Stochastic Dimension Reduction

Transcription

1 Stochastic Dimension Reduction Roger Ghanem University of Southern California Los Angeles, CA, USA Computational and Theoretical Challenges in Interdisciplinary Predictive Modeling Over Random Fields 12th Annual Red Raider Mini-Symposium October 26, 2012 Department of Mathematics and Statistics Texas Tech University Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

2 Motivation Objective is to characterize solution, u, of equations with stochastic coefficients on (Ω, Σ, P) Coefficients serve to construct an adapted Hilbert space: k = i k iξ i ξ i : (Ω, Σ(ξ i ), P) R i.i.d G = span{ξ 1,, ξ d }, L 2 (Ω) L 2 (Ω, Σ(G), P). THEN u(ξ) = u(ξ 1,, ξ d ) L 2 (Ω) and u(ξ) = i (u, ψ i) L 2 (Ω)ψ i (ξ) is a unique representation of u(ξ) Sandard PC machinery tries to identify the unique u(ξ) in R d. Number of terms in PC Expansion: M = (P+d)! P!d! Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

3 Motivation Objective is to characterize solution, u, of equations with stochastic coefficients on (Ω, Σ, P) Coefficients serve to construct an adapted Hilbert space: k = i k iξ i ξ i : (Ω, Σ(ξ i ), P) R i.i.d G = span{ξ 1,, ξ d }, L 2 (Ω) L 2 (Ω, Σ(G), P). THEN u(ξ) = u(ξ 1,, ξ d ) L 2 (Ω) and u(ξ) = i (u, ψ i) L 2 (Ω)ψ i (ξ) is a unique representation of u(ξ) Sandard PC machinery tries to identify the unique u(ξ) in R d. Number of terms in PC Expansion: M = (P+d)! P!d! Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

4 Motivation Objective is to characterize solution, u, of equations with stochastic coefficients on (Ω, Σ, P) Coefficients serve to construct an adapted Hilbert space: k = i k iξ i ξ i : (Ω, Σ(ξ i ), P) R i.i.d G = span{ξ 1,, ξ d }, L 2 (Ω) L 2 (Ω, Σ(G), P). THEN u(ξ) = u(ξ 1,, ξ d ) L 2 (Ω) and u(ξ) = i (u, ψ i) L 2 (Ω)ψ i (ξ) is a unique representation of u(ξ) Sandard PC machinery tries to identify the unique u(ξ) in R d. Number of terms in PC Expansion: M = (P+d)! P!d! Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

5 Motivation Objective is to characterize solution, u, of equations with stochastic coefficients on (Ω, Σ, P) Coefficients serve to construct an adapted Hilbert space: k = i k iξ i ξ i : (Ω, Σ(ξ i ), P) R i.i.d G = span{ξ 1,, ξ d }, L 2 (Ω) L 2 (Ω, Σ(G), P). THEN u(ξ) = u(ξ 1,, ξ d ) L 2 (Ω) and u(ξ) = i (u, ψ i) L 2 (Ω)ψ i (ξ) is a unique representation of u(ξ) Sandard PC machinery tries to identify the unique u(ξ) in R d. Number of terms in PC Expansion: M = (P+d)! P!d! Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

6 Pushing Uncertainty Through Model: Intrusive Stochastic Projection: Consider governing equation of general form: M ξ u = f. P Ψ k Ψ j M ξ u j = Ψ k f 0 k P, j=0 If M ξ = L i Ψ i (ξ)m i, results in a coupled system of equations: P j=0 i=0 L Ψ i Ψ j Ψ k M i u j = Ψ k f 0 k P. Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

7 Intrusive Stochastic Projection: Consider governing equation of general form: M ξ u = f. P Ψ k Ψ j M ξ u j = Ψ k f 0 k P, j=0 If M ξ = L i Ψ i (ξ)m i, results in a coupled system of equations: P L i=0 Ψ iψ j Ψ k M i u j = Ψ k f 0 k P. j=0 j Mjk u j = f k k Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

8 Intrusive Stochastic Projection: Resulting System of Equations c ijk = [ ξ i ψ j ψ k : Al 1 B A l = l C l D l ] Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

9 Intrusive Stochastic Projection: Resulting System of Equations c ijk = [ ξ i ψ j ψ k : Al 1 B A l = l C l D l ] c ijk = ψ i ψ j ψ k Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

