Finite Element Methods for Elliptic Least-Squares Problems with Inequality Constraints

具有不等式约束的椭圆最小二乘问题的有限元方法

• 批准号: 2208404
• 负责人: Susanne Brenner
• 金额: $ 36.13万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2208404&HistoricalAwards=false
• 关键词: Finite; Element; Methods; Elliptic; Least

Least-squares problems appear naturally in data fitting, where the parameters in a mathematical model are calibrated by minimizing the discrepancy (measured by a sum of squares) between the observed data and the output predicted by the model. They also appear naturally in solving nonlinear equations by optimization methods. The goal of this project is to develop novel numerical schemes for least-squares problems that appear in data fitting with infinitely many parameters, such as the determination of the flux of groundwater from the observed pressure, and for least-squares problems that appear in solving nonlinear equations with infinitely many unknowns, such as the equation for an optimal transport map. The equations in both settings are elliptic equations that describe steady-state problems in science and engineering, and a priori information on the underlying problems is included in the form of inequality constraints. The numerical schemes are based on finite element methods, one of the leading methodologies in computational engineering and science. The outcomes of this project will provide new tools for the optimal design process in engineering and materials science, and new methodologies for image processing and data science. The project provides research training opportunities for graduate students.Two classes of infinite dimensional least-squares problems with inequality constraints that involve elliptic partial differential equations will be investigated. The first class is concerned with elliptic distributed optimal control problems with pointwise state and control constraints. The second class is concerned with solving fully nonlinear elliptic boundary value problems with convexity constraints on the solutions. For the elliptic optimal control problems, novel finite element methods will be developed for problems with general cost functions that include point tracking problems for the state as a special case, problems with constraints on the gradient of the state, and problems constrained by elliptic equations with rough coefficients. For the fully nonlinear elliptic boundary value problems, finite element methods for their classical solutions will be investigated. They include equations of the Monge-Ampere type where the convexity of the solutions plays a key role, such as the first and second boundary value problems for the Monge-Ampere equations in two and three dimensions, and the Dirichlet boundary value problem for the prescribed Gaussian curvature equation in two dimensions. The 2-Hessin equation in three dimensions will also be treated, where the condition on the positivity of the Laplacian of the solution is the analog of the convexity condition on the solutions of the Monge-Ampere equations. A common theme for the research in these two classes of problems is the interplay among elliptic partial differential equations, optimization, and finite element technology such as discontinuous Galerkin methods, multiscale finite element methods, virtual element methods, and convexity enforcing finite element methods.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

最小二乘问题自然出现在数据拟合中，其中数学模型中的参数通过最小化观测数据与模型预测的输出之间的差异（通过平方和测量）来校准。 它们也自然地出现在用最优化方法求解非线性方程中。本项目的目标是为最小二乘问题开发新的数值方案，这些最小二乘问题出现在具有无限多参数的数据拟合中，例如从观测压力确定地下水的流量，以及为最小二乘问题出现在求解具有无限多未知数的非线性方程中，例如最佳运输图的方程。这两种设置中的方程都是描述科学和工程中稳态问题的椭圆方程，并且以不等式约束的形式包含有关潜在问题的先验信息。 数值方案是基于有限元方法，在计算工程和科学的领先方法之一。该项目的成果将为工程和材料科学的优化设计过程提供新的工具，并为图像处理和数据科学提供新的方法。本计画为研究生提供研究训练的机会，并将探讨两类包含椭圆型偏微分方程的不等式约束无穷维最小二乘问题。第一类是具有逐点状态和控制约束的椭圆分布最优控制问题。第二类是解带有凸性约束的完全非线性椭圆边值问题。对于椭圆型最优控制问题，新的有限元方法将开发的一般成本函数的问题，包括点跟踪问题的状态作为一个特殊情况下，与约束的问题的状态梯度，和椭圆方程约束的问题与粗糙系数。对于完全非线性椭圆边值问题，我们将研究其经典解的有限元方法。 它们包括Monge-Ampere型方程，其中解的凸性起着关键作用，例如二维和三维Monge-Ampere方程的第一和第二边值问题，以及二维高斯曲率方程的Dirichlet边值问题。 三维的2-Hessin方程也将被处理，其中解的Laplacian的正性条件类似于Monge-Ampere方程解的凸性条件。在这两类问题的研究中，一个共同的主题是椭圆型偏微分方程、优化和有限元技术之间的相互作用，例如间断Galerkin方法、多尺度有限元方法、虚元方法、该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准。

项目成果

An Interior Maximum Norm Error Estimate for the Symmetric Interior Penalty Method on Planar Polygonal Domains

平面多边形域对称内罚法的内最大范数误差估计

• 发表时间: 2022
• 期刊: Computational methods in applied mathematics
• 影响因子: 1.3
• 作者: Brenner, Susanne C;Sung, Li-yeng
• 通讯作者: Sung, Li-yeng

Multigrid methods for an elliptic optimal control problem with pointwise state constraints

• DOI: 10.1016/j.rinam.2023.100356
• 发表时间: 2023-02
• 期刊: Results in Applied Mathematics
• 影响因子: 2
• 作者: S. C. Brenner;Sijing Liu;L. Sung