Search

In this paper we introduce three new classes of nonnegative forms (or equivalently, symmetric tensors) and their extensions. The newly identified nonnegative symmetric tensors constitute distinctive convex cones in the space of general symmetric tensors (order six or above). For the special case of quartic forms, they collapse into the set of convex quartic homogeneous polynomial functions. We discuss the properties and applications of the new classes of nonnegative symmetric tensors in the context of polynomial and tensor optimization. Numerical experiments for solving certain polynomial optimization models based on the new class es of nonnegative symmetric tensors are presented. [ABSTRACT FROM AUTHOR]

A square matrix can be reduced to simpler form via similarity transformations. Here \simpler form" may refer to diagonal (when possible), triangular (Schur), or Hessenberg form. Sim- ilar reductions exist for matrix pencils if we consider general equivalence transformations instead of similarity transformations. For both matrices and matrix pencils, well-established algorithms are available for each reduction, which are useful in various applications. For matrix polynomials, uni- modular transformations can be used to achieve the reduced forms but we do not have a practical way to compute them. In this work we introduce a practical means to reduce a matrix polynomial with nonsingular leading coefficient to a simpler (diagonal, triangular, Hessenberg) form while preserving the degree and the eigenstructure. The key to our approach is to work with structure preserving similarity transformations applied to a linearization of the matrix polynomial instead of unimodular transformations applied directly to the matrix polynomial. As an application, we illustrate how to use these reduced forms to solve parameterized linear systems. [ABSTRACT FROM AUTHOR]

We present a recursive function approximation technique that does not require the storage of the arrival data stream. Our work is motivated by algorithms in stochastic optimization which require approximating functions in a recursive setting such as a stochastic approximation algorithm. The unique collection of these features in this technique is essential for nonlinear modeling of large data sets where the storage of the data becomes prohibitively expensive and in circumstances where our knowledge about a given query point increases as new information arrives. The algorithm presented here employs radial basis functions (RBFs) to provide locally adaptive parametric models (such as linear models). The local models are updated using recursive least squares and only store the statistical representative of the local approximations. The resulting scheme is very fast and memory efficient without compromising accuracy in comparison to methods well accepted as the standard and some advanced techniques used for functional data analysis in the literature. We motivate the algorithm using synthetic data and illustrate the algorithm on several real data sets. [ABSTRACT FROM AUTHOR]

We analyze a family of degenerate parabolic equations with linear growth Lagrangian having the form ut = div(φ(u)ψ(∇u/u)). Here |ψ| ≤ 1 and saturates at infinity. We present a simple and natural set of assumptions on the functions ψ, φ, under which (1) these equations fall into the framework provided by [F. Andreu, V. Caselles, and J. M. Mazón, Nonlinear Anal., 61 (2005), pp. 637-669, and J. Eur. Math. Soc. (JEMS), 7 (2005), pp. 361-393] and hence they are well posed, (2) we can ensure finite propagation speed for these models, (3) a Rankine-Hugoniot analysis on traveling fronts is also performed. On the particular case of φ(u) = u we get more detailed information on the spreading rate of compactly supported solutions and some interesting connections with optimal mass transportation theory. [ABSTRACT FROM AUTHOR]

We present an efficient numerical method for computing Fourier-Taylor expansions of (un)stable manifolds associated with hyperbolic periodic orbits. Three features of the method are that (1) we obtain accurate representation of the invariant manifold as well as the dynamics on the manifold, (2) it admits natural a posteriori error analysis, and (3) it does not require numerically integrating the vector field. Our approach is based on the parameterization method for invariant manifolds, and studies a certain partial differential equation which characterizes a chart map of the manifold. The method requires only that some mild nonresonance conditions hold. The novelty of the present work is that we exploit the Floquet normal form in order to efficiently compute the Fourier-Taylor expansion. A number of example computations are given including manifolds in phase space dimension as high as ten and manifolds which are two and three dimensional. We also discuss computations of cycle-to-cycle connecting orbits which exploit these manifolds. [ABSTRACT FROM AUTHOR]