10 Non-Intrusive Characterization We want: u(ξ) = i (u, ψ i ) L 2 (Ω)ψ i (ξ) Orthogonality of {ψ i } u i = (u, ψ i ) L 2 (Ω) = E{u(ξ)ψ i (ξ)} = Γ 1 u(ξ)ψ i (ξ)dµ(ξ) Γ d If ξ are independent and have density functions: u i = Γ 1 u(ξ)ψ i (ξ)f 1 (ξ 1 ) f d (ξ d )dξ 1 dξ d Γ d Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

11 Challenges: Intrusive Very large system of equations; Constructing K i and K jk. Non-Intrusive High-dimensional integration. Opportunities: Intrusive Linear Algebra, dimension reduction, adaptive refinement. Non-Intrusive Sparsity, anisotropy, dimension reduction. Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

12 First Idea: L 2 reduction replace the generic basis ψ i by a basis that is adapted to u H Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

13 Karhunen-Loeve and PC: Consider the polynomial chaos representation of the solution u: u = P u i ψ i, i=0 u, u i H Covariance Operator is nuclear, R u : H H (R u X, Y ) H = Solve eigenproblem: P (u i, X ) H (u i, Y ) H X, Y H i=1 (R u e, v) H = P (u i, e) H (u i, v) H = λ (e, v) H v H i=1 Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

14 Then we can represent PC coefficients on KL directions: KL u i = (u i, e j ) e j j=1 But u = P KL (u i, e j )e j ψ i = i=1 j=1 KL j=1 i=1 P KL (u i, e j )ψ i e j = λj η j e j j=1 Thus: η j = 1 λj P i=1 (u i, e j )ψ i j = 1,, KL Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

15 (N, P)-discretization: Error: ɛ j = η (N,P) j η (N,P) j = 1 λj P i=1 η j C j max i (u N i, e P j )ψ i P must be large enough to approximate R. (ui N, ej P ) (u i, e j ) N must be large enough to capture projection of u i on e j for the j that matter. Reduce global error - no control over local error. 2 Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

16 Coarse and fine meshes used in analysis: w w Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

17 Realization of elasticity over L-shaped domain: Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

18 Comparison of pdf for fine model and reduced-model: pdf of u(0.0,1.0) Full model analysis Reduced model, Mesh 1 Reduced model, Mesh 2 Reduced model, Mesh 3 Reduced model, Mesh 4 pdf of u(0.0,1.0), right tail 2 x Full model analysis Reduced model, Mesh 1 Reduced model, Mesh 2 Reduced model, Mesh 3 Reduced model, Mesh Horizontal displacement, u(0.0,1.0) Horizontal displacement, u(0.0,1.0) Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

19 Comparison of Stochatsic bases: ψ 1 ψ 2 ψ 3 ψ Mesh 5 Mesh 4 Mesh 3 Mesh 2 Mesh pdf ψi 1 pdf η ψ i PDF for 4 terms in PC basis η 1 PDF for first term in adapted basis. Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

20 Ghanem (USC) 5 Stochastic Dimenion Reduction Red Raider / 34 Muliscale Quadrature Main Idea Develop UQ for coupled models in Nuclear Reactor Technology. Adapt measure of approximation at every handshaking. Mitigate mixing of uncertainty at handshaking. Develop multiscale quadrature rules. Reduction The output from Model I is reduced using Karhunen-Loeve expansion. The joint PDF of the dominant KL variables is estimated and a corresponding orthogonal polynomials constructed.

21 Multiscale Quadrature η2 [ ] η2 [ ] Numerical quadrature in Model II can be developed relative to new measure: η1 [ ] Quadrature points relative to initial measure. Ghanem (USC) Quadrature points adapted measure. Stochastic Dimenion Reduction 0 η1 [ ] relative 2.5 to Red Raider / 34

22 Some analysis New basis is adapted to the solution, and not to the parameters. Complexity is driven by physics - not by parameters. Solution is still a random process - very high-dimensional. In many cases, the QoI is a functional of the solution, that is much less complex than solution. We are still spinning our wheels a lot for more than we need. Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

23 Some analysis New basis is adapted to the solution, and not to the parameters. Complexity is driven by physics - not by parameters. Solution is still a random process - very high-dimensional. In many cases, the QoI is a functional of the solution, that is much less complex than solution. We are still spinning our wheels a lot for more than we need. Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

