Jack Xin

Front Propagation in Heterogeneous Media

A review is presented of recent results on front propagation in reaction-diffusion-advection equations in homogeneous and heterogeneous media. Formal asymptotic expansions and heuristic ideas are used to motivate the results wherever possible. The fronts include constant-speed monotone traveling fronts in homogeneous media, periodically varying traveling fronts in periodic media, and fluctuating and fractal fronts in random media. These fronts arise in a wide range of applications such as chemical kinetics, combustion, biology, transport in porous media, and industrial deposition processes. Open problems are briefly discussed along the way.

A Convex Model for Nonnegative Matrix Factorization and Dimensionality Reduction on Physical Space

A collaborative convex framework for factoring a data matrix X into a nonnegative product AS , with a sparse coefficient matrix S, is proposed. We restrict the columns of the dictionary matrix A to coincide with certain columns of the data matrix X, thereby guaranteeing a physically meaningful dictionary and dimensionality reduction. We use l1, ∞ regularization to select the dictionary from the data and show that this leads to an exact convex relaxation of l0 in the case of distinct noise-free data. We also show how to relax the restriction-to-X constraint by initializing an alternating minimization approach with the solution of the convex model, obtaining a dictionary close to but not necessarily in X. We focus on applications of the proposed framework to hyperspectral endmember and abundance identification and also show an application to blind source separation of nuclear magnetic resonance data.

Existence of planar flame fronts in convective-diffusive periodic media

We prove the existence of planar travelling wave solutions in a reaction-diffusion-convection equation with combustion nonlinearity and self-adjoint linear part in Rn, n≧1. The linear part involves diffusion-convection terms and periodic coefficients. These travelling waves have wrinkled flame fronts propagating with constant effective speeds in periodic inhomogeneous media. We use the method of continuation, spectral theory, and the maximum principle. Uniqueness and monotonicity properties of solutions follow from a previous paper. These properties are essential to overcoming the lack of compactness and the degeneracy in the problem.

Multidimensional stability of traveling waves in a bistable reaction-diffusion equation, I

We prove that the planer traveling wave solutions of a bistable reaction–diffusion equation are stable in, povided the initial perturbation is small and localized in some sense. In order to obtain control on the perturbation globally in time, we estimate lower bounds of a Lyapunov functional using the maximum principle, and spectral theory.

Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media

We review the existence results of traveling wave solutions to the reaction-diffusion equations with periodic diffusion (convection) coefficients and combustion (bistable) nonlinearities. We prove that whenever traveling waves exist, the solutions of the initial value problem with either frontlike or pulselike data propagate with the constant effective speeds of traveling waves in all suitable directions. In the case of bistable nonlinearity and one space dimension, we give an example of nonexistence of traveling waves which causes “quenching” (“localization”) of wavefront propagation. Quenching (localization) only occurs when the variations of the media from their constant mean values are large enough. Our related numerical results also provide evidence for this phenomenon in the parameter regimes not covered by the analytical example. Finally, we comment on the role of the effective wave speeds in determining the effective wavefront equation (Hamilton-Jacobi equation) of the reactiondiffusion equations under the small-diffusion, fast-reaction limit with a formal geometric optics expansion.

Minimization of

We study minimization of the difference of

\ell_{1-2}

\ell_1

for Compressed Sensing

and

An Introduction to Fronts in Random Media

\ell_2

Modeling light bullets with the two-dimensional sine—Gordon equation

norms as a nonconvex and Lipschitz continuous metric for solving constrained and unconstrained compressed sensing problems. We establish exact (stable) sparse recovery results under a restricted isometry property (RIP) condition for the constrained problem, and a full-rank theorem of the sensing matrix restricted to the support of the sparse solution. We present an iterative method for