scholarly journals An Ensemble-Based Smoother with Retrospectively Updated Weights for Highly Nonlinear Systems

2007 ◽  
Vol 135 (1) ◽  
pp. 186-202 ◽  
Author(s):  
T. M. Chin ◽  
M. J. Turmon ◽  
J. B. Jewell ◽  
M. Ghil
Keyword(s):  
Monte Carlo ◽  
Nonlinear Systems ◽  
Ensemble Kalman Filter ◽  
Nonlinear Problems ◽  
Monte Carlo Algorithm ◽  
Operational Mode ◽  
Highly Nonlinear ◽  
Non Gaussian ◽  
Double Well Potential ◽  
Full Probability

Abstract Monte Carlo computational methods have been introduced into data assimilation for nonlinear systems in order to alleviate the computational burden of updating and propagating the full probability distribution. By propagating an ensemble of representative states, algorithms like the ensemble Kalman filter (EnKF) and the resampled particle filter (RPF) rely on the existing modeling infrastructure to approximate the distribution based on the evolution of this ensemble. This work presents an ensemble-based smoother that is applicable to the Monte Carlo filtering schemes like EnKF and RPF. At the minor cost of retrospectively updating a set of weights for ensemble members, this smoother has demonstrated superior capabilities in state tracking for two highly nonlinear problems: the double-well potential and trivariate Lorenz systems. The algorithm does not require retrospective adaptation of the ensemble members themselves, and it is thus suited to a streaming operational mode. The accuracy of the proposed backward-update scheme in estimating non-Gaussian distributions is evaluated by comparison to the more accurate estimates provided by a Markov chain Monte Carlo algorithm.

scholarly journals A parametric acceleration of multilevel Monte Carlo convergence for nonlinear variably saturated flow

2019 ◽  
Vol 24 (1) ◽  
pp. 311-331 ◽  
Author(s):  
Prashant Kumar ◽  
Carmen Rodrigo ◽  
Francisco J. Gaspar ◽  
Cornelis W. Oosterlee
Keyword(s):  
Monte Carlo ◽  
Variance Reduction ◽  
Nonlinear Problems ◽  
Picard Iteration ◽  
Richards Equation ◽  
Porous Media Flow ◽  
Soil Parameters ◽  
Multilevel Monte Carlo ◽  
Highly Nonlinear ◽  
Non Gaussian

AbstractWe present a multilevel Monte Carlo (MLMC) method for the uncertainty quantification of variably saturated porous media flow that is modeled using the Richards equation. We propose a stochastic extension for the empirical models that are typically employed to close the Richards equations. This is achieved by treating the soil parameters in these models as spatially correlated random fields with appropriately defined marginal distributions. As some of these parameters can only take values in a specific range, non-Gaussian models are utilized. The randomness in these parameters may result in path-wise highly nonlinear systems, so that a robust solver with respect to the random input is required. For this purpose, a solution method based on a combination of the modified Picard iteration and a cell-centered multigrid method for heterogeneous diffusion coefficients is utilized. Moreover, we propose a non-standard MLMC estimator to solve the resulting high-dimensional stochastic Richards equation. The improved efficiency of this multilevel estimator is achieved by parametric continuation that allows us to incorporate simpler nonlinear problems on coarser levels for variance reduction while the target strongly nonlinear problem is solved only on the finest level. Several numerical experiments are presented showing computational savings obtained by the new estimator compared with the original MC estimator.

Quasi-parallelized Multicanonical Monte Carlo Method for Highly Nonlinear Systems, with Application to All-optical Regeneration

2011 ◽  
Author(s):  
Taras I. Lakoba ◽  
Michael Vasilyev ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Nonlinear Systems ◽  
Highly Nonlinear ◽  
All Optical ◽  
Optical Regeneration

scholarly journals Ellipsoidal and Gaussian Kalman Filter Model for Discrete-Time Nonlinear Systems

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1168 ◽  
Author(s):  
Ligang Sun ◽  
Hamza Alkhatib ◽  
Boris Kargoll ◽  
Vladik Kreinovich ◽  
Ingo Neumann
Keyword(s):  
Kalman Filter ◽  
Nonlinear Systems ◽  
State Estimation ◽  
Discrete Time ◽  
Probability Distributions ◽  
Nonlinear Problems ◽  
The State ◽  
Filter Model ◽  
Highly Nonlinear ◽  
Correction Step

