Sous la co-tutelle de : LABORATOIRE DE MODÉLISATION ET SIMULATION MULTI ÉCHELLE CNRS UPEC UNIVERSITÉ PARIS-EST CRÉTEIL UPEM UNIVERSITÉ PARIS-EST MARNE-LA-VALLÉE Stochastic representation of random positive-definite tensor-valued properties: application to 3D anisotropic permeability random fields Johann Guilleminot Joint work with Christian Soize and Roger Ghanem Université Paris-Est Modélisation et Simulation Multi Echelle, MSME UMR 8208 CNRS University of Southern California Department of Aerospace and Mechanical Engineering October 6 2015 J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 1 / 26
Outline 1 Motivations 2 Algebraic Stochastic Representation 3 Application 4 Revisiting initial theory with SDEs 5 Conclusion J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 2 / 26
Motivations Central issue : modeling, generation and calibration of a matrix-valued random field (e.g. a permeability tensor field) - with various interpretations. Classical multiscale framework : HOMOGENIZATION Microscale Macroscale Mesoscale The medium is, hence, described through random fields of apparent properties. J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 3 / 26
Motivations Modeling of uncertainties, and identification solving an underdetermined inverse problem : Limited measurements are made on boundaries, and used to calibrate the model at any point. The aim here is to derive models : that are consistent from a mathematical standpoint ; that contain as much physics as possible ; that do not rely on too many parameters. J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 4 / 26
Algebraic Stochastic Representation Notation and overview of the construction Let {[A(x)], x Ω} be any M + n (R)-valued random field, indexed by Ω R d. Methodology : construct an information-theoretic stochastic model for the family of first order marginal distributions ; define {[A(x)], x Ω} through a pointwise nonlinear mapping T, acting on some underlying vector-valued Gaussian random field, such that it exhibits the above family of first-order marginal distributions. Relevant issues : how to define the marginal? how to construct mapping T? J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 5 / 26
Algebraic Stochastic Representation Reminder on Information Theory Assume that [A(x)] (x fixed) satisfies n c algebraic constraints : E{g i ([A(x)])} = f i (x), 1 i n c Information theory and Maximum entropy principle : with p [A(x)] = arg max S(p) p C ad (x) S(p) = p([a]) ln(p([a]))d[a] M + n (R) MaxEnt estimate for the probability density function : ) n c p [A(x)] ([A]) = 1 M + ([A]) n ( (R) c0 exp < L i(x), g i ([A]) > Hi i=1 J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 6 / 26
Algebraic Stochastic Representation Definition of RM : available information Constraints (no dependence on x for notational convenience) : (1) E {[A]} = [A] p [A] ([A]) d[a] = [A] (2) (3) E M + n (R) M + n (R) ln (det ([A])) p [A] ([A]) d[a] = β, β < + { ( ϕ kt [A] ϕ k) } 2 = s 2 kλ 2 k, k {1,..., m} (1) is related to the mean value, (2) to finiteness of second-order moments, and (3) are constraints on the spectrum. Let : [A(x)] = [L(x)][G(x)][L(x)] T with [G(x)] = [H(x)][H(x)] T and [H(x)] a lower triangular random matrix. J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 7 / 26
Algebraic Stochastic Representation Definition of RM : joint P.D.F. Assume m = n. For i fixed, r.v. {H il } i l=1 are jointly distributed : where ϕ is defined as : p Hi1,...,H ii (h) = 1 S(h) c 0 h ii n 1 i+2α(x) ϕ(h; µ i(x), τ i(x)) ϕ(h; µ i(x), τ i(x)) := exp µ i(x) Parameter µ i(x) satisfies : x α(x) : level of fluctuations. ( i l=1 h il 2 ) τ i(x) ( i l=1 h il 2 ) 2 R + c 1 g (n+2α(x) 1)/2 exp{ µ i(x)g τ i(x)g 2 } dg = 1 (1) x τ i(x) : variance of λ i (ordered statistics). J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 8 / 26
Algebraic Stochastic Representation Non linear mapping How to define mapping T? Rosenblatt transformation. Numerical approximation : KL decomposition (PCA) : Polynomial chaos expansion : N KL H (i) h (i) + λjη jv j N pce j=1 η pce z γ Ψ γ(u (i) ) Computation of the coefficients by the maximum likelihood method (first-order m.p.d.f., joint p.d.f.). γ=1 J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 9 / 26
