A path-based method for simulating large deviations and rare events in nonlinear lightwave systems (Q2822874)

This interesting paper is devoted to the problem of estimation of errors in nonlinear lightwave systems. It is known that the errors are assosiated with rare noise-induced large deviations of the signal. Such large deviations can be extremely rare in some physical systems. There are several methods designed for improving the simulation efficiency of these rare events. One of these methods of simulation of above mentioned phenomenon is the so called Monte Carlo (MC) method. Recently, a method applicable to arbitrary shaped pulses was published by \textit{G. M. Donovan} and \textit{W. L. Kath} [SIAM J. Appl. Math. 71, No. 3, 903--924 (2011; Zbl 1229.78029)]. In this case the most probable noise configurations can be identified by a combination of both the singular value decompositions (SVD) method and the so called cross-entropy (CE) method.NEWLINENEWLINEIn the present paper the authors pose a constrained optimization problem to identify the most probable noise configurations leading to errors in systems governed by the nonlinear Schrödinger equation (NLSE) NEWLINE\[NEWLINE \partial u/\partial z = (i/2)d(z)\partial^2u/\partial t^2+i|u|^2u+ \sum\limits_{n=1}^{N_a}s_n(t)\delta (z-z_n), NEWLINE\]NEWLINE where \(u(t,z)\) is the optical field envelope, \(z\) and \(t\) are dimensionless distance and retarded time, respectively. Note that the model includes varying dispersion, optical noise, and a receiver subsystem. The authors describe their method as for this purpose introduce an \(N\)-dimensional random variable \(\mathbf{X}=(X_1,\ldots ,X_N)\), and the event of interest defined by the map \(g(\mathbf{X})\leq \rho \), where \(\rho =\)const. There are problems, where \(\mathbf{X}\) play a role of independent and identically distributed (i.i.d.) zero-mean Gaussian random variables with the probability density function (PDF) \(p(\mathbf{x})\), \(p(\mathbf{x})\propto e^{-\|\mathbf{x}\|^2}\). After introducing the biasing distribution \(p^{\ast }(\mathbf{x})\), obtain \(p^{\ast }(\mathbf{x})=p(\mathbf{x}-\mathbf{x}^{\ast })\), where \(\mathbf{x}^{\ast }\) is the most probable location satisfying the constraint. In other words the later solves the basic problem: (\(\ast \)) \(\text{min}\|\mathbf{x}\|^2\), subject to \(g(\mathbf{x})\leq \rho \). Maximizing \(e^{-\|\mathbf{x}\|^2}\) clearly is equivalent to minimizing \(\|\mathbf{x}\|^2\). Here the minimum occurs on the constraint boundary, thus the above stated inequality in (\(\ast \)) should be replaced with the equality \(g(\mathbf{x}) = \rho \). An algorithm of interest for the approximate solving of (\(\ast \)) is invented: 1) Choose a sequence \(g(0)=\rho_0>\rho_1>\cdots >\rho_{j}=\rho \) of constant values such that \(\epsilon_j=\rho_{j-1}-\rho_j\ll 1\) for all \(j=0,1,\ldots ,J\). 2) Find the solution to \(\text{min}\|\mathbf{x}\|^2\) subject to \(g(\mathbf{x}) = \rho_0 \). 3) Compute an approximate solution to the problem (\(\ast \)) using the result of the previous step, \(\mathbf{x}_{j-1}\) as the starting point. 4) Stop if \(j=J\), otherwise, let \(j=j+1\) and go back to step 3). Some applications to lightwave systems are given. Simulation results are discussed as well.