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.

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.

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.

The Effect of Thermal Wall Properties on Natural Convection in Inclined Rectangular Cells

1982 ◽  
Vol 104 (1) ◽  
pp. 111-117 ◽  
Author(s):  
B. A. Meyer ◽  
J. W. Mitchell ◽  
M. M. El-Wakil
Keyword(s):  
Heat Transfer ◽  
Cell Wall ◽  
Natural Convection ◽  
Convective Heat Transfer ◽  
Convective Heat ◽  
Conductive Heat ◽  
Numerical Work ◽  
Aspect Ratios ◽  
Convection Problem ◽  
Natural Convection Problem

The effects of cell wall thickness and thermal conductivity on natural convective heat transfer within inclined rectangular cells was studied. The cell walls are thin, and the hot and cold surfaces are isothermal. The two-dimensional natural convection problem was solved using finite difference techniques. The parameters studied were cell aspect ratios (A) of 0.5 and 1, Rayleigh numbers (Ra) up to 105, a Prandtl number (Pr) of 0.72 and a tilt angle (φ) of 60 deg. These parameters are of interest in solar collectors. The numerical results are substantiated by experimental results. It was found that convection coefficients for cells with adiabatic walls are substantially higher than those for cells with conducting walls. Correlations are given for estimating the convective heat transfer across the cell and the conductive heat transfer across the cell wall. These correlations are compared with available experimental and numerical work of other authors.

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.

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