Application of Two Bayesian Filters to Estimate Unknown Heat Fluxes in a Natural Convection Problem

2012 ◽  
Vol 134 (9) ◽  
Author(s):  
Marcelo J. Colaço ◽  
Helcio R. B. Orlande ◽  
Wellington B. da Silva ◽  
George S. Dulikravich
Keyword(s):  
Natural Convection ◽  
Sequential Monte Carlo ◽  
Nonlinear Models ◽  
Probability Distributions ◽  
Heat Fluxes ◽  
Large Set ◽  
Sampling Importance Resampling ◽  
Filter Methods ◽  
Importance Resampling ◽  
Non Gaussian

Sequential Monte Carlo (SMC) or particle filter methods, which have been originally introduced in the beginning of the 1950s, became very popular in the last few years in the statistical and engineering communities. Such methods have been widely used to deal with sequential Bayesian inference problems in the fields like economics, signal processing, and robotics, among others. SMC methods are an approximation of sequences of probability distributions of interest, using a large set of random samples, named particles. These particles are propagated along time with a simple Sampling Importance distribution. Two advantages of this method are: they do not require the restrictive hypotheses of the Kalman filter, and they can be applied to nonlinear models with non-Gaussian errors. This paper uses two SMC filters, namely the SIR (sampling importance resampling filter) and the ASIR (auxiliary sampling importance resampling filter) to estimate a heat flux on the wall of a square cavity encasing a liquid undergoing natural convection. Measurements, which contain errors, taken at the boundaries of the cavity were used in the estimation process. The mathematical model as well as the initial condition are supposed to have some errors, which were taken into account in the probabilistic evolution model used for the filter. Also, the results using different grid sizes and patterns for the direct and inverse problems were used to avoid the so-called inverse crime. In these results, additional errors were considered due to the different location of the grid points used. The final results were remarkably good when using the ASIR filter.

Application of a Bayesian Filter to Estimate Unknown Heat Fluxes in a Natural Convection Problem

2011 ◽  
Author(s):  
Marcelo J. Colac¸o ◽  
Helcio R. B. Orlande ◽  
Wellington B. da Silva ◽  
George S. Dulikravich
Keyword(s):  
Natural Convection ◽  
Sequential Monte Carlo ◽  
Nonlinear Models ◽  
Probability Distributions ◽  
Heat Fluxes ◽  
Large Set ◽  
Filter Methods ◽  
Importance Resampling ◽  
Non Gaussian ◽  
Natural Convection Problem

Sequential Monte Carlo (SMC) or Particle Filter Methods, which have been originally introduced in the beginning of the 50’s, became very popular in the last few years in the statistical and engineering communities. Such methods have been widely used to deal with sequential Bayesian inference problems in fields like economics, signal processing, and robotics, among others. SMC Methods are an approximation of sequences of probability distributions of interest, using a large set of random samples, named particles. These particles are propagated along time with a simple Sampling Importance distribution. Two advantages of this method are: they do not require the restrictive hypotheses of the Kalman filter, and can be applied to nonlinear models with non-Gaussian errors. This papers uses a SMC filter, namely the ASIR (Auxiliary Sampling Importance Resampling Filter) to estimate a heat flux in the wall of a square cavity undergoing a natural convection. Measurements, which contain errors, taken at the boundaries of the cavity are used in the estimation process. The mathematical model, as well as the initial condition, are supposed to have some error, which are taken into account in the probabilistic evolution model used for the filter.

scholarly journals Sequential Monte Carlo Methods to Train Neural Network Models

2000 ◽  
Vol 12 (4) ◽  
pp. 955-993 ◽  
Author(s):  
J. F. G. de Freitas ◽  
M. Niranjan ◽  
A. H. Gee ◽  
A. Doucet
Keyword(s):  
Monte Carlo ◽  
Sequential Monte Carlo ◽  
Probability Distributions ◽  
Network Models ◽  
Computational Time ◽  
Optimization Strategy ◽  
Neural Network Models ◽  
Monte Carlo Techniques ◽  
Monte Carlo Algorithms ◽  
Importance Resampling

