Monday, January 21, 2008

Compressed Sensing: Nonlinear Approximation Techniques Using L1 and a Workshop.

So the L1 revolution is picking up and it will eventually help us solve linear particle transport equations. In the meantime, it looks like it is getting traction in the Applied Mathematics area interested in the solving of all kinds of PDEs, in particular the Navier-Stokes Equation and its simplified versions (1-D,...). In Survey of Nonlinear Approximation of PDEs in L1, Jean-Luc Guermond and Bojan Popov detail why L1 is becoming interesting for PDE solvers with 1-D and 2-D examples. They first point out that other areas are using L1 minimization techniques. While Compressed Sensing is one of them, they also mention CAD systems where L1 splines do better than L2 splines: "L1 splines are less oscillatory than L2 splines and L1 splines can compress data better than L2 splines." They then describe why in the PDE world, Galerkin methods are short in providing solution techniques (we already knew that in Neutron Transport !)

They first describe the attendant conformism that makes most of the engineering world live in L2 world (FEM).
and then provide what will sell the idea

Sweet. As they say in the concluding remarks: "We should not be afraid to work in Banach spaces. It is doable." It mirrors what I say to others when mentioning compressed sensing, "Come work on the dark side of the Force":-). Further reference on these results can be found in the following two papers:
  1. J.-.L. GUERMOND, A finite element technique for solving first order PDE's in Lp, SIAM J. Numer. Anal. 42 2 (2004) 714--737.
  2. J.-.L. GUERMOND, B. POPOV, Linear advection with ill-posed boundary conditions via L1-minimization, International Journal of Numerical Analysis and Modeling, 4:1 (2007) 39-47.

On a related note, Jean-Luc Guermond Bojan Popov and Ron DeVore organize a workshop on Nonlinear Approximation Techniques Using L1 at Texas A&M University on May 16th to 18th, 2008. It'll take place in the Blocker building, Room 120.

L1-based techniques are emerging in different fields of applied sciences. The objective of the workshop is to gather leading experts that are possibly unaware of each other's activity so that they exchange ideas and get a more global view of this new set of methods. The purpose of this workshop is also stimulate cross fertilization.

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