Bayesian updating of uncertanties in the description of heat and moisture transport in heterogenous materials

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Adaptive Modeling

and Simulation 2011

D. Aubry, P. Díez, B. Tie and N. Parés (Eds.)

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Adaptive Modeling and Simulation 2011

Proceedings of the V International Conference on Adaptive Modeling and Simulation (ADMOS 2011) held in Paris, France

6 – 8 June 2011

Edited by D. Aubry

École Centrale Paris, France P. Díez

Universitat Politècnica de Catalunya, Spain B. Tie

École Centrale Paris, France N. Parés

Universitat Politècnica de Catalunya, Spain

A publication of:

International Center for Numerical Methods in Engineering (CIMNE) Barcelona, Spain

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International Center for Numerical Methods in Engineering (CIMNE)

Gran Capitán s/n, 08034 Barcelona, Spain

www.cimne.com

Adaptive Modeling and Simulation 2011

D. Aubry, P. Díez, B. Tie and N. Parés (Eds.)

First edition, February 2012 © The authors

Printed by: Artes Gráficas Torres S.L., Huelva 9, 08940 Cornellà de Llobregat, Spain Depósito legal: B-xxxxxxxxxxx

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V International Conference on Adaptive Modeling and Simulation ADMOS 2011 D. Aubry and P. D´ıez (Eds)

BAYESIAN UPDATING OF UNCERTANTIES IN THE

DESCRIPTION OF HEAT AND MOISTURE TRANSPORT

IN HETEROGENOUS MATERIALS

B. ROSI ´Ca,∗_{, H. G. MATTHIES}a_{, A. LITVINENKO}a_{, O. PAJONK}a_{, A.}

KU ˘CEROV ´Ab _{AND J. S ´}_YKORAb,c a_{Institute of Scientific Computing}

Hans-Sommer Straße 65, 38106 Braunschweig, Germany e-mail: wire@tu-bs.de

b _{Department of Mechanics, Faculty of Civil Engineering}

Czech Technical University in Prague, Th´akurova 7, 166 29 Prague 6, Czech Republic e-mail:anicka@cml.fsv.cvut.cz

c _{Centre for Integrated Design of Advances Structures}

Th´akurova 7, 166 29 Prague 6, Czech Republic jan.sykora.1@fsv.cvut.cz

Key words: Karhunen-Lo`eve expansion, uncertainty updating, Bayesian inference, het-erogeneous materials, coupled heat and moisture transport

Abstract. The description of heterogenous material is here given within the probabilis-tic framework, where uncertain material properties in time and/or space are represented by stochastic processes and fields. For material with uncertain structure such as quarry, masonry etc., we study the coupled heat and moisture trasnport modelled by the K¨unzel equations. The transport coefficients defining the material behavior are nonlinear func-tions of structural responses — the temperature and moisture fields — and material properties.

In order to closely determine the mentioned parameters of such system we focus our attention on the solution of inverse problem via direct, non-sampling Bayesian update methods which combine the a priori information with the measurment data for the de-scription of the posterior distribution of parameters. Namely, we consider material param-eters, observations and forward operator as random. Since the measurments are always polluted by some kind of measurment error we modell it here by a Gaussian distribution. The new approach has shown to be effective and reliable in comparison to most methods, which take the form of integrals over the posterior and compute them by sampling, e.g. Markov chain Monte Carlo (MCMC). In addition, we compare our method with this and other Bayesian update methods

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B. Rosi´c∗_{, H. G. Matthies, A. Litvinenko, O. Pajonk, A. Ku˘cerova and J. S´}_ykora

1 INTRODUCTION

The heterogeneous material such as quarry masonary has random microstructure which cannot be modelled in efficient way via multiscale methods based on homogenization tech-niques such as for example in a case of regular masonary or composite materials. The properties of heterogenous material change in a space and/ or time, and thus are consid-ered as uncertain. Here the uncertainties are described by probabilistic methods based on white noise analysis. In other words, instead of assuming material properties to be constant in a homogenized medium, their more realistic random nature can be conserved by taking a finite number of random fields in the description. Numerical procedures are then developed to change/update the description once the materials have been manufac-tured, in order to take into account additional information, such as may be obtained for example from measurements. This results in an improved description of the uncertainties of the material behaviour called inverse problem.

The inverse problems are often settled in Bayesian framework setting, which offers a rigorous foundation for inference from noisy data and uncertain forward models, a natural mechanism for incorporating prior information, and a quantitative assessment of uncertainty in the inferred results summarising all available information.

