arXiv subjects

N. Sri Namachchivaya

Publications and source records attributed to N. Sri Namachchivaya.

8 recordsLinked to original sources

arXiv2023

Almost-Sure Stability of the Single Mode Solution of a Noisy Nonlinear Autoparametric System

For a pendulum suspended below a vibrating block with white noise forcing, the solution in which the pendulum remains vertical is called the single mode solution. When this solution becomes unstable there is energy transfer from the block to the pendulum, helping to absorb the vibrations of the block. We study the Lyapunov exponent governing the almost-sure stability of the process linearized along the single mode solution. The linearized equation is excited by a combination of white and colored noise processes, which makes the evaluation of the Lyapunov exponent non trivial. We obtain an explicit third order asymptotic expression for the Lyapunov exponent as the intensity of the white noise forcing tends to zero.

math.PR↗

arXiv2022

Quantitative Convergence of the Filter Solution for Multiple Timescale Nonlinear Systems with Coarse-Grain Correlated Noise

In this paper we prove a rate of convergence for the continuous time filtering solution of a multiple timescale correlated nonlinear system to a lower dimensional filtering equation in the limit of large timescale separation. Correlation is assumed to occur between the slow signal and observation processes. Convergence is almost sure in the weak topology. An asymptotic expansion of the dual process for the solution to the Zakai equation, and probabilistic representation using backward doubly stochastic differential equations is leveraged to prove the result.

math.PR↗

arXiv2021

Compressed Compressor

In this paper, we present a data-driven reduced order model of viscous Moore-Greitzer (MG) partial differential equation (PDE) by threading together ideas from principal component analysis (PCA) and autoencoder neural networks to sparse regression and compressed sensing. Numerical simulation of the infinite dimensional viscous MG system is reduced into low dimensional data using PCA and autoencoder neural networks based reduced order modelling (ROM) approaches. Based on the observation that MG equations close to bifurcations have a sparse representation (normal form) with respect to high-dimensional polynomial spaces, we use the Sparse Identification of Dynamical Systems (SINDy) algorithm which uses a collection of all monomials as sampling matrix and the LASSO algorithm to recover a system of sparse two ordinary differential equations (ODEs) with cubic nonlinearities. The discovered governing equations can be used to fully recover the original system dynamics up to 98.9% accuracy. When dimensional reduction is performed along the dataset's principal components, the resulting low dimensional differential equations will be consistent and have some resemblance to the normal form structure. Additionally, a new nonlinear behaviour is exhibited in viscous MG equations during rotating stall instability past the Hopf bifurcation point.

math.DS↗

arXiv2020

Approximation of the Filter Equation for Multiple Timescale, Correlated, Nonlinear Systems

This paper considers the approximation of the continuous time filtering equation for the case of a multiple timescale (slow-intermediate, and fast scales) that may have correlation between the slow-intermediate process and the observation process. The signal process is considered fully coupled, taking values in $\mathbb{R^{m}} \times \mathbb{R^{n}}$ and without periodicity assumptions on coefficients. It is proved that in the weak topology, the solution of the filtering equation converges in probability to a solution of a lower dimensional averaged filtering equation in the limit of large timescale separation. The method of proof uses the perturbed test function approach (method of corrector) to handle the intermediate timescale in showing tightness and characterization of limits. The correctors are solutions of Poisson equations.

math.PR↗

arXiv2016

Perturbations of linear delay differential equations at the verge of instability

The characteristic equation for a linear delay differential equation (DDE) has countably infinite roots on the complex plane. This paper considers linear DDEs that are on the verge of instability, i.e. a pair of roots of the characteristic equation lie on the imaginary axis of the complex plane, and all other roots have negative real parts. It is shown that, when small noise perturbations are present, the probability law of the dynamics can be approximated by the probability law of a one dimensional stochastic differential equation (SDE) without delay. This is advantageous because equations without delay are easier to simulate and one-dimensional SDE are analytically tractable. When the perturbations are also linear, it is shown that the stability depends on a specific complex number. The theory is applied to study oscillators with delayed feedback. Some errors in other articles that use multiscale approach are pointed out.

math.PR↗

arXiv2014

Nonlinear and additive white noise perturbations of linear delay differential equations at the verge of instability: an averaging approach

The characteristic equation for a linear delay differential equation (DDE) has countably infinite roots on the complex plane. We deal with linear DDEs that are on the verge of instability, i.e. a pair of roots of the characteristic equation (eigenvalues) lie on the imaginary axis of the complex plane, and all other roots have negative real parts. We show that, when the system is perturbed by small noise, under an appropriate change of time scale, the law of the amplitude of projection onto the critical eigenspace is close to the law of a certain one-dimensional stochastic differential equation (SDE) without delay. Further, we show that the projection onto the stable eigenspace is small. These results allow us to give an approximate description of the delay-system using an SDE (without delay) of just one dimension. The proof is based on the martingale problem technique.

math.PR↗

arXiv2013

Dimensional reduction in nonlinear filtering: A homogenization approach

We propose a homogenized filter for multiscale signals, which allows us to reduce the dimension of the system. We prove that the nonlinear filter converges to our homogenized filter with rate $\sqrt{\varepsilon}$. This is achieved by a suitable asymptotic expansion of the dual of the Zakai equation, and by probabilistically representing the correction terms with the help of BDSDEs.

math.PR↗

arXiv2012

Particle filtering in high-dimensional chaotic systems

We present an efficient particle filtering algorithm for multiscale systems, that is adapted for simple atmospheric dynamics models which are inherently chaotic. Particle filters represent the posterior conditional distribution of the state variables by a collection of particles, which evolves and adapts recursively as new information becomes available. The difference between the estimated state and the true state of the system constitutes the error in specifying or forecasting the state, which is amplified in chaotic systems that have a number of positive Lyapunov exponents. The purpose of the present paper is to show that the homogenization method developed in Imkeller et al. (2011), which is applicable to high dimensional multi-scale filtering problems, along with important sampling and control methods can be used as a basic and flexible tool for the construction of the proposal density inherent in particle filtering. Finally, we apply the general homogenized particle filtering algorithm developed here to the Lorenz'96 atmospheric model that mimics mid-latitude atmospheric dynamics with microscopic convective processes.

physics.data-an↗