Increasing the laser power is essential to improve the sensitivity of interferometric gravitational wave detectors. However, optomechanical parametric instabilities can set a limit to that power. It is of major importance to understand and characteri
ze the many parameters and effects that influence these instabilities. Here, we model with a high degree of precision the optical and mechanical modes that are involved in these parametric instabilities, such that our model can become predictive. As an example, we perform simulations for the Advanced Virgo interferometer (O3 configuration). In particular we compute mechanical modes losses by combining both on-site measurements and finite element analysis with unprecedented level of detail and accuracy. We also study the influence on optical modes and parametric gains of mirror finite size effects, and mirror deformations due to thermal absorption. We show that these effects play an important role if transverse optical modes of order higher than four are involved in the instability process.
Solutions to the stochastic wave equation on the unit sphere are approximated by spectral methods. Strong, weak, and almost sure convergence rates for the proposed numerical schemes are provided and shown to depend only on the smoothness of the drivi
ng noise and the initial conditions. Numerical experiments confirm the theoretical rates. The developed numerical method is extended to stochastic wave equations on higher-dimensional spheres and to the free stochastic Schrodinger equation on the unit sphere.
We analyse a splitting integrator for the time discretization of the Schrodinger equation with nonlocal interaction cubic nonlinearity and white noise dispersion. We prove that this time integrator has order of convergence one in the $p$-th mean sens
e, for any $pgeq1$ in some Sobolev spaces. We prove that the splitting schemes preserves the $L^2$-norm, which is a crucial property for the proof of the strong convergence result. Finally, numerical experiments illustrate the performance of the proposed numerical scheme.
This article analyses the convergence of the Lie-Trotter splitting scheme for the stochastic Manakov equation, a system arising in the study of pulse propagation in randomly birefringent optical fibers. First, we prove that the strong order of the nu
merical approximation is 1/2 if the nonlinear term in the system is globally Lipschitz. Then, we show that the splitting scheme has convergence order 1/2 in probability and almost sure order 1/2- in the case of a cubic nonlinearity. We provide several numerical experiments illustrating the aforementioned results and the efficiency of the Lie-Trotter splitting scheme. Finally, we numerically investigate the possible blowup of solutions for some power-law nonlinearities.
We analyze the qualitative properties and the order of convergence of a splitting scheme for a class of nonlinear stochastic Schrodinger equations driven by additive It^o noise. The class of nonlinearities of interest includes nonlocal interaction cu
bic nonlinearities. We show that the numerical solution is symplectic and preserves the expected mass for all times. On top of that, for the convergence analysis, some exponential moment bounds for the exact and numerical solutions are proved. This enables us to provide strong orders of convergence as well as orders of convergence in probability and almost surely. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme.
We perform a numerical analysis of a class of randomly perturbed {H}amiltonian systems and {P}oisson systems. For the considered additive noise perturbation of such systems, we show the long time behavior of the energy and quadratic Casimirs for the
exact solution. We then propose and analyze a drift-preserving splitting scheme for such problems with the following properties: exact drift preservation of energy and quadratic Casimirs, mean-square order of convergence one, weak order of convergence two. These properties are illustrated with numerical experiments.
This article presents and analyses an exponential integrator for the stochastic Manakov equation, a system arising in the study of pulse propagation in randomly birefringent optical fibers. We first prove that the strong order of the numerical approx
imation is $1/2$ if the nonlinear term in the system is globally Lipschitz-continuous. Then, we use this fact to prove that the exponential integrator has convergence order $1/2$ in probability and almost sure order $1/2$, in the case of the cubic nonlinear coupling which is relevant in optical fibers. Finally, we present several numerical experiments in order to support our theoretical findings and to illustrate the efficiency of the exponential integrator as well as a modified version of it.
We show that applying any deterministic B-series method of order $p_d$ with a random step size to single integrand SDEs gives a numerical method converging in the mean-square and weak sense with order $lfloor p_d/2rfloor$.As an application, we derive
high order energy-preserving methods for stochastic Poisson systems as well as further geometric numerical schemes for this wide class of Stratonovich SDEs.
The paper deals with numerical discretizations of separable nonlinear Hamiltonian systems with additive noise. For such problems, the expected value of the total energy, along the exact solution, drifts linearly with time. We present and analyze a ti
me integrator having the same property for all times. Furthermore, strong and weak convergence of the numerical scheme along with efficient multilevel Monte Carlo estimators are studied. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme.
In this paper, we study the spectrum and the eigenvectors of radial kernels for mixtures of distributions in $mathbb{R}^n$. Our approach focuses on high dimensions and relies solely on the concentration properties of the components in the mixture. We
give several results describing of the structure of kernel matrices for a sample drawn from such a mixture. Based on these results, we analyze the ability of kernel PCA to cluster high dimensional mixtures. In particular, we exhibit a specific kernel leading to a simple spectral algorithm for clustering mixtures with possibly common means but different covariance matrices. We show that the minimum angular separation between the covariance matrices that is required for the algorithm to succeed tends to $0$ as $n$ goes to infinity.