In recent studies [2, 9, 10, 12, 14, 21], the Bayesian estimates of the posterior density are taking the forms of integrals, computed via asymptotic, deterministic or sampling methods. The most often used technique represents a Markov chain Monte Carlo (MCMC) method [7, 10, 14], which takes the posterior distribution for the asymptotic one. This method has been improved by introducing the stochastic spectral finite element method [15] into the approximation of the prior distribution and corresponding observations[12, 14]. This group of methods is based on Bayes’s formula itself. Another group belongs to so-called ’linear Bayesian’ [8] methods, which update the functionals of the random variables. The simplest known version represents the Kalman-type method [6, 1, 3, 22]. However these methods require a large number of samples in order to obtain satisfying results. On the other hand, methods like the extended Kalman filter run into closure problems for non-linear models [5]. Due to this, they are not suitable for high dimensional problems and another procedure has to be introduced.

In this paper the identification problem is cast in a linear Bayesian framework based on ’white-noise’ analysis [11], starting with a probabilistic model for the uncertain properties with the prior assumption in a form of a lognormal random field with some covariance function. The system output due to specified external loading is measured in certain locations, and from this we sequentially update the description of the unknown fields. The method is direct and doesn’t involve sampling at any stage.

2 THE PROBLEM SETTING

The system model or so-called forward model is described by an operator A, which describes the relation between the model parameters q, external influence f , and the

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system state u [18]:

A(u; q) = f. (1)

This further defines a well-posed problem such that there exists a unique “solution” u satisfying:

u = S(q; f ), (2)

where S is the solution operator describing the explicit relationship between u and the model parameters q. We further define an observation operator Y relating the complete model response u to an observation y as

y = Y (q; u) = Y (q; S(q; f )). (3)

However, the measurements are in practice always disturbed by some kind of error ε, which determines the difference between the actual value ˆy of a measured quantity and the observed value z

z = ˆy + ε. (4)

The random elements in ε are assumed to be independent of the uncertainty in the model parameters q.

3 THE COUPLED HEAT AND MOISTURE TRANSFER

The subject of this paper is to perfom uncertainty updating in the K¨unzel coupled heat and moisture transport in heterogenous material [20, 12] with uncertain structure as quarry masonry, described by the energy balance equation:

dH dΘ

dΘ

dt =∇[λ∇Θ] + hv∇[δp∇ϕpsat(Θ)] (5)

and the equation of the mass conservation: dw

dϕ dϕ

dt =∇[Dϕ∇ϕ] + ∇[δp∇ϕpsat(Θ)]. (6)

The problem is fully described by following material parameters: thermal conductivity λ, evaporation enthalpy of water hv, water vapor permeability δp, water vapor saturation

preassure psat, luiquid conduction coefficient Dϕ, and the total entalpy of building material

H. These parameters depend on the set q of eight material characteristics which will be modelled as random fields.

In order to simplify the problem we assume that the described random fields are fully correlated, which is more realistic than the full spatial independence. In the same way we choose the corresponding prior probability distribution functions via maximum entropy principle.

Thus, the particular material parameter q is modeled by defining q(x) for each x _{∈ G} as a random variables q(x) : Ω _{→ G on a suitable probability space (Ω, B, P) in some}

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bounded admissible region _{G. As a consequence, q : G × Ω → R is a random field and one} may identify Ω with the set of all possible values of q or with the space of all real-valued functions on G. Alternatively, q(x, ω) can be seen as a collection of real-valued random variables indexed by x_{∈ G.}

3.1 Stochastic Finite Element Method

The problem is described by a strong form of K¨unzel equations [20], which furter may be transformed to a corresponding variational formulation. The discretisation techniques for the weak problem are then used for both spaces, deterministic and stochastic one. The spatial discretisation is based on the known finite methods such as finite element or finite volume method.

Taking a finite element ansatz {φn(x)}Nn=1 [19, 4] as a corresponding subspace, the

solution may be discretised by:

u(x, ω) =

N

n=1

un(ω)φn(x), (7)

where the coefficients _{un(ω)} are now random variables, further represented as a

func-tions of independent and uncorrelated random variables, in this case the Hermitian basis {Hα}JZ and Gaussian random variables θ:

un(θ(ω)) =

α∈JZ u(α)

n Hα(θ(ω)), (8)

where JZ represents the finite set of multiindexes with cardianlity Z.

The stochastic discretisation is done by expanding the corresponding fields into the Karhunen-Lo`eve expansion (KLE)

q = ∞ j=0 √_χ jψjξj(θ), (9)

where ψj represent the eigenvector and ξj(θ) the random variables. The eigenvectors

describe the fluctuation of material property within the studied domain _{G. They are} obtained as the eigenfunctions of the Fredholm integral equation with the covariance function Cq as the integral kernel:

G

Cq(x, y)ψj(x)dy = χjψj(x) (10)

where χj are positive eigenvalues ordered in a descending order.

