Numerical Analysis of Selected Variational and Quasi-variational Inequalities

选定变分和拟变分不等式的数值分析

• 批准号: 1418784
• 负责人: Abner Salgado
• 金额: $ 13.43万
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-15 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1418784&HistoricalAwards=false
• 关键词: Numerical; Analysis; Selected; Variational; Quasi

Unilateral and constrained phenomena are ubiquitous in science and engineering. The quantities that govern a physical process are often subject to constraints: Be it of mechanical, physical, thermodynamical or practical nature. Examples of these can be impenetrability conditions (two bodies cannot be at the same place at the same time), the fact that mass cannot be negative and that entropy cannot decrease or simply the fact that we are not able to produce more than a fixed amount of forcing. In addition, many of these constraints might depend on the quantities of interest themselves (e.g. friction). One final example is mechanical contact which is, in fact, the only mechanism through which we can exert any mechanical action on another body. These examples show that the derivation of realistic models for the description of these phenomena is of fundamental importance in applications. However these models will be, as a rule, nonlinear. The development and analysis of numerical techniques for approximating the solution of these problems is of practical relevance.The models that govern constrained phenomena carry the name of variational and quasi-variational inequalities. This proposal aims at the study of numerical techniques for a collection of variational and quasi-variational inequalities which have not been considered before and to which the classical ideas and techniques do not apply. They include: variational inequalities for nonlocal operators, evolution problems for degenerate parabolic equations and the general study of time discretization techniques for quasi-variational inequalities. The numerical methods that will result from this project will be of interest to a wide range of practitioners, since the proposed problems have a wide range of applications. For instance, obstacle problems with fractional diffusion appear in control theory, fluid mechanics and even finance; the degenerate parabolic equations that will be considered find applications in imaging and materials science; the prototypical example of a quasi-variational inequality is friction, but they also arise in game theory - when trying to find Nash points - and in control theory when dealing with impulse controls.

单边和约束现象在科学和工程中普遍存在。控制物理过程的量通常受到限制：无论是机械、物理、热力学还是实际性质。例如，不可穿透条件（两个物体不能同时位于同一地点）、质量不能为负且熵不能减少的事实，或者仅仅是我们无法生产超过固定数量的事实的强迫。此外，许多这些约束可能取决于感兴趣的数量本身（例如摩擦力）。最后一个例子是机械接触，事实上，这是我们可以对另一个物体施加任何机械作用的唯一机制。这些例子表明，推导用于描述这些现象的现实模型在应用中至关重要。然而，这些模型通常是非线性的。用于逼近解决这些问题的数值技术的开发和分析具有实际意义。控制约束现象的模型具有变分和拟变分不等式的名称。该提案旨在研究一系列变分和准变分不等式的数值技术，这些不等式以前没有被考虑过，并且经典思想和技术不适用于这些不等式。它们包括：非局部算子的变分不等式、简并抛物型方程的演化问题以及拟变分不等式的时间离散技术的一般研究。该项目产生的数值方法将引起广泛的从业者的兴趣，因为所提出的问题具有广泛的应用范围。例如，分数扩散的障碍问题出现在控制理论、流体力学甚至金融领域；将考虑在成像和材料科学中找到应用的简并抛物线方程；准变分不等式的典型例子是摩擦，但它们也出现在博弈论中（试图找到纳什点时）以及控制论中处理脉冲控制时。