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.

最小二乘问题在数据拟合中自然出现，其中数学模型中的参数是通过最大程度地减少观察到的数据与模型预测的输出之间的差异（通过平方之和测量的）来校准的。 它们也自然地通过优化方法求解非线性方程。该项目的目的是为最小二乘问题开发新的数值方案，这些方案与无限多个参数拟合时出现的问题，例如从观察到的压力中确定地下水的通量，以及在求解非线性方程中使用无限多个未知数的最小值方程中出现的问题，例如最佳传输图的方程式。两种设置中的方程式都是椭圆方程，描述了科学和工程中的稳态问题，并且有关基本问题的先验信息以不平等约束的形式包括在内。 数值方案基于有限元方法，这是计算工程和科学领域的主要方法之一。该项目的结果将为工程和材料科学的最佳设计过程以及图像处理和数据科学的新方法提供新的工具。该项目为研究生提供了研究培训机会。将研究涉及椭圆形偏微分方程的不平等限制的无限维度最小二乘问题。头等舱与椭圆形分布式最佳控制问题有关，具有点状状态和控制约束。第二类涉及解决解决方案上的凸度约束的完全非线性椭圆边界值问题。对于椭圆形的最佳控制问题，将针对具有一般成本功能的问题开发新的有限元方法，其中包括将国家跟踪问题作为特殊情况，对国家梯度的限制问题以及受椭圆方程（具有粗糙系数的椭圆方程）的问题。对于完全非线性的椭圆边界值问题，将研究其经典解决方案的有限元方法。 它们包括Monge-Ampere类型的方程式，其中解决方案的凸度起着关键作用，例如在两个和三维中的Monge-Ampere方程的第一和第二个边界值问题，以及在二维中规定的高斯曲率方程的Dirichlet边界值问题。 在三个维度上的2-新方程也将得到处理，其中溶液的拉普拉斯（Laplacian）阳性的条件是Monge-Ampere方程解决方案的凸状条件的类似物。这两个类别问题研究的一个共同主题是椭圆形部分微分方程，优化和有限元技术之间的相互作用，例如不连续的盖尔金方法，多尺度有限元方法，虚拟元素方法，虚拟元素方法以及凸性强制执行有限元的方法，这些方法反映了NSF的合法任务和范围的范围。 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