The objectives of this project were to develop new hybrid analytical/computational methods that are capable of simulating the rare events that are the determining factors of the performance of lightwave systems and devices. These methods use the following: (1) analytical techniques, such as perturbation and asymptotic methods, to guide numerical simulations using importance sampling; and (2) adaptive numerical methods, such as the multicanonical Monte Carlo and cross-entropy methods, to perform the simulation of rare events when guiding analytical models are not available. The above methods can be used to evaluate the performance of specific optical systems and devices, including ultra-high-precision optical clocks based upon mode-locked fiber lasers, and optical clocks and other devices based upon hybrid opto-electronic oscillators. In each case, the goal is to use the methods to develop models that can accurately predict the performance of these devices, as well as determine the failure modes that are the limiting factors in their performance. The author has developed methods based upon soliton perturbation theory and importance sampling to simulate rare events in lightwave systems, including mode-locked laser systems. A key step to using the methods based upon soliton perturbation theory is to use an approximate version of the system dynamics to determine the locations in the large-dimensional state space that most contribute to the desired rare events (e.g., errors). In this method, calculus of variations applied to the approximate system allows the most significant rare events to be located, and then fully detailed importance-sampled Monte-Carlo simulations in the vicinity of these locations properly determines the probabilities of these rare events and corrects for any errors made by the approximations in determining the system dynamics.
Pt. I. Analytical methods. On the IST for discrete nonlinear Schrödinger systems and polarization shift for discrete vector solitons / M.J. Ablowitz, B. Prinari, A.D. Trubatch -- Soliton solutions of coupled nonlinear Klein-Gordon equations / T. Alagesan -- Characteristic initial value problems for integrable hyperbolic reductions of Einstein's equations / G.A. Alekseev -- Discrete sine-Gordon equation / M. Boiti [und weitere] -- Integrable and non-integrable equations with peakons / A. Degasperis, D.D. Holm, A.N.W. Hone -- Solution of a free boundary problem for a nonlinear diffusion-convection equation / S. De Lillo, M.C. Salvatori, G. Sanchini -- Iterative construction of solutions for a nonisospectral problem in 2 + 1 dimensions / P.G. Estevez -- Discrete breathers close to the anticontinuum limit: existence and wave scattering / S. Flach [und weitere] -- Complex Toda chain - an integrable universal model for adiabatic N-soliton interactions! / V.S. Gerdjikov -- On the reductions and scattering data for the generalized Zakharov-Shabat systems / G.G. Grahovski -- Bilinear representation for the modified nonlinear Schrödinger equations and their quantum potential deformations / J.H. Lee, O.K. Pashaev -- Noncommutative Burgers' equations / L. Martina, O.K. Pashaev -- On the quasi-classical [sumbol]-dressing method / B. Konopelchenko, A. Moro -- New solvable matrix integrals - U(n) case / A. Yu. Orlov -- Integrable hydrodynamic chains / M.V. Pavlov -- KPII: new results and open problems / A.K. Pogrebkov -- A workmate for KdV / P.C. Sabatier -- Space-time lattice for operator Schrödinger equation / A. Spire, V.V. Konotop, L. Vazquez -- On isomonodromy deformations for the ZS-AKNS flows / D. Wu -- pt. II. Symmetry properties, Hamiltonian methods and group theoretical methods. New symmetry reductions for a lubrication model / M.S. Bruzón [und weitere] -- Quantum solitons for quantum information and quantum computing / R.K. Bullough, M. Wadati -- Solving renormalization group equations by recursion relations / A. Cafarella, C. Corianò, M. Guzzi -- A tri-Hamiltonian route to spectral curves / L. Degiovanni, G. Magnano -- Construction of real forms of complexified Hamiltonian dynamical systems / V.S. Gerdjikov [und weitere] -- Integrable and super-integrable systems in classical and quantum mechanics / M. Giordano [und weitere] -- Non-commuting coordinates in vortex dynamics and in the Hall effect, related to "exotic" Galilean symmetry / P.A. Horváthy -- Structure of multi-meron knot action / L.S. Isaev, A.P. Protogenov -- Compatible nonlocal Poisson brackets of hydrodynamic type and integrable reductions of the Lamé equations / O.I. Mokhov -- Pseudoanti-Hermiticity in QQM, time-reversal and Kramers degeneracy / G. Scolarici -- On the integrability of supersymmetric equations / P. Tempesta, R.A. Leo, G. Soliani
The main focus of this dissertation is the application of importance sampling (IS) to calculate the probabilities associated with rare events in nonlinear, large-dimensional lightwave systems that are driven by noise, including models for fiber-based optical communication system and mode-locked lasers. Throughout the last decade, IS has emerged as a valuable tool for improving the efficiency of simulating rare events in such systems. In particular, it has shown great success in simulating various sources of transmission impairments found in optical communication systems, with examples ranging from large polarization fluctuations resulting from randomly varying fiber birefringence to large pulse-width fluctuations resulting from imperfections in the optical fiber. In many cases, the application of IS is guided by a low-dimensional reduction of the system dynamics. Combining the low-dimensional reduction with Monte Carlo simulations of the original system has been shown to be an extremely effective scheme for computing, for example, the probability with which a pulse deviates significantly from its initial form due to a random forcing. In the context of nonlinear optics, this might represent a transmission error where the propagation model is the nonlinear Schrödinger equation (NLSE) with additive or multiplicative noise. A shortcoming of this method is that the efficiency of the IS technique depends strongly on the accuracy of the low-dimensional reduction used to guide the simulations. These low-dimensional reductions are often derived from a formal perturbation theory, referred to as soliton perturbation theory (SPT) for the case of soliton propagation under the forced NLSE. As demonstrated