Over the last few years, a number of dynamical density functional theories (DDFT)
have been developed for describing the dynamics of the one-body density of both
colloidal and atomic fluids. The DDFT is capable of describing the dynamics of the
fluid down to the scale of the individual fluid particles. DDFT is particularly
successful for colloidal fluids, for which one may assume that the particles have
stochastic equations of motion and from the resulting Fokker-Plank equation one is
able to derive the DDFT. I will give an overview of the DDFT and show applications
to various inhomogeneous fluid dynamical phenomena such as colloidal sedimentation
and evaporative dewetting of nanoparticle suspensions, which exhibit pattern
formation.

In this talk, I will discuss the existence of a unique global weak solution
to the general Ericksen-Leslie system in $R^2$, which is smooth away from possiblyfinite many singular times, for any initial data. This is a joint work with Jinrui Huang and Fanghua Lin.

We show that the set of limit-periodic Schroedinger operators with
purely absolutely continuous spectrum is dense in the space of
limit-periodic
Schroedinger operators in arbitrary dimensions. This result was previously
known only in dimension one.
The proof proceeds through the non-perturbative construction of
limit-periodic
extended states. The proof relies on a new estimate of the probability (in
quasi-momentum) that the Floquet Bloch operators have only simple
eigenvalues.

In the first part, we discuss penalty decomposition (PD) methods for solving
a more general class of $l_0$ minimization in which a sequence of penalty
subproblems are solved by a block coordinate descent (BCD) method. Under
some suitable assumptions, we establish that any accumulation point of the
sequence generated by the PD methods satisfies the first-order optimality conditions
of the problems. Furthermore, for the problems in which the $l_0$ part is the only
nonconvex part, we show that such an accumulation point is a local minimizer of the
problems. Finally, we test the performance of the PD methods by applying them to sparse
logistic regression, sparse inverse covariance selection, and compressed sensing
problems. The computational results demonstrate that our methods generally
outperform the existing methods in terms of solution quality and/or speed.  

In the second part, we consider $l_0$ regularized convex cone programming problems.
In particular, we first propose an iterative hard thresholding (IHT) method and
its variant for solving $l_0$ regularized box constrained convex programming. We
show that the sequence generated by these methods converges to a local minimizer.
Also, we establish the iteration complexity of the IHT method for finding an
$\epsilon$-local-optimal solution. We then propose a method for solving $l_0$
regularized convex cone programming by applying the IHT method to its quadratic
penalty relaxation and establish its iteration complexity for finding an
$\epsilon$-approximate local minimizer. Finally, we propose a variant of this
method in which the associated penalty parameter is dynamically updated, and
show that every accumulation point is a local minimizer of the problem.