24 Some analysis New basis is adapted to the solution, and not to the parameters. Complexity is driven by physics - not by parameters. Solution is still a random process - very high-dimensional. In many cases, the QoI is a functional of the solution, that is much less complex than solution. We are still spinning our wheels a lot for more than we need. Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

25 Some analysis New basis is adapted to the solution, and not to the parameters. Complexity is driven by physics - not by parameters. Solution is still a random process - very high-dimensional. In many cases, the QoI is a functional of the solution, that is much less complex than solution. We are still spinning our wheels a lot for more than we need. Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

26 Some analysis New basis is adapted to the solution, and not to the parameters. Complexity is driven by physics - not by parameters. Solution is still a random process - very high-dimensional. In many cases, the QoI is a functional of the solution, that is much less complex than solution. We are still spinning our wheels a lot for more than we need. Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

27 Second Idea: Probabilistic reduction In many instances, the QoI is h(u) where h is some nonlinear functional. if the QoI is a single random variable, describing it in terms of more than one random variables seems to be a waste of bandwidth. Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

28 Motivation Objective is to characterize solution, u, of equations with stochastic coefficients on (Ω, Σ, P) Coefficients serve to construct an adapted Hilbert space: k = i k iξ i ξ i : (Ω, Σ(ξ i ), P) R i.i.d G = span{ξ 1,, ξ d }, L 2 (Ω) L 2 (Ω, Σ(G), P). THEN u(ξ) = u(ξ 1,, ξ d ) L 2 (Ω) and u(ξ) = i (u, ψ i) L 2 (Ω)ψ i (ξ) is a unique representation of u(ξ) Sandard PC machinery tries to identify the unique u(ξ) in R d. BUT u(πξ) d = u(ξ) π G (permutation group on R d ) Other representations in R d that have the same distribution. Perhaps we are working too hard. Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

29 Motivation Objective is to characterize solution, u, of equations with stochastic coefficients on (Ω, Σ, P) Coefficients serve to construct an adapted Hilbert space: k = i k iξ i ξ i : (Ω, Σ(ξ i ), P) R i.i.d G = span{ξ 1,, ξ d }, L 2 (Ω) L 2 (Ω, Σ(G), P). THEN u(ξ) = u(ξ 1,, ξ d ) L 2 (Ω) and u(ξ) = i (u, ψ i) L 2 (Ω)ψ i (ξ) is a unique representation of u(ξ) Sandard PC machinery tries to identify the unique u(ξ) in R d. BUT u(πξ) d = u(ξ) π G (permutation group on R d ) Other representations in R d that have the same distribution. Perhaps we are working too hard. Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

30 Motivation Objective is to characterize solution, u, of equations with stochastic coefficients on (Ω, Σ, P) Coefficients serve to construct an adapted Hilbert space: k = i k iξ i ξ i : (Ω, Σ(ξ i ), P) R i.i.d G = span{ξ 1,, ξ d }, L 2 (Ω) L 2 (Ω, Σ(G), P). THEN u(ξ) = u(ξ 1,, ξ d ) L 2 (Ω) and u(ξ) = i (u, ψ i) L 2 (Ω)ψ i (ξ) is a unique representation of u(ξ) Sandard PC machinery tries to identify the unique u(ξ) in R d. BUT u(πξ) d = u(ξ) π G (permutation group on R d ) Other representations in R d that have the same distribution. Perhaps we are working too hard. Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

31 Motivation Objective is to characterize solution, u, of equations with stochastic coefficients on (Ω, Σ, P) Coefficients serve to construct an adapted Hilbert space: k = i k iξ i ξ i : (Ω, Σ(ξ i ), P) R i.i.d G = span{ξ 1,, ξ d }, L 2 (Ω) L 2 (Ω, Σ(G), P). THEN u(ξ) = u(ξ 1,, ξ d ) L 2 (Ω) and u(ξ) = i (u, ψ i) L 2 (Ω)ψ i (ξ) is a unique representation of u(ξ) Sandard PC machinery tries to identify the unique u(ξ) in R d. BUT u(πξ) d = u(ξ) π G (permutation group on R d ) Other representations in R d that have the same distribution. Perhaps we are working too hard. Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

