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Coarse-grained models for PDEs with random coefficients C. Grigo - - PowerPoint PPT Presentation

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Coarse-grained models for PDEs with random coefficients C. Grigo and P.-S. Koutsourelakis Continuum Mechanics Group Department of Mechanical Engineering Technical University of Munich SIAM CSE Atlanta 1 Mar 2017 SIAM CSE 17 Mar 1 st ,

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Coarse-grained models for PDEs with random coefficients

  • C. Grigo and P.-S. Koutsourelakis

Continuum Mechanics Group Department of Mechanical Engineering Technical University of Munich SIAM CSE Atlanta

1 Mar 2017

SIAM CSE 17 — Mar 1st, 2017 Reduced order modeling of SPDE’s 1/26

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Stochastic PDE with random coefficients

Stochastic PDE: K (x, λ(x, ξ))u(x, λ(x, ξ)) = f(x), +B.C.

Figure: Random process λ(x, ξ) leads to random solutions u(x, ξ).

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Outline

1 The Full-Order Model 2 A generative Bayesian surrogate model

Model training

3 Sample problem: 2D stationary heat equation

Model specifications Feature functions

4 Results 5 Summary

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The Full-Order Model (FOM)

Discretize K (x, λ(x, ξ))u(x, λ(x, ξ)) = f(x), +B.C. to a set of algebraic equations rf(U f, λf(ξ)) = 0 Usually large (Nequations ∼ millions) Expensive, repeated evaluations for UQ (and various deterministic tasks, e.g. optimization/control, inverse problems)

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Surrogate models

Idea: Replace FOM U f = U f(λf) by cheaper, yet inaccurate input-output map U f = f(λf; θ) based

  • n training data D =
  • U (i)

f , λ(i) f

N

i=1

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Surrogate models

Idea: Replace FOM U f = U f(λf) by cheaper, yet inaccurate input-output map U f = f(λf; θ) based

  • n training data D =
  • U (i)

f , λ(i) f

N

i=1

Problem: High dimensional uncertainties λf - learning direct functional mapping (e.g. PCE [Gahem, Spanos 1991] , GP [Rasmussen 2006], neural nets [Bishop 1995]) will fail

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Surrogate models

Idea: Replace FOM U f = U f(λf) by cheaper, yet inaccurate input-output map U f = f(λf; θ) based

  • n training data D =
  • U (i)

f , λ(i) f

N

i=1

Problem: High dimensional uncertainties λf - learning direct functional mapping (e.g. PCE [Gahem, Spanos 1991] , GP [Rasmussen 2006], neural nets [Bishop 1995]) will fail Solution: Coarse-grained model: Use models based on coarser discretization of PDE, U c = U c(λc)

SIAM CSE 17 — Mar 1st, 2017 Reduced order modeling of SPDE’s 5/26

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Surrogate models

Idea: Replace FOM U f = U f(λf) by cheaper, yet inaccurate input-output map U f = f(λf; θ) based

  • n training data D =
  • U (i)

f , λ(i) f

N

i=1

Problem: High dimensional uncertainties λf - learning direct functional mapping (e.g. PCE [Gahem, Spanos 1991] , GP [Rasmussen 2006], neural nets [Bishop 1995]) will fail Solution: Coarse-grained model: Use models based on coarser discretization of PDE, U c = U c(λc) Question: Relation between U f and coarse output U c, but also relation between fine/coarse inputs λf, λc

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Coarse-graining of SPDE’s

Retain as much as possible information on U f during coarse-graining, i.e. Information bottleneck [Tishby, Pereira, Bialek, 1999] maxθ I(λc, U f; θ) s.t. I(λf, λc; θ) ≤ I0

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Concept: Coarse grain random field λ,. . .

Probabilistic mapping λf → λc : pc(λc|λf, θc) Goal: Prediction of U f, not reconstruction of λf!

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. . . solve ROM and reconstruct U f from U c

λc → U c: solve rc(U c, λc) = 0 Decode via coarse-to-fine map U c → U f : pcf(U f|U c, θcf)

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Graphical Bayesian model

Figure: Bayesian network defining ¯ p(U f|λf, θc, θcf).

¯ p(U f|λf, θc, θcf) =

  • pcf(U f|U c, θcf)p(U c|λc)pc(λc|λf, θc)dU cdλc

=

  • pcf(U f|U c(λc), θcf)pc(λc|λf, θc)dλc.