Application Context Darcean flows through random porous media 1 : v = 1 µ [K(x)] p Various fields of applications : Geomechanics : oil industry, etc. Composite manufacturing : resin transfer molding, etc. Let and [K] = [ 1.05 0 0 0.95 ( ) h11 0 [H] = h 21 h 22 ] 1. See Guilleminot, J., Soize, C. and Ghanem, R. G., IJNAMG, 36(13), 2012. J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 10 / 26
Application Influence of parametrization 8 7 6 5 PDF 4 3 2 1 0 0.4 0.6 0.8 1 1.2 1.4 1.6 Principal permeability FIGURE 1: Plot of the p.d.f. of λ x (red line) and λ y (blue line), in 10 10 m 2, for τ x = τ y = 10 2 and α = 20 (dash-dot line), α = 50 (solid line) and α = 80 (dashed line). J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 11 / 26
Application Influence of parametrization PDF 15 10 τ = 1 τ = 10 τ = 10 2 τ = 10 3 τ = 10 4 PDF 16 14 12 10 8 τ = 1 τ = 10 τ = 10 2 τ = 10 3 τ = 10 4 5 6 4 2 0 0.8 1 1.2 1.4 1.6 1.8 2 Principal permeability λ x 0 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 Principal permeability λ y FIGURE 2: Plot of the p.d.f. of λ x (left) and λ y (right), for α = 60 and different values of τ x = τ y = τ. J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 12 / 26
Application Influence of parametrization PDF 18 16 14 12 10 8 6 4 2 0 0.4 0.6 0.8 1 1.2 1.4 1.6 Principal permeability PDF 18 16 14 12 10 8 6 4 2 0 0.4 0.6 0.8 1 1.2 1.4 1.6 Principal permeability FIGURE 3: Plot of the p.d.f. of the first (red dashed line) and second (blue solid line) stochastic principal permeability (in 10 10 m 2 ). Left : α = 80, τ x = 10 4 and τ y = 1. Right : α = 80, τ x = 1 and τ y = 10 4. J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 13 / 26
Application Defining T with the chaos expansion 10 0 10 1 10 2 PDF 10 3 10 4 10 5 10 6 4 2 0 2 4 ρ (1) FIGURE 4: Plot (in semi-log scale) of the p.d.f. of random variable ρ (1) : target (thin solid line) and 10th-order chaos projection (thick solid line). J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 14 / 26
Application Defining T with the chaos expansion 10 0 10 0 10 2 10 2 PDF PDF 10 4 10 4 10 6 3 2 1 0 1 2 3 4 (1) ρ 1 10 6 4 2 0 2 4 (2) ρ 2 FIGURE 5: Plot (in semi-log scale) of the p.d.f. p ηj (2). Reference : thin solid line ; Estimation based on chaos expansion (4th-order expansion) : thick solid line. Left : j = 1 ; Right : j = 2. J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 15 / 26
Application Example 20 1.02 20 1.02 15 1 0.98 15 1 0.98 y 10 0.96 0.94 y 10 0.96 0.94 5 0 0 5 10 15 20 x 0.92 0.9 5 0 0 5 10 15 20 x 0.92 0.9 0.88 FIGURE 6: Plot of a realisation of random fields x λ x(x) (left) and x λ y(x) (right). Two important applications : in geomechanics : orientation of cracks ; in composite manufacturing : local architecture of the preform (racetracking channels, etc.). J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 16 / 26
Revisiting initial theory with SDEs Revisiting the initial formulation The previous derivations involve PCEs, and require a reference generator to be available : can we circumvent these drawbacks? Let p be the target marginal p.d.f., and let : p(u) := c exp ( Φ(u)), u S R q, (2) with Φ the potential. In our case, for the i-th row (q = i) : Φ(u) = log (1 S(u)) β(x) log (u i) + µ i(x) ( i with β(x) := n 1 i + 2α(x). Two cases of practical interest : unconstrained case, S = R q ; constrained case, S R q (recall that H ii > 0 a.s.). l=1 u l 2 ) + τ i(x) ( i l=1 u l 2 ) 2 J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 17 / 26
Revisiting initial theory with SDEs The sampling scheme in a nutshell (S = R q ) Let {W(r, x), r R +, x Ω} be a Gaussian random field, with values in R q, such that : for x fixed in R d, {W(r, x), r R + } is a normalized Wiener process ; for r fixed R +, {W(r, x), x R d } is a colored Gaussian field. Let F be the family of ISDEs, indexed by x, such that : r 0, { du(r, x) = V(r, x) dr dv(r, x) = ( uφ(u(r, x)) η ) 2 V(r, x) dr + η dw(r, x) (3) with given initial conditions. Under some assumptions on Φ : Summary 2 : family of ergodic homogeneous diffusion processes ; law lim U(r, x) = A(x). (4) r + definition of the associated family of invariant measures. 2. See Guilleminot, J., Soize, C., SIAM MULTISCALE MODEL. SIMUL., 11(3), 2013. J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 18 / 26