After the discretisation is done and the corresponding problem is linearized one arrives at the standard stochastic linear problem already studied in many papers before [12, 15, 16].

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3.2 THE UPDATE PROCEDURE

The primary computational challenge is extracting information from the posterior den-sity of parameters. The estimates done in this paper are based on the forward problem solved in stochastic Galerkin way and the update procedure done in a pure deterministic way, as well as by sampling procedures.

By subjecting the system Eq. 5 and Eq. 6 to varying external influences we observe the output, and update our knowledge of the range of possible parameter values q from these observations. Thus, the probabilistic identification problem is cast in a functional approximation setting — the best known of which is the polynomial chaos expansion (PCE) — and the linear Bayes form of updating. In this way the identification process can be carried out completely deterministically. In the case where the original problem was a deterministic identification task the method additionally provides a quantification of the remaining uncertainty in a Bayesian setting. But it can also be used as an identification procedure in an originally probabilistic setting.

If we now view the parameters q due to the uncertainty as a random variable (with values in the space of admissible log-conductivity fields) and want to approximate this random variable with our previous knowledge — the prior random variable qf and the

measurements z, then the minimum mean square estimator is one frequently used approx-imation. The estimator is also the minimum variance estimator, i.e. a linear Bayesian update with the variance of the difference as loss function. In the case of a linear problem and Gaussian random variables it is well known in the guise of the Kalman filter. Taking the projection formula of the estimator [18, 17]:

qa(ω) = qf(ω) + K(z(ω)− y(ω)), (11)

with so called ’Kalman’ gain

K = Cqfy(Cy+ C)

−1 ₍₁₂₎

and projecting it onto the polynomial chaos space, one obtains:

qa= qf + (K⊗ I) (z − y). (13)

in tensorial notation, where

qa= β∈JZ qβ a⊗ eβ. (14) Here qβ

a denote the coefficients of polynomial chaos expansion, while eβ the canonical

basis in RZ_{. The covariances between the corresponding variables (given in index) are}

denoted by C.

This is contreasted to the Markov chain Monte Carlo method [12], formulating the update procedure on the measure how good a forward operator A explains the data z, i.e. on the likelihood function. Using Bayes’s rule:

p(q_{|z) = const p(z|q)p}q(q), (15)

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which denotes the a posteriori probability density for the model parameter, given an a priori distribution pq(q) of parameters and an uncertain observation of the data z

repre-sented by conditional probability distribution p(z|q) where the forward operator relying observable quantities z and model parameter q is included. The a posteriori information in the space of model parameters is given by the marginal probability density:

πq(q) =

p(q_{|z)dz = const p}q(q)L(q) (16)

where data z enters through the likelihood function L(q), which gives a measure of how good a model is in explaining the data z. Since we explicitely know the a priori distribu-tion, we also assume that we are able to obtain as many samples of the prior probability density pq(q) as we wish. The problem at hand is to sample the conjunction of the

known a priori density function and the likelihood function L(q). The evaluation of the likelihood function requires evaluation of the forward model, which may be done in two different ways: by Galerkin projection and stochastic finite element methods or sampling techniques such as Monte Carlo Markov Chain technique (MCMC). MCMC exploration of the reduced-dimensionality posterior still requires repeated solutions of the forward model, once for each proposed move of the Markov chain. The acceleration of the Bayesian infer-ence is done by using stochastic spectral methods to propagate prior uncertainty through the forward problem. These methods effectively create a surrogate posterior containing polynomial chaos (PC) representations of the forward model outputs as it is described in [13].

4 CONCLUSIONS

A linear Bayesian estimation of unknown parameters is formulated in a purerly de-terministic and algebraic way, without the need for any kind of sampling techniques like MCMC. The regularisation of the ill-posed problem is achieved by introduction of a priori information approximated by a combination of KLE and polynomial chaos expansions truncated to a finite number of terms. This representation enters a stochastic forward model, solved by a Galerkin procedure and reduction methods.

The suggested method is tested on the model of heat transport equation under steady state conditions, and then on the coupled model.

The identification experiment, characterised by a sequential update process, is per-formed for a “deterministic truth”with increasing spatial compability. In this way we have shown the influence of additional information and loading conditions on the up-date process. The presented linear Bayesian upup-date does not need any linearity in the forward model, and it can readily update non-Gaussian uncertainties. In addition, it accommodates noisy measurements and skewed RVs.

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