32 Motivation Objective is to characterize solution, u, of equations with stochastic coefficients on (Ω, Σ, P) Coefficients serve to construct an adapted Hilbert space: k = i k iξ i ξ i : (Ω, Σ(ξ i ), P) R i.i.d G = span{ξ 1,, ξ d }, L 2 (Ω) L 2 (Ω, Σ(G), P). THEN u(ξ) = u(ξ 1,, ξ d ) L 2 (Ω) and u(ξ) = i (u, ψ i) L 2 (Ω)ψ i (ξ) is a unique representation of u(ξ) Sandard PC machinery tries to identify the unique u(ξ) in R d. BUT u(πξ) d = u(ξ) π G (permutation group on R d ) Other representations in R d that have the same distribution. Perhaps we are working too hard. Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

33 Focus on QoI Consider square-integrable nonlinear functionals h : H R: h def = h(u(ξ)) = h 0 + α =1 h α ψ α (ξ) + α >1 h α ψ α (ξ) Inverse CDF: A mapping from a Gaussian variable ˆξ to h can be constructed as follows: [ )] h = d def ĥ(ˆξ) = µ 1 Φ (ˆξ Expand as: ĥ(ˆξ) = ĥ 0 + ĥ 1 ˆξ + i ĥiψ i (ˆξ) Thus a one-dimensional expansion in terms of a Gaussian variable exists. It only matches probability measure of h. Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

34 Focus on QoI Consider square-integrable nonlinear functionals h : H R: h def = h(u(ξ)) = h 0 + α =1 h α ψ α (ξ) + α >1 h α ψ α (ξ) Inverse CDF: A mapping from a Gaussian variable ˆξ to h can be constructed as follows: [ )] h = d def ĥ(ˆξ) = µ 1 Φ (ˆξ Expand as: ĥ(ˆξ) = ĥ 0 + ĥ 1 ˆξ + i ĥiψ i (ˆξ) Thus a one-dimensional expansion in terms of a Gaussian variable exists. It only matches probability measure of h. Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

35 Focus on QoI Consider square-integrable nonlinear functionals h : H R: h def = h(u(ξ)) = h 0 + α =1 h α ψ α (ξ) + α >1 h α ψ α (ξ) Inverse CDF: A mapping from a Gaussian variable ˆξ to h can be constructed as follows: [ )] h = d def ĥ(ˆξ) = µ 1 Φ (ˆξ Expand as: ĥ(ˆξ) = ĥ 0 + ĥ 1 ˆξ + i ĥiψ i (ˆξ) Thus a one-dimensional expansion in terms of a Gaussian variable exists. It only matches probability measure of h. Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

36 Focus on QoI Introduce a new Gaussian rv η 1 : ĥ 1 η 1 = α =1 h α ψ α = d h i ξ i i=1 Then: h = h 0 + ĥ 1 η 1 + α >1 h α ψ α (ξ) Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

37 Let: A be an isometry and ξ be Gaussian. Then u(aξ) has the same probability measure as u(ξ). If: ξ is Gaussian, then η = Aξ has the same probability measure as ξ. Thus η is a basis for the Gaussian Hilbert space spanned by ξ. Then: Hermite Polynomials in η span the same space as Hermite polynomials in ξ - namely: L 2 (Ω, Σ(ξ), P) = L 2 (Ω, Σ(η), P). Choose: A so that ĥ1η 1 = α =1 h αψ α Then in L 2 : h(ξ) = h 0 + ĥ1η 1 + h α ψ α (η) = h 0 + ĥ1η 1 + α >1 α >1 α=( α,0,,0) + h α ψ α (η 1 ) α >1 α ( α,0,,0) h α ψ α (η) Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

38 Let: A be an isometry and ξ be Gaussian. Then u(aξ) has the same probability measure as u(ξ). If: ξ is Gaussian, then η = Aξ has the same probability measure as ξ. Thus η is a basis for the Gaussian Hilbert space spanned by ξ. Then: Hermite Polynomials in η span the same space as Hermite polynomials in ξ - namely: L 2 (Ω, Σ(ξ), P) = L 2 (Ω, Σ(η), P). Choose: A so that ĥ1η 1 = α =1 h αψ α Then in L 2 : h(ξ) = h 0 + ĥ1η 1 + h α ψ α (η) = h 0 + ĥ1η 1 + α >1 α >1 α=( α,0,,0) + h α ψ α (η 1 ) α >1 α ( α,0,,0) h α ψ α (η) Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

