MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

Many Body Quantum Chaos

This editorial remembers Shmuel Fishman, one of the founding fathers of the research field "quantum chaos", and puts into context his contributions to the scientific community with respect to the twelve papers that form the special issue

Large deviations for rare realizations of dynamical systems

A central problem in uncertainty quantification is how to characterize the impact that our incomplete knowledge about models has on the predictions we make from them. This question naturally lends itself to a probabilistic formulation, by making the unknown model parameters random with given statistics. In the following this approach is used in concert with tools from large deviation theory (LDT) and optimal control to estimate the probability that some observables in a dynamical system go above a large threshold after some time, given the prior statistical information about the system’s parameters and/or its initial conditions. It is established under which conditions the extreme events occur in a predictable way, as the minimizer of the LDT action functional, i.e. the instanton. In the first physical application, the appearance of rogue waves in a long-crested deep sea is investigated. First, the leading order equations are derived for the wave statistics in the framework of wave turbulence (WT), showing that the theory cannot go beyond Gaussianity, although it remains the main tool to understand the energetic transfers. It is shown how by applying our LDT method one can use the incomplete information contained in the spectrum (with the Gaussian statistics of WT) as prior and supplement this information with the governing nonlinear dynamics to reliably estimate the probability distribution of the sea surface elevation far in the tail at later times. Our results indicate that rogue waves occur when the system hits unlikely pockets of wave configurations that trigger large disturbances of the surface height. The rogue wave precursors in these pockets are wave patterns of regular height but with a very specific shape that is identified explicitly, thereby potentially allowing for early detection. Finally, the first experimental evidence of hydrodynamic instantons is presented using data collected in a long wave flume, elevating the instanton description to the role of a unifying theory of extreme water waves. Other applications of the method are illustrated: To the nonlinear Schrödinger equation with random initial conditions, relevant to fiber optics and integrable turbulence, and to a rod with random elasticity pulled by a time-dependent force. The latter represents an interesting nonequilibrium statistical mechanics setup with a strongly out-of-equilibrium transient (absence of local thermodynamic equilibrium) and a small number of degrees of freedom (small system), showing how the LDT method can be exploited to solve optimal-protocol problems

SIMULATING SEISMIC WAVE PROPAGATION IN TWO-DIMENSIONAL MEDIA USING DISCONTINUOUS SPECTRAL ELEMENT METHODS

We introduce a discontinuous spectral element method for simulating seismic wave in 2- dimensional elastic media. The methods combine the flexibility of a discontinuous finite element method with the accuracy of a spectral method. The elastodynamic equations are discretized using high-degree of Lagrange interpolants and integration over an element is accomplished based upon the Gauss-Lobatto-Legendre integration rule. This combination of discretization and integration results in a diagonal mass matrix and the use of discontinuous finite element method makes the calculation can be done locally in each element. Thus, the algorithm is simplified drastically. We validated the results of one-dimensional problem by comparing them with finite-difference time-domain method and exact solution. The comparisons show excellent agreement