here, such reduction methodsare often inadequate in their description of the pulse's dynamics. In particular, the interaction between a propagating pulse and dispersive radiation leads to a radiation-induced drift in a pulse's phase, which is largely unaccounted for in the reduced systems currently in use. The first part of this dissertation is devoted to understanding the interaction between a pulse and dispersive radiation, leading to the derivation of an improved reduced system based on a variational approach. Once this system is derived and verified numerically, it serves as the basis for an improved IS method that incorporates the dynamics of the radiation, which is subsequently extended to more realistic propagation models. Of particular interest is the case of the NLSE with a periodic modulation of the dispersion constant, referred to as dispersion management (DM), and a related model where this modulation is averaged to give an autonomous, nonlocal equation. Following the nomenclature commonly use in literature, the former (nonautonomous) equation will be referred to as the NLSE+DM and the latter (autonomous) equation as the DMNLSE. A complicating aspect of these more realistic models is that, unlike the NLSE, exact solutions only exist as numerical objects rather than as closed-form solutions, which introduces an addition source of error in the derivation of a reduced system for the pulse dynamics. In the second part of this dissertation, the IS method is extended to the calculation of phase-slip probabilities in mode-locked lasers (MLL). Realistic models for pulse propagation in MLL include the dissipative effects of gain and loss, in addition to nonlocal saturation effects. As a result most of the reduced systems derived for pulse dynamics are extremely complicated, which diminishes their applicability as guides for IS simulations. Therefore, a MLL operating in the soliton propagation regime is considered, where the effects of gain, loss and saturation are treated perturbatively. A simple reduced system for the pulse dynamics is derived for this MLL model, allowing the IS technique to be effectively applied.
The theory of solitons involves a broad variety of mathematical methods and appears in many areas of physics, technology, biology, and pure and applied mathematics. In this book, emphasis is placed on both theory (considering mathematical approaches for classical and quantum nonlinear systems — both continuous and discrete) and experiment (with special discussions on high bit rate optical communications and pulse dynamics in optical materials).
While the term Big Data is open to varying interpretation, it is quite clear that the Volume, Velocity, and Variety (3Vs) of data have impacted every aspect of computational science and its applications. The volume of data is increasing at a phenomenal rate and a majority of it is unstructured. With big data, the volume is so large that processing it using traditional database and software techniques is difficult, if not impossible. The drivers are the ubiquitous sensors, devices, social networks and the all-pervasive web. Scientists are increasingly looking to derive insights from the massive quantity of data to create new knowledge. In common usage, Big Data has come to refer simply to the use of predictive analytics or other certain advanced methods to extract value from data, without any required magnitude thereon. Challenges include analysis, capture, curation, search, sharing, storage, transfer, visualization, and information privacy. While there are challenges, there are huge opportunities emerging in the fields of Machine Learning, Data Mining, Statistics, Human-Computer Interfaces and Distributed Systems to address ways to analyze and reason with this data. The edited volume focuses on the challenges and opportunities posed by "Big Data" in a variety of domains and how statistical techniques and innovative algorithms can help glean insights and accelerate discovery. Big data has the potential to help companies improve operations and make faster, more intelligent decisions. - Review of big data research challenges from diverse areas of scientific endeavor - Rich perspective on a range of data science issues from leading researchers - Insight into the mathematical and statistical theory underlying the computational methods used to address big data analytics problems in a variety of domains
Rare event simulation involves using Monte Carlo methods to estimate probabilities of unlikely events and to understand the dynamics of a system conditioned on a rare event. An established class of algorithms based on large deviations theory and control theory constructs provably asymptotically efficient importance sampling estimators. Dynamic importance sampling is one these algorithms in which the choice of biasing distribution adapts in the course of a simulation according to the solution of an Isaacs partial differential equation or by solving a sequence of variational problems. However, obtaining the solution of either problem may be expensive, where the cost of solving these problems may be even more expensive than performing simple Monte Carlo exhaustively. Deterministic couplings induced by transport maps allows one to relate a complex probability distribution of interest to a simple reference distribution (e.g. a standard Gaussian) through a monotone, invertible function. This diverts the complexity of the distribution of interest into a transport map. We extend the notion of transport maps between probability distributions on Euclidean space to probability distributions on path space following a similar procedure to Itô’s coupling. The contraction principle is a key concept from large deviations theory that allows one to relate large deviations principles of different systems through deterministic couplings. We convey that with the ability to computationally construct transport maps, we can leverage the contraction principle to reformulate the sequence of variational problems required to implement dynamic importance sampling and make computation more amenable. We apply this approach to simple rotorcraft models. We conclude by outlining future directions of research such as using the coupling interpretation to accelerate rare event simulation via particle splitting, using transport maps to learn large deviations principles, and accelerating inference of rare events.