Revisiting initial theory with SDEs Tackling the constrained case : S R q 3 Let Ψ be the target potential : Ψ(u) := log(1 S(u)) + Z(u), u S. (5) Step 1 : define the potential Ψ ɛ, 0 < ɛ 1, through Ψ ɛ(u) := log (1 ɛ S(u)) + Z(u), u V S, (6) with 1 ɛ S the regularized indicator function (lim ɛ 0 1 ɛ S = 1 S), S V R q, such that Ψ ɛ satisfies all the assumptions related to the SDE analysis ; lim ɛ 0 Ψ ɛ = Ψ in S. For all x in Ω : Then consider the asymptotic case ɛ 0. law lim Uɛ(r, x) = A r + ɛ(x) (7) 3. See Guilleminot, J., Soize, C., SIAM J. SCI. COMPUT., 36(6), 2014. J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 19 / 26
Revisiting initial theory with SDEs Step 2 : definition of an adaptive scheme (adapted from Lamberton & Lemaire (2005)) Let (γ k) k 1 be the sequence : γ k := γ 0 k 1/τ, k 1, (8) with γ 0 > 0 and τ N. Let { {χ k x, k 1} } be the family of processes defined x Ω as : χ k x := 2 V(Uk ɛ(x), V k ɛ(x)), k 0, x Ω. (9) b(u k ɛ(x), V k ɛ(x) 2 1 Define the step sequence ( γ k+1) k 0 as : γ k+1 := min x Ω ( γ k+1 χ k x ), k 0, (10) with γ 0 = γ 0. Scheme : X k+1 ɛ (x) = D SV ( X k ɛ(x), W k+1 (x), γ k+1 ), k 0. (11) with X 0 ɛ(x) = x 0 a.s. for all x in Ω (x 0 = (u 0, v 0) S R q ). J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 20 / 26
Revisiting initial theory with SDEs The computational cost is essentially governed by : the rate of convergence towards the stationary solution ; the gradient computation ; the sampling algorithm for the underlying Gaussians. Current applications deal with more than 100,000 points (100,000 SDEs are integrated simultaneously). Various techniques were benchmarked for the sampling of Gaussians, such as circular embedding ; direct factorization-based methods (Cholesky) ; Krylov subspace methods ; spectral methods. However, it is still an open issue for large-scale simulations and for arbitrary covariance kernels. J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 21 / 26
Revisiting initial theory with SDEs Application 1 Let p be defined as : p(u) := c exp { Ψ(u)}, u S, (12) where Ψ(u) := log (1 S(u)) + 1 [K]u, u, u S, (13) 2 with S := [ 0.5, 0.5] [ 0.5, 0.5] and [K] := diag(1, 1). FIGURE 7: Graph of the indicator function 1 ɛ S for ɛ = 0.04 (left) and ɛ = 0.01 (right). J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 22 / 26
Revisiting initial theory with SDEs Sample paths of {(U k 1, U k 2), k k stat} : FIGURE 8: Sample paths for ɛ = 0.04 (left) and ɛ = 0.01 (right). J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 23 / 26
Revisiting initial theory with SDEs Application 2 Let the target potential be defined as (nonlinear term) : with Ψ(u) := log (1 S(u)) (λ 1) log (det([mat(u)])) + q 1 + 2λ 2 q u k(k+1)/2, (14) S := { u R n [Mat(u)] M + q (R) }. (15) k=1 Let ρ k be the r.v. with values in R + defined as : ( ) ρ k := min S [Mat(U k )], (16) with S([A]) the spectrum of [A] M S q(r) (q = 2 hereinafter). J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 24 / 26
Revisiting initial theory with SDEs FIGURE 9: Sample paths of {ρ k, k k stat} (red circles, left axis) and { γ k, k k stat} (black line, right axis), obtained for ɛ = 0.01, γ 0 = 2 6 and τ = 10 6. J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 25 / 26
Conclusion Concluding comments Construction of random field models that are consistent with mathematics and physics : based on Information Theory ; for Darcean flows through porous media, it allows for the local architecture of the pore space to be accounted for (at least to some extent). Some parts of the model were revisited using a transformation with memory, and involve an adaptive procedure. Numerical investigations on permeability fields are currently under progress. Thank You J. Guilleminot johann.guilleminot@u-pem.fr Workshop MoMas October 6 2015 26 / 26