Collaborative Research: Numerical Methods for Fully and Implicitly Nonlinear Equations

本项目的主要目的是进一步研究Monge-Ampère、S等完全非线性椭圆型方程的数值解，以推广在NSF Grant DMS-0412267的支持下所做的工作。这些研究的主要发现是，在数据正则化(这并不总是必要的)之后，在适当的Hilbert空间中选择适当的最小二乘公式可以得到能够计算经典解的稳健解方法，或者如果不存在经典解，则可以得到广义解。本项目的目标是：(1)改进用于求解最小二乘问题的迭代方法的性能。这将需要开发新的算法来解决从最小二乘问题的分解获得的许多(每个网格点一个)低维非线性特征值问题，因为这导致了许多小但复杂的约束特征值问题。(Ii)演示这些新算法在各种测试问题(Monge-Ampère、Pucci、GaussCurvature、sigma-2，对于Pucci问题的2维、3维甚至4维，使用并行化)上的有效性。(Iii)确定当算子中的系数在空间中周期性或随机变化时，某些完全非线性椭圆型方程在二维中观察到的显著的齐次化性质是否在更高的维度上持续存在。(Iv)最终将上述方法(或其近似式)应用于一些隐含的非线性偏微分方程组的求解，这些偏微分方程用非光滑的微分几何来模拟折叠现象。推动这些研究的是这样一个事实：完全非线性的椭圆型方程在材料科学、非线性弹性、流体力学、大气科学、非线性弹性、电子和结构工程(天线、汽车形状等)的形状设计、金融、应用和理论物理、微分几何等领域发挥着重要的作用。相关的数学问题已经产生了大量的文献。相比之下，从计算的角度来看，这些问题被认为很难解释为什么计算数学家和应用数学家在数值解上没有取得重大进展。该项目的目标之一是缩小关注完全非线性椭圆型方程的不同群体之间的差距，使他们每个人都能向其他人学习，树立跨学科科学的榜样。这样的努力也将使科学和工程专业人员和学生受益，通过出版物、专门的网站和研究生课程、在会议上的演讲，当然还有一些研究生的直接参与。它还将激发其他科学家对这些重要领域的贡献。由于以前由首席研究人员及其同事开发的计算方法目前用于科学与工程、学术和工业的许多领域，因此可以预期，由该合作项目产生的结果和产品也将做出类似的努力。

The main goal of this project is to further investigate the numerical solution of fully nonlinear elliptic equations such as Monge-Ampère?s, in order to extend previous work done with the support of NSF Grant DMS-0412267. The main findings of these previous investigations are that, after regularization of the data (which is not always necessary), well-chosen least-squares formulations in appropriate Hilbert spaces lead to robust solution methods able to compute classical solutions, or generalized ones if classical solutions do not exist. The objectives of the present project are: (i) To improve the performances of the iterative methods used to solve the least-squares problems. This will require the development of novel algorithms to solve the many (one per grid point) low dimensional nonlinear eigenvalue problems obtained from the decomposition of the least squares problems, since this results in a number of small but intricate constrained eigenvalue problems. (ii) To demonstrate the effectiveness of these new algorithms on a variety of test problems (Monge-Ampère, Pucci, Gaussian curvature, sigma-2, in dimension 2, 3, and even 4 for the Pucci problem, using parallelization). (iii) To determine whether the remarkable homogenization properties observed in two-dimensions for some of these fully nonlinear elliptic equations when a coefficient in the operator varies periodically or randomly in space persist in higher dimensions. (iv) To apply, ultimately, the above methodology (or close variant of it) to the solution of some implicitly nonlinear partial differential equations from non-smooth differential geometry that model folding phenomena.What motivates these investigations is the fact that fully nonlinear elliptic equations play an important role in areas as diverse as material sciences, nonlinear elasticity, fluid mechanics, atmospheric sciences, nonlinear elasticity, shape design in electrical and structural engineering (antennas, car shape,?), finance, applied and theoretical physics, differential geometry and others. The related mathematical problems have generated a large literature. In contrast these problems have the reputation to be difficult from a computational standpoint explaining why the computational and applied mathematicians have not made significant progress on their numerical solution. One of the goals of this project is to close the gap between the various communities concerned with fully nonlinear elliptic equations so that each of them will learn from the others, setting an example of interdisciplinary science. Such an effort will also benefit science and engineering professionals and students, via publications, dedicated web sites and post-graduate courses, lectures at conferences, and of course direct involvement for some graduate students. It will also stimulate the contributions of other scientists to these important areas. Since computational methods developed previously by the Principal Investigators and their associates are currently used in many areas of Science and Engineering, Academia and Industry, one can expect a similar endeavor for the results and products originating from this collaborative project.