L1-based Approximation Techniques for PDEs

L1近似方法最近获得了发展势头，由于深刻的理论结果将L1与数据的稀疏表示联系起来。这一特性是压缩感知技术的起源。与压缩感知运动的发展平行，研究人员在2005年开始了一项由美国国家科学基金会赞助的研究计划，目的是开发一种新的基于L1最小化的非线性近似技术，用于求解第一阶非线性问题，阶非线性微分方程在这个研究项目中，研究人员证明了基于L1的方法是近似一阶非线性微分方程的非常有效的工具，这些微分方程的解是不连续的或具有尖锐的界面。 在即将到来的研究计划中，研究人员将做以下工作：（i）开发计算L1-极小的快速算法;（ii）将他们的方法扩展到时间依赖的非线性一阶微分方程;（iii）开发基于L1的表面重建，数据增强，图像恢复/去模糊技术。 关键是要依靠压缩感知的精神稀疏。 由于L1引入的强非线性和缺乏平滑性，该项目的计算和理论方面都非常具有挑战性。该研究项目将在工程领域（机械、航空航天、海洋等）产生广泛影响，环境科学、物理学、石油工程。 提出一种新的鲁棒近似技术来解决非线性问题，发展冲击或尖锐的界面，最终将有利于控制或处理这种类型的现象仍然是一个巨大的挑战的科学和工程的每一个领域。研究人员开发的算法也将有助于解决诸如从分散数据重建表面、人脸识别、数据恢复和增强等问题。该计划的最后一个方面可能有助于国家安全。

L1-approximation methods have recently gained momentum due to profound theoretical results relating L1 to sparse representation of data.This property is at the origin of the compressed sensing technique.In parallel to the development of compressed sensing movement, the investigators started in 2005 a NSF-sponsored research program to develop a new nonlinear approximation technique based on L1 minimization for solving first-order nonlinear differential equations. In this research program the investigators proved that L1-based methods are very efficient tools for approximating first-order nonlinear differential equations whose solutions are discontinuous or have sharp interfaces. In the forthcoming research program the investigators will do the following: (i) develop fast algorithms for computing L1-minimizers; (ii) extend their methodology to time-dependent nonlinear first-order differential equations; (iii) develop L1-based techniques for surface reconstruction, data enhancing, image recovery/de-blurring.The outcome of this project will be a computational framework radically different from the existing mainstream techniques. The key is to rely on sparsity in the spirit of compressed sensing. Both the computational and theoretical aspects of the project are very challenging because of the strong nonlinearity and lack of smoothness introduced by L1. This research project will have a broad impact in engineering (mechanical, aerospace, ocean, etc.), environmental sciences, geophysics, petroleum engineering. Proposing a novel robust approximation technique for solving nonlinear problems developing shock or sharp interfaces will eventually benefit every areas of science and engineering where controlling or dealing with this type of phenomenon is still an enormous challenge. The algorithms developed by the investigators will also be useful for solving problems like surface reconstruction from scattered data, face recognition, data recovery and enhancing. This last aspect of the program could be useful for national security purposes.