Moment Varieties of Gaussian Mixtures

The points of a moment variety are the vectors of all moments up to some order of a family of probability distributions. We study this variety for mixtures of Gaussians. Following up on Pearson's classical work from 1894, we apply current tools from computational algebra to recover the parameters from the moments. Our moment varieties extend objects familiar to algebraic geometers. For instance, the secant varieties of Veronese varieties are the loci obtained by setting all covariance matrices to zero. We compute the ideals of the 5-dimensional moment varieties representing mixtures of two univariate Gaussians, and we offer a comparison to the maximum likelihood approach.Comment: 17 pages, 2 figure

On the Short-Time Compositional Stability of Periodic Multilayers

The short-time stability of concentration profiles in coherent periodic multilayers consisting of two components with large miscibility gap is investigated by analysing stationary solutions of the Cahn-Hilliard diffusion equation. The limits of the existence and stability of periodic concentration profiles are discussed as a function of the average composition for given multilayer period length. The minimal average composition and the corresponding layer thickness below which artificially prepared layers dissolve at elevated temperatures are calculated as a function of the multilayer period length for a special model of the composition dependence of the Gibbs free energy. For period lengths exceeding a critical value, layered structures can exist as metastable states in a certain region of the average composition. The phase composition in very thin individual layers, comparable with the interphase boundary width, deviates from that of the corresponding bulk phase.Comment: 29 pages including 7 figures, to be published in Thin Solid Film

Use of cohesive elements in fatigue analysis

Cohesive laws describe the resistance to incipient separation of material surfaces. A cohesive finite element is formulated on the basis of a particular cohesive law. Cohesive elements are placed at the boundary between adjacent standard volume finite elements to model fatigue damage that leads to fracture at the separation of the element boundaries per the cohesive law. In this work, a cohesive model for fatigue crack initiation is taken to be the irreversible loadingunloading hysteresis that represents fatigue damage occuring due to cyclic loads leading to the initiation of small cracks. Various cohesive laws are reviewed and one is selected that incorporates a hysteretic cyclic loading that accounts for energetic dissipative mechanisms. A mathematical representation is developed based on an exponential effective load-separation cohesive relationship. A three-dimensional cohesive element is defined using this compliance relationship integrated at four points on the mid-surface of the area element. Implementation into finite element software is discussed and particular attention is applied to numerical convergence issues as the inflection point between loading and 'unloading in the cohesive law is encountered. A simple example of a displacementcontrolled fatigue test is presented in a finite element simulation. Comments are made on applications of the method to prediction of fatigue life for engineering structures such as pressure vessels and piping

Solitary waves due to x(2):x(2) cascading

Solitary waves in materials with a cascaded x(2):x(2) nonlinearity are investigated, and the implications of the robustness hypothesis for these solitary waves are discussed. Both temporal and spatial solitary waves are studied. First, the basic equations that describe the x(2):x(2) nonlinearity in the presence of dispersion or diffraction are derived in the plane-wave approximation, and we show that these equations reduce to the nonlinear Schrödinger equation in the limit of large phase mismatch and can be considered a Hamiltonian deformation of the nonlinear Schrödinger equation. We then proceed to a comprehensive description of all the solitary-wave solutions of the basic equations that can be expressed as a simple sum of a constant term, a term proportional to a power of the hyperbolic secant, and a term proportional to a power of the hyperbolic secant multiplied by the hyperbolic tangent. This formulation includes all the previously known solitary-wave solutions and some exotic new ones as well. Our solutions are derived in the presence of an arbitrary group-velocity difference between the two harmonics, but a transformation that relates our solutions to zero-velocity solutions is derived. We find that all the solitary-wave solutions are zero-parameter and one-parameter families, as opposed to nonlinear-Schrödinger-equation solitons, which are a two-parameter family of solutions. Finally, we discuss the prediction of the robustness hypothesis that there should be a two-parameter family of solutions with solitonlike behavior, and we discuss the experimental requirements for observation of solitonlike behavior.Peer ReviewedPostprint (published version

