484 research outputs found
An integrated approach to global synchronization and state estimation for nonlinear singularly perturbed complex networks
This paper aims to establish a unified framework to handle both the exponential synchronization and state estimation problems for a class of nonlinear singularly perturbed complex networks (SPCNs). Each node in the SPCN comprises both 'slow' and 'fast' dynamics that reflects the singular perturbation behavior. General sector-like nonlinear function is employed to describe the nonlinearities existing in the network. All nodes in the SPCN have the same structures and properties. By utilizing a novel Lyapunov functional and the Kronecker product, it is shown that the addressed SPCN is synchronized if certain matrix inequalities are feasible. The state estimation problem is then studied for the same complex network, where the purpose is to design a state estimator to estimate the network states through available output measurements such that dynamics (both slow and fast) of the estimation error is guaranteed to be globally asymptotically stable. Again, a matrix inequality approach is developed for the state estimation problem. Two numerical examples are presented to verify the effectiveness and merits of the proposed synchronization scheme and state estimation formulation. It is worth mentioning that our main results are still valid even if the slow subsystems within the network are unstable
Spectral methods for multiscale stochastic differential equations
This paper presents a new method for the solution of multiscale stochastic
differential equations at the diffusive time scale. In contrast to
averaging-based methods, e.g., the heterogeneous multiscale method (HMM) or the
equation-free method, which rely on Monte Carlo simulations, in this paper we
introduce a new numerical methodology that is based on a spectral method. In
particular, we use an expansion in Hermite functions to approximate the
solution of an appropriate Poisson equation, which is used in order to
calculate the coefficients of the homogenized equation. Spectral convergence is
proved under suitable assumptions. Numerical experiments corroborate the theory
and illustrate the performance of the method. A comparison with the HMM and an
application to singularly perturbed stochastic PDEs are also presented
Coarse Stability and Bifurcation Analysis Using Stochastic Simulators: Kinetic Monte Carlo Examples
We implement a computer-assisted approach that, under appropriate conditions,
allows the bifurcation analysis of the coarse dynamic behavior of microscopic
simulators without requiring the explicit derivation of closed macroscopic
equations for this behavior. The approach is inspired by the so-called
time-step per based numerical bifurcation theory. We illustrate the approach
through the computation of both stable and unstable coarsely invariant states
for Kinetic Monte Carlo models of three simple surface reaction schemes. We
quantify the linearized stability of these coarsely invariant states, perform
pseudo-arclength continuation, detect coarse limit point and coarse Hopf
bifurcations and construct two-parameter bifurcation diagrams.Comment: 26 pages, 5 figure
Estimation and control of non-linear and hybrid systems with applications to air-to-air guidance
Issued as Progress report, and Final report, Project no. E-21-67
Spectral Methods for Multiscale Stochastic Differential Equations
This paper presents a new method for the solution of multiscale stochastic differential equations at the diffusive time scale. In contrast to averaging-based methods, e.g., the heterogeneous multiscale method (HMM) or the equation-free method, which rely on Monte Carlo simulations, in this paper we introduce a new numerical methodology that is based on a spectral method. In particular, we use an expansion in Hermite functions to approximate the solution of an appropriate Poisson equation, which is used in order to calculate the coefficients of the homogenized equation. Spectral convergence is proved under suitable assumptions. Numerical experiments corroborate the theory and illustrate the performance of the method. A comparison with the HMM and an application to singularly perturbed stochastic PDEs are also presented
Less Emphasis on Difficult Layer Regions: Curriculum Learning for Singularly Perturbed Convection-Diffusion-Reaction Problems
Although Physics-Informed Neural Networks (PINNs) have been successfully
applied in a wide variety of science and engineering fields, they can fail to
accurately predict the underlying solution in slightly challenging
convection-diffusion-reaction problems. In this paper, we investigate the
reason of this failure from a domain distribution perspective, and identify
that learning multi-scale fields simultaneously makes the network unable to
advance its training and easily get stuck in poor local minima. We show that
the widespread experience of sampling more collocation points in high-loss
layer regions hardly help optimize and may even worsen the results. These
findings motivate the development of a novel curriculum learning method that
encourages neural networks to prioritize learning on easier non-layer regions
while downplaying learning on harder layer regions. The proposed method helps
PINNs automatically adjust the learning emphasis and thereby facilitate the
optimization procedure. Numerical results on typical benchmark equations show
that the proposed curriculum learning approach mitigates the failure modes of
PINNs and can produce accurate results for very sharp boundary and interior
layers. Our work reveals that for equations whose solutions have large scale
differences, paying less attention to high-loss regions can be an effective
strategy for learning them accurately.Comment: 22 page
34th Midwest Symposium on Circuits and Systems-Final Program
Organized by the Naval Postgraduate School Monterey California. Cosponsored by the IEEE Circuits and Systems Society.
Symposium Organizing Committee: General Chairman-Sherif Michael, Technical Program-Roberto Cristi, Publications-Michael Soderstrand, Special Sessions- Charles W. Therrien, Publicity: Jeffrey Burl, Finance: Ralph Hippenstiel, and Local Arrangements: Barbara Cristi
Rational Construction of Stochastic Numerical Methods for Molecular Sampling
In this article, we focus on the sampling of the configurational
Gibbs-Boltzmann distribution, that is, the calculation of averages of functions
of the position coordinates of a molecular -body system modelled at constant
temperature. We show how a formal series expansion of the invariant measure of
a Langevin dynamics numerical method can be obtained in a straightforward way
using the Baker-Campbell-Hausdorff lemma. We then compare Langevin dynamics
integrators in terms of their invariant distributions and demonstrate a
superconvergence property (4th order accuracy where only 2nd order would be
expected) of one method in the high friction limit; this method, moreover, can
be reduced to a simple modification of the Euler-Maruyama method for Brownian
dynamics involving a non-Markovian (coloured noise) random process. In the
Brownian dynamics case, 2nd order accuracy of the invariant density is
achieved. All methods considered are efficient for molecular applications
(requiring one force evaluation per timestep) and of a simple form. In fully
resolved (long run) molecular dynamics simulations, for our favoured method, we
observe up to two orders of magnitude improvement in configurational sampling
accuracy for given stepsize with no evident reduction in the size of the
largest usable timestep compared to common alternative methods
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