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.