Time evolutions of scalar field perturbations in DD-dimensional Reissner-Nordstr\"om Anti-de Sitter black holes

Reissner-Nordstr\"om Anti-de Sitter (RNAdS) black holes are unstable against the charged scalar field perturbations due to the well-known superradiance phenomenon. We present the time domain analysis of charged scalar field perturbations in the RNAdS black hole background in general dimensions. We show that the instabilities of charged scalar field can be explicitly illustrated from the time profiles of evolving scalar field. By using the Prony method to fit the time evolution data, we confirm the mode that dominates the long time behavior of scalar field is in accordance with the quasinormal mode from the frequency domain analysis. The superradiance origin of the instability can also be demonstrated by comparing the real part of the dominant mode with the superradiant condition of charged scalar field. It is shown that all the unstable modes are superradiant, which is consistent with the analytical result in the frequency domain analysis. Furthermore, we also confirm there exists the rapid exponential growing modes in the RNAdS case, which makes the RNAdS black hole a good test ground to investigate the nonlinear evolution of superradiant instability.Comment: 15 pages, 7 figure

Applications of an exact counting formula in the Bousso-Polchinski Landscape

The Bousso-Polchinski (BP) Landscape is a proposal for solving the Cosmological Constant Problem. The solution requires counting the states in a very thin shell in flux space. We find an exact formula for this counting problem which has two simple asymptotic regime one of them being the method of counting low $\Lambda$ states given originally by Bousso and Polchinski. We finally give some applications of the extended formula: a robust property of the Landscape which can be identified with an effective occupation number, an estimator for the minimum cosmological constant and a possible influence on the KKLT stabilization mechanism.Comment: 43 pages, 11 figures, 2 appendices. We have added a new section (3.4) on the influence of the fraction of non-vanishing fluxes in the KKLT mechanism. Other minor changes also mad

A Multi-Layer Line Search Method to Improve the Initialization of Optimization Algorithms (Preprint submitted to Optimization Online)

We introduce a novel metaheuristic methodology to improve the initialization of a given deterministic or stochastic optimization algorithm. Our objective is to improve the performance of the considered algorithm, called core optimization algorithm, by reducing its number of cost function evaluations, by increasing its success rate and by boosting the precision of its results. In our approach, the core optimization is considered as a suboptimization problem for a multi-layer line search method. The approach is presented and implemented for various particular core optimization algorithms: Steepest Descent, Heavy-Ball, Genetic Algorithm, Differential Evolution and Controlled Random Search. We validate our methodology by considering a set of low and high dimensional benchmark problems (i.e., problems of dimension between 2 and 1000). The results are compared to those obtained with the core optimization algorithms alone and with two additional global optimization methods (Direct Tabu Search and Continuous Greedy Randomized Adaptive Search). These latter also aim at improving the initial condition for the core algorithms. The numerical results seem to indicate that our approach improves the performances of the core optimization algorithms and allows to generate algorithms more efficient than the other optimization methods studied here. A Matlab optimization package called ”Global Optimization Platform” (GOP), implementing the algorithms presented here, has been developed and can be downloaded at: http://www.mat.ucm.es/momat/software.ht

Low rank updates in preconditioning the saddle point systems arising from data assimilation problems

The numerical solution of saddle point systems has received a lot of attention over the past few years in a wide variety of applications such as constrained optimization, computational fluid dynamics and optimal control, to name a few. In this paper, we focus on the saddle point formulation of a large-scale variational data assimilation problem, where the computations involving the constraint blocks are supposed to be much more expensive than those related to the (1, 1) block of the saddle point matrix. New low-rank limited memory preconditioners exploiting the particular structure of the problem are proposed and analysed theoretically. Numerical experiments performed within the Object-Oriented Prediction System are presented to highlight the relevance of the proposed preconditioners