Georgia Institute of Technology College of Sciences

Nonlinear Resonance Analysis (NRA) is a natural next step after Fourieranalysis developed for linear PDEs. The main subject of NRA isevolutionary nonlinear PDEs, possessing resonant solutions. Importance ofNRA is due to its wide application area -- from climatepredictability to cancer diagnostic to breaking of the wing of an aircraft.In my talk I plan to give a brief overview of the methods and resultsavailable in NRA, and illustrate it with some examples from fluid mechanics.In particular, it will be shown how1) to use a general method of q-class decomposition for computing resonantmodes for a variety of physically relevant dispersion functions;2) to construct NR-reduced models for numerical simulations basing on theresonance clustering; theoretical comparision with Galerkin-like models willbe made and illustrated by the results of some numerical simulations withnonlinear PDE.3) to employ NR-reduced models for interpreting of real-life phenomena (inthe Earth`s atmosphere) and results of laboratory experiments with watertanks.A short presentation of the software available in this area will be given.

Solutions of differential equations with singular source terms easily becomenon-smooth or even discontinuous. High order approximations of suchsolutions yield the Gibbs phenomenon. This results in the deterioration ofhigh order accuracy. If the problem is nonlinear and time-dependent it mayalso destroy the stability. In this presentation, we focus on thedevelopment of high order methods to obtain high order accuracy rather thanregularization methods. Regularization yields a good stability condition,but may lose the desired accuracy. We explain how high order collocationmethods can be used to enhance accuracy, for which we will adopt severalmethods including the Green’s function approach and the polynomial chaosmethod. We also present numerical issues associated with the collocationmethods. Numerical results will be presented for some differential equationsincluding the nonlinear sine-Gordon equation and the Zerilli equation.

 Tight frame is a generalization of orthonormal basis. It  inherits most good properties of orthonormal basis but gains more  robustness to represent signals of intrests due to the redundancy. One can  construct tight frame systems under which signals of interests have sparse  representations. Such tight frames include translation invariant wavelet,  framelet, curvelet, and etc. The sparsity of a signal under tight frame systems has three different formulations, namely, the analysis-based sparsity, the synthesis-based one, and the balanced one between them. In this talk, we discuss Bregman algorithms for finding signals that are sparse under tight frame systems with the above three different formulations. Applications of our algorithms include image inpainting, deblurring, blind deconvolution, and cartoon-texture decomposition. Finally, we apply the linearized Bregman, one of the Bregman algorithms, to solve the problem of matrix completion, where we want to find a low-rank matrix from its incomplete entries. We view the low-rank matrix as a sparse vector under an adaptive linear transformation which depends on its singular vectors. It leads to a singular value thresholding (SVT) algorithm.

Schwarz algorithms have experienced a second youth over the lastdecades, when distributed computers became more and more powerful andavailable. In the classical Schwarz algorithm the computational domain is divided into subdomains and Dirichlet continuity is enforced on the interfaces between subdomains. Fundamental convergence results for theclassical Schwarzmethods have been derived for many partial differential equations. Withinthis frameworkthe overlap between subdomains is essential for convergence. More recently, Optimized Schwarz Methods have been developed: based on moreeffective transmission conditions than the classical Dirichlet conditions at theinterfaces between subdomains, such algorithms can be used both with and without overlap. On the other hand, such algorithms show greatly enhanced performance compared to the classical Schwarz method. I will present a survey of Optimized Schwarz Methods for the numerical approximation of partial differential equation, focusing mainly on heterogeneous convection-diffusion and electromagnetic problems.