We discuss a novel strategy for training neural networks using sequential Monte Carlo algorithms and propose a new hybrid gradient descent/sampling importance resampling algorithm (HySIR). In terms of computational time and accuracy, the hybrid SIR is a clear improvement over conventional sequential Monte Carlo techniques. The new algorithm may be viewed as a global optimization strategy that allows us to learn the probability distributions of the network weights and outputs in a sequential framework. It is well suited to applications involving on-line, nonlinear, and nongaussian signal processing. We show how the new algorithm outperforms extended Kalman filter training on several problems. In particular, we address the problem of pricing option contracts, traded in financial markets. In this context, we are able to estimate the one-step-ahead probability density functions of the options prices.

On the Probability Distribution of Mooring Line Tensions in a Directional Environment

2011 ◽  
Author(s):  
Jan Mathisen ◽  
Siril Okkenhaug ◽  
Kjell Larsen
Keyword(s):  
Time Series ◽  
Probability Distributions ◽  
Extreme Value Distribution ◽  
Line Tension ◽  
Extreme Value ◽  
Mooring Line ◽  
Large Set ◽  
Short Term ◽  
Reliability Method ◽  
Line Design

A joint probabilistic model of the metocean environment is assembled, taking account of wind, wave and current and their respective heading angles. Mooring line tensions are computed in the time domain, for a large set of short-term stationary conditions, intended to span the domain of metocean conditions that contribute significantly to the probabilities of high tensions. Weibull probability distributions are fitted to local tension maxima extracted from each time series. Long time series of 30 hours duration are used to reduce statistical uncertainty. Short-term, Gumbel extreme value distributions of line tension are derived from the maxima distributions. A response surface is fitted to the distribution parameters for line tension, to allow interpolation between the metocean conditions that have been explicitly analysed. A second order reliability method is applied to integrate the short-term tension distributions over the probability of the metocean conditions and obtain the annual extreme value distribution of line tension. Results are given for the most heavily loaded mooring line in two mooring systems: a mobile drilling unit and a production platform. The effects of different assumptions concerning the distribution of wave heading angles in simplified analysis for mooring line design are quantified by comparison with the detailed calculations.

Non-Gaussian probability distributions of solar wind fluctuations

1994 ◽  
Vol 12 (12) ◽  
pp. 1127-1138 ◽  
Author(s):  
E. Marsch ◽  
C. Y. Tu
Keyword(s):  
Solar Wind ◽  
Probability Distributions ◽  
Time Lag ◽  
Small Scale ◽  
Time Lags ◽  
Mhd Turbulence ◽  
Theoretical Concepts ◽  
Fluid Turbulence ◽  
Long Time ◽  
Non Gaussian

Abstract. The probability distributions of field differences ∆x(τ)=x(t+τ)-x(t), where the variable x(t) may denote any solar wind scalar field or vector field component at time t, have been calculated from time series of Helios data obtained in 1976 at heliocentric distances near 0.3 AU. It is found that for comparatively long time lag τ, ranging from a few hours to 1 day, the differences are normally distributed according to a Gaussian. For shorter time lags, of less than ten minutes, significant changes in shape are observed. The distributions are often spikier and narrower than the equivalent Gaussian distribution with the same standard deviation, and they are enhanced for large, reduced for intermediate and enhanced for very small values of ∆x. This result is in accordance with fluid observations and numerical simulations. Hence statistical properties are dominated at small scale τ by large fluctuation amplitudes that are sparsely distributed, which is direct evidence for spatial intermittency of the fluctuations. This is in agreement with results from earlier analyses of the structure functions of ∆x. The non-Gaussian features are differently developed for the various types of fluctuations. The relevance of these observations to the interpretation and understanding of the nature of solar wind magnetohydrodynamic (MHD) turbulence is pointed out, and contact is made with existing theoretical concepts of intermittency in fluid turbulence.