In this paper, we propose a new technique—called Ellipsoidal and Gaussian Kalman filter—for state estimation of discrete-time nonlinear systems in situations when for some parts of uncertainty, we know the probability distributions, while for other parts of uncertainty, we only know the bounds (but we do not know the corresponding probabilities). Similarly to the usual Kalman filter, our algorithm is iterative: on each iteration, we first predict the state at the next moment of time, and then we use measurement results to correct the corresponding estimates. On each correction step, we solve a convex optimization problem to find the optimal estimate for the system’s state (and the optimal ellipsoid for describing the systems’s uncertainty). Testing our algorithm on several highly nonlinear problems has shown that the new algorithm performs the extended Kalman filter technique better—the state estimation technique usually applied to such nonlinear problems.

scholarly journals Nonlinear Parameter Estimation: Comparison of an Ensemble Kalman Smoother with a Markov Chain Monte Carlo Algorithm

2012 ◽  
Vol 140 (6) ◽  
pp. 1957-1974 ◽  
Author(s):  
Derek J. Posselt ◽  
Craig H. Bishop
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Parameter Uncertainty ◽  
Posterior Probability ◽  
Monte Carlo Algorithm ◽  
Parameter Estimates ◽  
Posterior Probability Distribution ◽  
Kalman Smoother ◽  
Non Gaussian

Abstract This paper explores the temporal evolution of cloud microphysical parameter uncertainty using an idealized 1D model of deep convection. Model parameter uncertainty is quantified using a Markov chain Monte Carlo (MCMC) algorithm. A new form of the ensemble transform Kalman smoother (ETKS) appropriate for the case where the number of ensemble members exceeds the number of observations is then used to obtain estimates of model uncertainty associated with variability in model physics parameters. Robustness of the parameter estimates and ensemble parameter distributions derived from ETKS is assessed via comparison with MCMC. Nonlinearity in the relationship between parameters and model output gives rise to a non-Gaussian posterior probability distribution for the parameters that exhibits skewness early and multimodality late in the simulation. The transition from unimodal to multimodal posterior probability density function (PDF) reflects the transition from convective to stratiform rainfall. ETKS-based estimates of the posterior mean are shown to be robust, as long as the posterior PDF has a single mode. Once multimodality manifests in the solution, the MCMC posterior parameter means and variances differ markedly from those from the ETKS. However, it is also shown that if the ETKS is given a multimode prior ensemble, multimodality is preserved in the ETKS posterior analysis. These results suggest that the primary limitation of the ETKS is not the inability to deal with multimodal, non-Gaussian priors. Rather it is the inability of the ETKS to represent posterior perturbations as nonlinear functions of prior perturbations that causes the most profound difference between MCMC posterior PDFs and ETKS posterior PDFs.

Non-Gaussian Linearization Method for Stochastic Parametrically and Externally Excited Nonlinear Systems

1992 ◽  
Vol 114 (1) ◽  
pp. 20-26 ◽  
Author(s):  
R. J. Chang
Keyword(s):  
Monte Carlo ◽  
Nonlinear System ◽  
Nonlinear Systems ◽  
Stochastic System ◽  
Linearization Method ◽  
Weighted Sum ◽  
Gaussian Density ◽  
Parametric Noise ◽  
Non Gaussian ◽  
Parametric And External Excitations

A new practical non-Gaussian linearization method is developed for the problem of the dynamic response of a stable nonlinear system under both stochastic parametric and external excitations. The non-Gaussian linearization system is derived through a non-Gaussian density that is constructed as the weighted sum of undetermined Gaussian densities. The undetermined Gaussian parameters are then derived through solving a set of nonlinear algebraic moment relations. The method is illustrated by a Duffing-type stochastic system with/without parametric noise excited term. The accuracy in predicting the stationary and nonstationary variances by the present approach is compared with some exact solutions and Monte Carlo simulations.

Mathematical modeling of conversion of non-Gaussian random processes, signals and noise in linear and nonlinear systems

2018 ◽  
pp. 66-78
Author(s):  
V. M. Artyushenko ◽  
V. I. Volovach
Keyword(s):  
Mathematical Modeling ◽  
Nonlinear Systems ◽  
Random Processes ◽  
Mathematical Transformation ◽  
Positive And Negative Feedback ◽  
Non Linear Systems ◽  
Linear And Nonlinear ◽  
Non Gaussian ◽  
Linear And Nonlinear Systems ◽  
Inertial Systems

The questions connected with mathematical modeling of transformation of non-Gaussian random processes, signals and noise in linear and nonlinear systems are considered and analyzed. The mathematical transformation of random processes in linear inertial systems consisting of both series and parallel connected links, as well as positive and negative feedback is analyzed. The mathematical transformation of random processes with polygamous density of probability distribution during their passage through such systems is considered. Nonlinear inertial and non-linear systems are analyzed.

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