Georgia Institute of Technology College of Sciences

A fundamental problem in compressed sensing (CS) is to reconstruct a sparsesignal under a few linear measurements far less than the physical dimensionof the signal. Currently, CS favors incoherent systems, in which any twomeasurements are as little correlated as possible. In reality, however, manyproblems are coherent, in which case conventional methods, such as L1minimization, do not work well. In this talk, I will present a novelnon-convex approach, which is to minimize the difference of L1 and L2 norms(L1-L2) in order to promote sparsity. Efficient minimization algorithms areconstructed and analyzed based on the difference of convex functionmethodology. The resulting DC algorithms (DCA) can be viewed as convergentand stable iterations on top of L1 minimization, hence improving L1 consistently. Through experiments, we discover that both L1 and L1-L2 obtain betterrecovery results from more coherent matrices, which appears unknown intheoretical analysis of exact sparse recovery. In addition, numericalstudies motivate us to consider a weighted difference model L1-aL2 (a>1) todeal with ill-conditioned matrices when L1-L2 fails to obtain a goodsolution. An extension of this model to image processing will be alsodiscussed, which turns out to be a weighted difference of anisotropic andisotropic total variation (TV), based on the well-known TV model and naturalimage statistics. Numerical experiments on image denoising, imagedeblurring, and magnetic resonance imaging (MRI) reconstruction demonstratethat our method improves on the classical TV model consistently, and is onpar with representative start-of-the-art methods.

In 2007, Honda, Kazez, and Matic defined an invariant of contact 3-manifolds with convex boundaries using sutured Heegaard Floer homology (SHF). Last year, Steven Sivek and I defined an analogous contact invariant using sutured Monopole Floer homology (SMF). In this talk, I will describe work with Sivek to prove that these two contact invariants are identified by an isomorphism relating the two sutured theories. This has several interesting consequences. First, it gives a proof of invariance for the contact invariant in SHF which does not rely on the relative Giroux correspondence between contact structures and open books (something whose proof has not yet been written down in full). Second, it gives a proof that the combinatorially computable invariants of Legendrian knots in Heegaard Floer homology can obstruct Lagrangian concordance.

I will present a new approach for computing boundary integrals that are defined on implicit interfaces, without the need of explicit parameterization. A key component of this approach is a volume integral which is identical to the integral over the interface. I will show results applying this approach to simulate interfaces that evolve according to Mullins-Sekerka dynamics used in certain phase transition problems. I will also discuss our latest results in generalization of this approach to summation of unstructured point clouds and regularization of hyper-singular integrals.