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

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是通过最小化观测数据与模型预测输出之间的差异（通过平方和测量）来校准的。它们在用最优化方法求解非线性方程时也很自然地出现。该项目的目标是为具有无限多个参数的数据拟合中出现的最小二乘问题开发新的数值格式，例如从观测压力确定地下水通量，以及解决具有无限多个未知数的非线性方程中出现的最小二乘问题，例如最优运输图方程。这两种情况下的方程都是描述科学和工程中的稳态问题的椭圆方程，并且关于潜在问题的先验信息以不等式约束的形式包含。数值格式基于有限元方法，这是计算工程和科学中的主要方法之一。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。本项目为研究生提供研究训练机会。研究了两类具有不等式约束的无限维最小二乘问题，这些问题涉及椭圆型偏微分方程。第一类研究具有点态约束和控制约束的椭圆型分布最优控制问题。第二类是求解具有凸性约束的完全非线性椭圆型边值问题。对于椭圆型最优控制问题，将开发新的有限元方法来解决具有一般成本函数的问题，其中包括作为特殊情况的状态点跟踪问题，状态梯度约束问题以及具有粗糙系数的椭圆方程约束问题。对于完全非线性椭圆型边值问题，将研究其经典解的有限元方法。它们包括蒙日-安培型方程，其中解的凹凸性起着关键作用，例如二维和三维蒙日-安培方程的第一和第二边值问题，以及二维规定高斯曲率方程的Dirichlet边值问题。我们还将讨论三维空间中的2-Hessin方程，其解的拉普拉斯函数为正的条件类似于monage - ampere方程解的凸性条件。这两类问题研究的共同主题是椭圆偏微分方程、优化和有限元技术之间的相互作用，如不连续伽辽金方法、多尺度有限元方法、虚元方法和凸性强化有限元方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.