39 Let: A be an isometry and ξ be Gaussian. Then u(aξ) has the same probability measure as u(ξ). If: ξ is Gaussian, then η = Aξ has the same probability measure as ξ. Thus η is a basis for the Gaussian Hilbert space spanned by ξ. Then: Hermite Polynomials in η span the same space as Hermite polynomials in ξ - namely: L 2 (Ω, Σ(ξ), P) = L 2 (Ω, Σ(η), P). Choose: A so that ĥ1η 1 = α =1 h αψ α Then in L 2 : h(ξ) = h 0 + ĥ1η 1 + h α ψ α (η) = h 0 + ĥ1η 1 + α >1 α >1 α=( α,0,,0) + h α ψ α (η 1 ) α >1 α ( α,0,,0) h α ψ α (η) Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

40 Let: A be an isometry and ξ be Gaussian. Then u(aξ) has the same probability measure as u(ξ). If: ξ is Gaussian, then η = Aξ has the same probability measure as ξ. Thus η is a basis for the Gaussian Hilbert space spanned by ξ. Then: Hermite Polynomials in η span the same space as Hermite polynomials in ξ - namely: L 2 (Ω, Σ(ξ), P) = L 2 (Ω, Σ(η), P). Choose: A so that ĥ1η 1 = α =1 h αψ α Then in L 2 : h(ξ) = h 0 + ĥ1η 1 + h α ψ α (η) = h 0 + ĥ1η 1 + α >1 α >1 α=( α,0,,0) + h α ψ α (η 1 ) α >1 α ( α,0,,0) h α ψ α (η) Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

41 Let: A be an isometry. Then u(aξ) has the same probability measure as u(ξ). If: ξ is Gaussian, then η = Aξ has the same probability measure as ξ. Thus η is a basis for the Gaussian Hilbert space spanned by ξ. Then: Hermite Polynomials in η span the same space as Hermite polynomials in ξ (namely: L 2 (Ω, Σ(ξ), P). Choose: A so that ĥ1η 1 = α =1 h αψ α Then in L 2 : h(ξ) = h 0 + ĥ1η 1 + h α ψ α (η) = h 0 + ĥ1η 1 + α >1 α >1 α=( α,0,,0) + h α ψ α (η 1 ) α >1 α ( α,0,,0) h α ψ α (η) Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

42 One-Dimensional Approximation of Solution Projection of the solution on L 2 (Ω, Σ(η 1 ), P): h(ξ) = h 0 + ĥ 1 η 1 + h α ψ α (η 1 ) α >1 Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

43 Implementation Recipe Compute linear components in expansion of QoI: η 1 = i w iξ i. Construct isometry A with η 1 as leading direction. Construct projection operators for η = Aξ: f (ξ)ψ(η). Solve for the representation of solution with respect to η 1. If an L 2 characterization is required, then evaluate components with respect to full η basis. Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

44 Implementation Numerical Effort: 1 Evaluate: ψ i (ξ)ψ j (η)ψ k (η) = ψ i (ξ)ψ j (Aξ)ψ k (Aξ) This is the multi-dimensional integral of a scalar polynomial function. Function evaluations are very inexpensive. These evaluations are massively parallelizable 2 Discover the linear terms of the QoI in a non-intrusive fashion. Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

45 Numerical Example Plane Stress; Random Young s Modulus. Quantity of Interest: X-Displacement at location (0.25,0). Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

46 Using leading dimension of new basis. Probability density function Full Dimension Reduced 1D Model Displacement in x direction at (0.25, 0.00) Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

47 Behavior in L 2 PC Coefficients of the Solution Projected in L 2 on η 1 vs. PC Coefficients of the Inverse CDF Operator. Maginitude of PC Coeffs in log scale xid10 p1 pk3 etap10 p50 etad1 1-dimensional approximation Inverse-CDF approximation xid10 p4 pk3 Nataf etad Index of PC Coeffs Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

48 Optimal Dimension Varies over the Domain Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

49 Nonlinear QoI Von Mises Stress at a Point: σ v = σ 2 1 σ 1σ 2 + σ full basis reduced basis 20 pdf Von Mises stress at (0.2245,0.1101) Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34

50 Conclusions The geometric structure provides a very rich context to describe complicated objects (stochastic processes and white noise). Some quantities of interest are simple, and that simplicity can be discovered within the richer mathematical structure. If the QoI are scalars, and if we merely care about an L 1 characterization, then 1-d representations exist and the question can be reformulated as to discover them. Additional physical/empirical constraints can be reflected in the construction of A. Ghanem (USC) Stochastic Dimenion Reduction Red Raider / 34