scholarly journals On the physical constraints for the exceeding probability of deep water rogue waves

2021 ◽  
Author(s):  
Saulo Mendes ◽  
Alberto Scotti ◽  
Paul Stansell
Keyword(s):  
Sample Size ◽  
Rogue Wave ◽  
Probability Distributions ◽  
Rogue Waves ◽  
Uneven Distribution ◽  
Large Set ◽  
Sea State ◽  
Sampling Variability ◽  
The North ◽  
Wave Heights

<p><strong>(manuscript accepted into Applied Ocean Research https://www.researchgate.net/publication/344786014)</strong></p><p><strong>Abstract</strong></p><p>Nearly four decades have elapsed since the first efforts to obtain a realistic narrow-banded model for extreme wave crests and heights were made, resulting in a couple of dozen different exceeding probability distributions. These models reflect results of numerical simulations and storm records measured from oil platforms, buoys, and more recently, satellite data. Nevertheless, no consensus has been achieved in either deterministic or operational approaches. Typically, distributions found in the literature analyze a very large set of waves with large variations in sea-state parameters while neglecting homogeneous smaller samples, such that we lack a suitable definition for the sample size and homogeneity of sea variables, also known as sampling variability (Bitner-Gregersen et al., 2020). Naturally, a possible consequence of such sample size inconsistency is the apparent disagreement between several studies regarding the prediction of rogue wave occurrence, as some studies can report less rogue wave heights while others report more rogue waves or the same statistics predicted by Longuet-Higgins (1952), sometimes a combination of the three in the very same study (Stansell, 2004; Cherneva et al., 2005). In this direction, we have obtained a dimensionless parameter capable of measuring how large the deviations in sea state variables can be so that accuracy in wave statistics is preserved.  In particular, we have defined which samples are too heterogeneous to create an accurate description of the uneven distribution of rogue wave likelihood among different storms (Stansell, 2004). Though the literature is rich in physical bounds for single waves, here we describe empirical physical limits for the ensemble of waves (such as the significant steepness) devised to bound these variables within established and prospective wave distributions. Furthermore, this work supplies a combination of sea state parameters that provide guidance on the influence of sea states influence on rogue wave occurrence. Based on these empirical limits, we conjecture a mathematical model for the dependence of the expected maximum of normalized wave heights and crests on the sea state parameters, thus explaining the uneven distribution of rogue wave likelihood among different storms collected by infrared laser altimeters of the North Alwyn oil platform discussed in Stansell (2004). Finally, we demonstrate that for heights and crests beyond 90% of their thresholds (H>2H<sub>1/3</sub> for heights), the exceeding probability becomes stratified, i.e. they resemble layers of probability curves according to each sea state, suggesting the existence of a dynamical definition for rogue waves rather than purely statistical.</p><p> </p><p><strong>References</strong></p><p>Bitner-Gregersen, E. M., Gramstad, O., Magnusson, A., Malila, M., 2020. Challenges in description of nonlinear waves due to sampling variability. J. Mar. Sci. Eng. 8, 279.</p><p>Longuet-Higgins, M., 1952. On the statistical distribution of the heights of sea waves. Journal of Marine Research 11, 245–265.</p><p>Stansell, P., 2004. Distribution of freak wave heights measured in the north sea. Appl. Ocean Res. 26, 35–48.</p><p>Cherneva, Z., Petrova, P., Andreeva, N., Guedes Soares, C., 2005. Probability distributions of peaks, troughs and heights of wind waves measured in the black sea coastal zone. Coastal Engineering 52, 599–615.</p>

Natural Convection in a Horizontal Cavity Partially Occupied by a Porous Media Saturated by a Coolant Liquid

