My recent research focuses on the numerical analysis of compressive algorithms (CA) for sparse solution of equations and inverse
problems. In particular, I study algorithms for problems in
- signal/image processing,
- modern data analysis, and
- adaptive numerical solution of PDE's
that exploit parsimonious expansions via redundant discretizations, adaptive compression, and
randomness.
Compressive algorithms are a new approach to efficient computing and take advantage of the property of solutions of certain
PDE's and variational problems to be characterized by few major features which are recovered by adaptive nonlinear iterations.
CA are fast, tend to use minimal number of degrees of freedom, and are simple. Their numerical analysis is challenging. CA are
very successfully applied in several applied problems.
The relevant mathematics includes applied harmonic analysis, functional analysis, probability theory, discrete and convex
geometry, convex optimization, and calculus of variations. The main numerical techniques include iterative thresholding algorithms,
operator compression, random alternating projections, subspace correction, and domain decomposition methods.
Poster, October 2007 [ .pdf ]
Habilitationsschrift, January 2008 [ .pdf (c.a. 9 MB)] |