Nonlinear PDE's, Numerical Analysis, and Applications; October 2-3, 2015; Pittsburgh, PA

会议：非线性偏微分方程，数值分析与应用，2015年10月2-3日，匹兹堡大学。偏微分方程（PDEs）在理解、模拟和预测科学和工程中发生的各种现象方面起着关键作用。然而，即使是简单的问题，精确的数学解是无法获得的，并且需要计算方法来构建近似解。因此，迫切需要发展数值方法，并从理论上证明所得近似的质量和可靠性。理论分析除了为数值方法提供依据外，还经常为提高效率、精度和能力的新方法的发展提供见解。这个为期两天的会议将为初级和高级研究人员提供一个论坛，讨论非线性偏微分方程数值方法的前沿结果和应用。会议的目的是提供一个非正式的环境，让年轻的研究人员和顶尖的数值分析专家可以见面、合作和发展新的想法。此外，会议还将向研究生和博士后介绍数值偏微分方程这一活跃领域。完全非线性二阶偏微分方程的计算方法的构建、实现和分析是相对较新的，但在数值分析和科学计算中却是关键的研究领域。这些问题出现在许多应用领域，包括气象学、宇宙学、几何光学、微分几何、最优运输、经济学、图像处理和网格生成。这些问题构成了最难以数值近似的偏微分方程之一，其离散化方面的突破只是在最近15年才出现。虽然在数值全非线性二阶偏微分方程方面取得了一些进展，但仍然存在需要适当解决的基本挑战。例子包括在现实网格条件下具有单调性的非线性伽辽金方法的构造，一般非线性问题族的鲁棒数值方法，非标准边界条件的强加，以及由此产生的非线性代数系统的快速求解。本次会议将聚集该领域的领导者，讨论最新的研究趋势和未来的研究方向。

Conference: Nonlinear PDEs, Numerical Analysis, and Applications, October 2-3, 2015, University of Pittsburgh. Partial differential equations (PDEs) play a key role in the understanding, simulation and prediction of various phenomena occurring in the sciences and engineering. Yet even for simple problems, exact mathematical solutions are unattainable, and computational methods are necessary to construct approximate solutions. Therefore there is a critical need to develop numerical methods and to theoretically justify the quality and reliability of the resulting approximations. In addition to providing justification of the numerical methods, the theoretical analysis often gives insight for the development of new methods with improved efficiency, accuracy and capabilities. This two-day conference will create a forum for junior and senior researchers to discuss cutting-edge results and applications of numerical methods for nonlinear PDEs. The aim of the conference is to provide an informal setting in which young researchers and leading experts in numerical analysis can meet, collaborate, and develop new ideas. In addition, the conference will expose graduate students and postdocs on this active field of numerical PDEs.The construction, implementation, and analysis of computational methods for fully nonlinear second order partial differential equations are relatively new, yet critical research areas in numerical analysis and scientific computing. Such problems arise in many application areas including meteorology, cosmology, geometric optics, differential geometry, optimal transport, economics, imagine processing and mesh generation. These problems constitute one of the most difficult classes of PDEs to approximate numerically, and breakthroughs in their discretization have only appeared within the last 15 years. While there have been several advances in numerical fully nonlinear second order PDEs, there still remain fundamental challenges that need to be properly addressed. Examples include the construction of nonlinear Galerkin methods with monotonicity properties under realistic mesh conditions, robust numerical methods for general families of nonlinear problems, imposition of non-standard boundary conditions, and fast solvers of the resulting non-linear algebraic systems. This conference will gather leaders in the field to discuss state-of-the-art research trends and directions of future research.