# Rongjie Lai : Compressed Modes and Compressed Plane Waves for Variational PDEs

$\ell_1$ regularization for sparsity has played important role in recent developments in many fields including signal processing, statistics, optimization. The concept of sparsity is usually for the coefficients (i.e., only a small set of coefficients are nonzero) in a well-chosen set of modes (e.g. a basis or dictionary) for representation of the corresponding vectors or functions. In this talk, I will discuss our recent work on a new use of sparsity-promoting techniques to produce Â?compressed modes/compress plane waves" - modes that are sparse and localized in space - for efficient solutions of constrained variational problems in mathematics and physics. In particularly, I will focus on L1 regularized variational Schrodinger equations for creating spatially localized modes and orthonormal basis, which can efficiently represent localized functions and has promising potential to a variety of applications in many fields such as signal processing, solid state physics, materials science, etc. (This is a joint work with Vidvuds Ozolins, Russel Caflisch and Stanley Osher)

**Category**: Applied Math and Analysis**Duration**: 01:34:52**Date**: March 24, 2014 at 4:25 PM**Views**: 104-
**Tags:**seminar, Applied Math And Analysis Seminar

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