1. Application of the Proper Orthogonal Decomposition for Bayesian Estimation of flow parameters in porous medium
Zbigniew Buliński*, Helcio R.B. Orlande**, Tomasz Krysiński*, Łukasz Ziółkowski*, Sebastian Werle**
* Institute of Thermal Technology, Silesian University of Technology, Poland
** Department of Mechanical Engineering, Politecnica/COPPE
Federal University of Rio de Janeiro – UFRJ, Rio de Janeiro, Brasil
New Trends in Parameter Identification for Mathematical Models 2017
30.10 – , Rio de Janeiro, Brasil

2. Motivation
According to EU regulations biomass is treated as a renewable source of Energy which might be used directly (burning) or indirectly (gasification),
The European Environment Agency estimates that around 235 Mtoe/year of biomass could be made available in EU by 2020 without harming the environment (toe ≈ 42GJ),
Effective utilisation of this source of energy needs reliable mathematical model of heat and fluid flow,
In some applications mathematical model of biomass burning or gasifiaction may be based on the porous medium formulation (fixed bed),
Porous medium (material) is a solid material (solid matrix) containing interconnected voids which filled with a fluid – two phase system,
Usually model parameters are unknown.
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3. Mathematical modelling of porous material
Direct modelling
Detailed geometrical model of the porous material
Huge computational cost
Detailed solution field
Velocity field
Mesh size ~2.2 M cells
Domain size: 3x3x3 mm
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4. Mathematical modeling of porous material
Volume (sometimes also time) averaged flow equations
Darcy Law
Forchheimer’s equation (for isotropic porous medium)
Hsu and Cheng, 1990, Vafai and Kim, 1990 refinement
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5. Mathematical modeling of porous material
Volume (sometimes also time) averaged flow equations
Darcy Law
Forchheimer’s equation (for isotropic porous medium)
Hsu and Cheng, 1990, Vafai and Kim, 1990 refinement
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6. Problem formulation Vector of unknown parameters Measured data
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7. Statistical inverse problem formulation
Main features of the Statistical inverse problem formulation
All unknown quantities (estimated parameters) and input data (measurements) are treated as random variables,
Knowledge on the realisation of all these quantities is given in the form of the probability distribution,
Solution of the considered problem is given in a form of probability distribution of unknown quantities,
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8. Bayesian inverse problem formulation
Bayes formula
Likelihood distribution
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9. Bayesian inverse problem formulation
Prior distribution
Gaussian prior
Uniform prior
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10. Bayesian inverse problem formulation
Posterior distribution of unknown parameters
No explicit analytical formulation of the distribution is known, obtaining distribution moments needs numerical integration usually by sampling.
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11. Metropolis-Hastings algorithm
1. Draw a Candidate Point K+ from a known proposal distribution p(K+,Ki-1),
2. Calculate the acceptance factor:
3. Generate a random number R that is uniformly distributed on (0,1),
4. Accept candidate if R < a, set Ki = K+,otherwise set Ki = Ki-1,
5. Return to step 1.
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12. Metropolis-Hastings algorithm
Proposal distribution – random walk
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13. Posterior distribution of unknown parameters
Exact values
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14. Posterior distribution of unknown parameters
Porosity
Pore diameter
Exact values
Computation time ~72h
Reduced Order Model
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15. Proper Orthogonal Decomposition (POD)
Single snapshot yi is discretized response (image) of the field under investigation for given spatial coordinates and time
System
Snapshot
Matrix of snapshots
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16. Proper Orthogonal Decomposition (POD)
Steady state flow problem – a snapshot represents discretised pressure field inside computational domain
Snapshot
Matrix of snapshots
System
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17. Proper Orthogonal Decomposition (POD)
Steady state flow problem – a snapshot represents discretised pressure field inside computational domain
System
Snapshot
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18. Proper Orthogonal Decomposition (POD)
Retrieve structure (anatomy) of the snapshot matrix by SVD
Matrix of amplitudes
Singular vectors resolving the response space
Product describing influence of input space
May be interpreted as a variables separation
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19. Proper Orthogonal Decomposition (POD)
Truncate the input space basis
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20. Proper Orthogonal Decomposition (POD)
Approximation of the amplitude matrix with a set of basis functions
Finally, response of the system for any set of input parameters K can be calculated with trained POD-RBF network
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21. Proper Orthogonal Decomposition (POD)
Number of snapshots – full factorization
Snapshots number 11x11
Snapshots number 21x21
Snapshots number 31x31
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22. Proper Orthogonal Decomposition (POD)
Number of snapshots – full factorization
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23. Proper Orthogonal Decomposition (POD)
Type of approximation function
IMQD
MQD
Gauss
Tspline
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24. Reduced Order Model – POD
Pressure field - exact model
Pressure field – reduced POD model
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25. Reduced Order Model – POD
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26. Approximation Error Model (AEM)
Posterior distribution
Modified likelihood distribution
Prior distribution
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27. Posterior distribution of unknown parameters (AEM)
Exact values
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28. Posterior distribution of unknown parameters (AEM)
Porosity
Pore diameter
Exact values
Computation time 5 min
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29. Summary
Bayesian inverse methodology was developed to estimate porosity and pore diameter of the porous material based on the pressure drop measurements,
Reduced Order Model based on the Proper Order Decomposition was developed to reduce computational time
Approximation Error Model (AEM) was used to incorporate approximation errors into the inference procedure giving promising results,
Developed procedure is going to be applied to real-life problem of estimation of flow parameters for the biomass.
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30. The financial support of The National Science Centre, Poland, within the grant UMO-2014/15/D/ST8/02767  is gratefully acknowledged herewith. 
Investigation of heat and mass transfer in porous media with chemical reactions with use of statistical inverse methodology
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