Application of the Proper Orthogonal Decomposition for Bayesian

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 – 03. 11. 2017, Rio de Janeiro, Brasil

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 ≈ 42 GJ), § 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. 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

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: 3 x 3 x 3 mm 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

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 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

Problem formulation Vector of unknown parameters Measured data 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

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, 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

Bayesian inverse problem formulation Bayes formula Likelihood distribution 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

Bayesian inverse problem formulation Prior distribution Gaussian prior Uniform prior 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

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. 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

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. 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

Metropolis-Hastings algorithm Proposal distribution – random walk 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

Posterior distribution of unknown parameters Exact values 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

Posterior distribution of unknown parameters Porosity Pore diameter Exact values Computation time ~72 h Reduced Order Model 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

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 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

Proper Orthogonal Decomposition (POD) Steady state flow problem – a snapshot represents discretised pressure field inside computational domain Snapshot Matrix of snapshots System 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

Proper Orthogonal Decomposition (POD) Steady state flow problem – a snapshot represents discretised pressure field inside computational domain System Snapshot 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

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 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

Proper Orthogonal Decomposition (POD) Truncate the input space basis 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

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 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

Proper Orthogonal Decomposition (POD) Number of snapshots – full factorization Snapshots number 11 x 11 Snapshots number 21 x 21 Snapshots number 31 x 31 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

Proper Orthogonal Decomposition (POD) Number of snapshots – full factorization 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

Proper Orthogonal Decomposition (POD) Tspline Gauss MQD IMQD Type of approximation function 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

Reduced Order Model – POD Pressure field - exact model Pressure field – reduced POD model 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

Reduced Order Model – POD 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

Approximation Error Model (AEM) Posterior distribution Modified likelihood distribution Prior distribution 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

Posterior distribution of unknown parameters (AEM) Exact values 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

Posterior distribution of unknown parameters (AEM) Exact values Porosity Pore diameter Computation time 5 min 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

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. 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil

The financial support of The National Science Centre, Poland, within the grant UMO-2014/15/D/ST 8/02767 is gratefully acknowledged herewith. Investigation of heat and mass transfer in porous media with chemical reactions with use of statistical inverse methodology zbigniew. bulinski@polsl. pl 30. 10 – 03. 11. 2017, Rio de Janeiro, Brasil