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- H-selfadjoint matrices (1)
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- VARMA(1,1) model (1)
- chemical master equation (1)
- error of finite state projections (1)
- existence of solutions and moments (1)
- first order eigenvalue perturbation theory (1)
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Trajectory- or mesh-based methods for analyzing the dynamical behavior of large
molecules tend to be impractical due to the curse of dimensionality - their computational cost increases
exponentially with the size of the molecule. We propose a method to break the curse by a
novel square root approximation of transition rates, Monte Carlo quadrature and a discretization
approach based on solving linear programs. With randomly sampled points on the molecular energy
landscape and randomly generated discretizations of the molecular conguration space as our initial
data, we construct a matrix describing the transition rates between adjacent discretization regions.
This transition rate matrix yields a Markov State Model of the molecular dynamics. We use Perron
cluster analysis and coarse-graining techniques in order to identify metastable sets in conguration
space and approximate the transition rates between the metastable sets. Application of our method
to a simple energy landscape on a two-dimensional conguration space provides proof of concept and
an example for which we compare the performance of dierent discretizations. We show that the
computational cost of our method grows only polynomially with the size of the molecule. However,
nding discretizations of higher-dimensional conguration spaces in which metastable sets can be
identied remains a challenge.

Enforcing solvability of a nonlinear matrix equation and estimation of multivariate ARMA time series
(2013)

The matrix equation $X+AX^{-1}A^T=B$, arising in parameter estimation of certain time series models,
is solvable only for certain values of the matrices $A,B$.
We present a numerical method to modify $A,B$ in order to make the matrix equation solvable.
Since solvability depends on the location of the eigenvalues of the palindromic matrix polynomial $\lambda^2 A+\lambda B+A^T$,
our method works by moving those eigenvalues to specified locations using first order spectral perturbation theory.
The method is heuristic but works in practice, as is supported by several compelling numerical examples.
These examples arise from parameter estimation of a common time series model, the multivariate ARMA(1,1).

This paper deals with the effect of generic but structured low rank perturbations on the Jordan structure and sign
characteristic of matrices that have structure in an indefinite inner product space.
The paper is a follow-up of earlier papers in which the effect of rank one perturbations
was considered. Several results that are in contrast to the case of unstructured low rank
perturbations of general matrices are presented here.

The chemical master equation is a fundamental equation in chemical kinetics. It underlies the classical reaction-rate equations and takes stochastic effects into account. In this paper we give a simple argument showing that the solutions of a large class of chemical master equations are bounded in weighted $\ell_1$-spaces and possess high-order moments. This class includes all equations in which no reactions between two or more already present molecules and further external reactants occur that add mass to the system. As an illustration for the implications of this kind of regularity, we analyze the effect of truncating the state space. This leads to an error analysis for the finite state projections of the chemical master equation, an approximation that forms the basis of many numerical methods.

Physics of short optical pulses is an important and active research area in nonlinear optics. In what follows we theoretically consider the most extreme representatives of short pulses that contain only several oscillations of electromagnetic field. Description of such pulses is traditionally based on envelope equations and slowly varying envelope approximation, despite the fact that the envelope is not “slow” and, moreover, there is no clear definition of such a “fast” envelope. This happens due to another paradoxical feature: the standard (envelope) generalized nonlinear Schrödinger equation yields very good correspondence to numerical solutions of full Maxwell equations even for few-cycle pulses, a thing that should not be. In what follows we address ultrashort optical pulses using Hamiltonian framework for nonlinear waves. As it appears, the standard optical envelope equation is just a reformulation of general Hamiltonian equations. In a sense, no approximations are required, this is why the generalized nonlinear Schrödinger equation is so effective. Moreover, the Hamiltonian framework greatly contributes to our understanding of “fast” envelope, ultrashort solitons, stability and radiation of optical pulses. Even the inclusion of dissipative terms is possible making the Hamiltonian approach an universal theoretical tool also in extreme nonlinear optics.

We consider scattering of small-amplitude dispersive waves at an intense optical soliton which constitutes a nonlinear perturbation of the refractive index. Specifically, we consider a single-mode optical fiber and a group velocity matched pair: an optical soliton and a nearly perfectly reflected dispersive wave, a fiber-optical analogue of the event horizon. By combining (i) an adiabatic approach that is used in soliton perturbation theory and (ii) scattering theory from Quantum Mechanics, we give a quantitative account for the evolution of all soliton parameters. In particular, we quantify the increase in the soliton peak power that may result in spontaneous appearance of an extremely large, so-called champion soliton. The presented adiabatic theory agrees well with the numerical solutions of the pulse propagation equation. Moreover, for the first time we predict the full frequency band of the scattered dispersive waves and explain an emerging caustic structure in the space-time domain.

We consider the phenomenon of an optical soliton controlled (e.g. amplified) by a much weaker second pulse which is efficiently scattered at the soliton. An important problem in this context is to quantify the small range of parameters at which the interaction takes place. This has been achieved by using adiabatic ODEs for the soliton characteristics, which is much faster than an empirical scan of the full propagation equations for all parameters in question.

We report the cancellation of the soliton self-frequency shift in nonlinear optical fibers. A soliton which interacts with a group velocity matched low intensity dispersive pump pulse, experiences a continuous blue-shift in frequency, which counteracts the soliton self- frequency shift due to Raman scattering. The soliton self-frequency shift can be fully compensated by a suitably prepared dispersive wave. We quantify this kind of soliton-dispersive wave interaction by an adiabatic approach and demonstrate that the compensation is stable in agreement with numerical simulations.