored as the penalty parameter tends to zero. The main theoretical result of the paper is the convergence of the penalty based numerical solutions to the solution of the constrained inequality problem as the mesh-size and the penalty parameter approach zero simultaneously but independently. The …

A minimax approach for inverse variational inequalities

In this work, we characterize the existence of solution for a certain variational inequality by means of a classical minimax theorem. In addition, we propose a numerical algorithm for the solution of an inverse problem associated with a variational inequality. To this end we state a collage-type result in this variational framework.

深度域分解方法用于求解变分不等式问题

Deep Domain Decomposition Method for Solving the Variational Inequality Problems

本文提出一种结合PINN与域分解方法的深度域分解方法，用于高效求解椭圆变分不等式问题，通过残差自适应训练策略提升模型学习能力，实验表明该方法具有高精度和稳定性。

非局部变分等式与不等式问题的正则性与逼近分析

Regularity and approximation analyses of nonlocal variational equality and inequality problems

本文研究由非局部积分算子驱动的线性与障碍问题，通过非局部向量微积分将其转化为弱形式，证明了最优正则性结果，并分析了有限元近似收敛性。

隐式数值方案用于最优市场做市问题中的Hamilton-Jacobi-Bellman拟变分不等式

Implicit Numerical Scheme for the Hamilton-Jacobi-Bellman Quasi-Variational Inequality in the Optimal Market-Making Problem with Alpha Signal

本文提出隐式数值方案和策略迭代算法，用于解决最优市场做市问题中的HJBQVI，确保无条件稳定性并收敛到唯一粘性解。

近端与收缩方法结合松弛惯性与修正项用于求解混合变分不等式问题

Proximal and Contraction method with Relaxed Inertial and Correction Terms for Solving Mixed Variational Inequality Problems

本文提出了一种结合松弛惯性与修正项的近端与收缩方法，用于高效求解混合变分不等式问题，并通过实验验证了其有效性。

沙堆生长建模中出现的拟变分不等式问题

A Quasi-Variational Inequality Problem Arising in the Modeling of Growing Sandpiles

针对沙堆表面演化模型中的拟变分不等式问题，提出正则化混合形式并证明解的存在性，同时通过数值方法有效计算沙堆表面和沙流。