2006 ◽  
Author(s):  
Fakhreddine S. Oueslati ◽  
Rachid Bennacer ◽  
Habib Sammouda ◽  
Ali Belghith
Keyword(s):  
Heat Transfer ◽  
Porous Media ◽  
Natural Convection ◽  
Flow Structure ◽  
Boussinesq Approximation ◽  
Density Variation ◽  
Heat Fluxes ◽  
Complex Flow ◽  
Coupled Equations ◽  
Complex Flow Structure

The natural convection is studied in a cavity witch the lower half is filled with a porous media that is saturated with a first fluid (liquid), and the upper is filled with a second fluid (gas). The horizontal borders are heated and cooled by uniform heat fluxes and vertical ones are adiabatic. The formulation of the problem is based on the Darcy-Brinkman model. The density variation is taken into account by the Boussinesq approximation. The system of the coupled equations is resolved by the classic finite volume method. The numerical results show that the variation of the conductivity of the porous media influences strongly the flow structure and the heat transfer as well as in upper that in the lower zones. The effect of conductivity is conditioned by the porosity which plays a very significant roll on the heat transfer. The structures of this flow show that this kind of problem with specific boundary conditions generates a complex flow structure of several contra-rotating two to two cells, in the upper half of the cavity.

scholarly journals On uniform asymptotic risk of averaging GMM estimators

10.3982/qe711 ◽  
2019 ◽  
Vol 10 (3) ◽  
pp. 931-979 ◽  
Author(s):  
Xu Cheng ◽  
Zhipeng Liao ◽  
Ruoyao Shi
Keyword(s):  
Nonlinear Models ◽  
Sufficient Conditions ◽  
Risk Difference ◽  
Upper And Lower Bounds ◽  
Quadratic Loss ◽  
Finite Sample ◽  
Moment Conditions ◽  
Gmm Estimator ◽  
Asymptotic Risk ◽  
Non Gaussian

This paper studies the averaging GMM estimator that combines a conservative GMM estimator based on valid moment conditions and an aggressive GMM estimator based on both valid and possibly misspecified moment conditions, where the weight is the sample analog of an infeasible optimal weight. We establish asymptotic theory on uniform approximation of the upper and lower bounds of the finite‐sample truncated risk difference between any two estimators, which is used to compare the averaging GMM estimator and the conservative GMM estimator. Under some sufficient conditions, we show that the asymptotic lower bound of the truncated risk difference between the averaging estimator and the conservative estimator is strictly less than zero, while the asymptotic upper bound is zero uniformly over any degree of misspecification. The results apply to quadratic loss functions. This uniform asymptotic dominance is established in non‐Gaussian semiparametric nonlinear models.

scholarly journals Distributed simultaneous localization and mapping for mobile robot networks via hybrid dynamic belief propagation

2017 ◽  
Vol 13 (8) ◽  
pp. 155014771772671
Author(s):  
Jiuqing Wan ◽  
Shaocong Bu ◽  
Jinsong Yu ◽  
Liping Zhong
Keyword(s):  
Mobile Robot ◽  
Posterior Probability ◽  
Belief Propagation ◽  
Nonlinear Models ◽  
Simultaneous Localization And Mapping ◽  
Factor Graph ◽  
Posterior Probability Distribution ◽  
Localization And Mapping ◽  
Nonparametric Techniques ◽  
Non Gaussian

This article proposes a hybrid dynamic belief propagation for simultaneous localization and mapping in the mobile robot network. The positions of landmarks and the poses of moving robots at each time slot are estimated simultaneously in an online and distributed manner, by fusing the odometry data of each robot and the measurements of robot–robot or robot–landmark relative distance and angle. The joint belief state of all robots and landmarks is encoded by a factor graph and the marginal posterior probability distribution of each variable is inferred by belief propagation. We show how to calculate, broadcast, and update messages between neighboring nodes in the factor graph. Specifically, we combine parametric and nonparametric techniques to tackle the problem arisen from non-Gaussian distributions and nonlinear models. Simulation and experimental results on publicly available dataset show the validity of our algorithm.

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