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Model training

Maximum likelihood:

θ∗

c

θ∗

cf

  • = arg max

θc,θcf N

  • i=1

log ¯ p(U (i)

f |λ(i) f , θc, θcf)

Maximum posterior: θ∗

c

θ∗

cf

  • = arg max

θc,θcf N

  • i=1

log ¯ p(U (i)

f |λ(i) f , θc, θcf) + log p(θc, θcf)

Data: λ(i)

f

∼ p(λ(i)

f ),

U (i)

f

= U f(λ(i)

f ).

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Expectation - Maximization

¯ p(U (i)

f |λ(i) f , θc, θcf) =

  • pcf(U (i)

f |U c(λ(i) c ), θcf)pc(λ(i) c |λ(i) f , θc)dλ(i) c

→ Likelihood contains N integrals over N latent variables λc → Use Expectation-Maximization algorithm [Dempster, Laird, Rubin 1977] : find lower bound log(¯ p(U (i)

f |λ(i) f , θc, θcf))

≥

  • q(i)(λ(i)

c ) log

  • pcf(U (i)

f |U c(λ(i) c ), θcf)pc(λ(i) c |λ(i) f , θc)

q(i)(λ(i)

c )

  • dλ(i)

c

= F (i)(θ; q(i)

t (λ(i) c )),

where θ = [θc, θcf].

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Expectation-Maximization algorithm

Maximize iteratively:

E-step: Find optimal q(i)

t (λ(i) c ) given current estimate θt of optimal θ

and compute expectation values (MCMC, VI, EP) M-step: Maximize lower bound Ft(θ) =

i F (i) t (θ; q(i) t (λ(i) c )) w.r.t. θ to

find θt+1

Figure: Expectation-Maximization algorithm illustration

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Expectation-Maximization algorithm

Maximize iteratively:

E-step: Find optimal q(i)

t (λ(i) c ) given current estimate θt of optimal θ

and compute expectation values (MCMC, VI, EP) M-step: Maximize lower bound Ft(θ) =

i F (i) t (θ; q(i) t (λ(i) c )) w.r.t. θ to

find θt+1

Figure: Expectation-Maximization algorithm illustration

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Expectation-Maximization algorithm

Maximize iteratively:

E-step: Find optimal q(i)

t (λ(i) c ) given current estimate θt of optimal θ

and compute expectation values (MCMC, VI, EP) M-step: Maximize lower bound Ft(θ) =

i F (i) t (θ; q(i) t (λ(i) c )) w.r.t. θ to

find θt+1

Figure: Expectation-Maximization algorithm illustration

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Expectation-Maximization algorithm

Maximize iteratively:

E-step: Find optimal q(i)

t (λ(i) c ) given current estimate θt of optimal θ

and compute expectation values (MCMC, VI, EP) M-step: Maximize lower bound Ft(θ) =

i F (i) t (θ; q(i) t (λ(i) c )) w.r.t. θ to

find θt+1

Figure: Expectation-Maximization algorithm illustration

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Expectation-Maximization algorithm

Maximize iteratively:

E-step: Find optimal q(i)

t (λ(i) c ) given current estimate θt of optimal θ

and compute expectation values (MCMC, VI, EP) M-step: Maximize lower bound Ft(θ) =

i F (i) t (θ; q(i) t (λ(i) c )) w.r.t. θ to

find θt+1

Figure: Expectation-Maximization algorithm illustration

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Sample problem: 2D heat equation with random coefficients

∇x(−λ(x, ξ(x))∇xT(x, ξ(x))) = 0, +B.C. where ξ(x) ∼ GP(0, cov(xi, xj)) with cov(xi, xj) = exp

  • −|xi − xj|2

l2

  • ,

and λ(x, ξ(x)) =

  • λhi,

if ξ(x) > c, λlo,

  • therwise

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FOM data samples

Figure: Data samples for φhi = 0.35, l = 0.098, c = 100

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FOM data samples

Figure: Data samples for φhi = 0.35, l = 0.098, c = 100

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FOM data samples

Figure: Data samples for φhi = 0.35, l = 0.098, c = 100

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FOM data samples

Figure: Data samples for φhi = 0.35, l = 0.098, c = 100

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Model specifications

λf → λc = ezc : zc,k =

Nfeatures

  • j=1

θc,jϕj(λf,k) + σkZk, Zk ∼ N(0, 1), U c → U f : pcf(U f|U c(zc), θcf) = N(U f|W U c(zc), S) with feature functions ϕi, coarse-to-fine projection W , diagonal covariance S = diag(s).

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Feature functions ϕi(λf,k)

Any function ϕi : (R+)dim(λf,k) → R admissible Could/should be guided by physical insight: Effective-medium approximations

Self-consistent approximation (SCA)[Bruggeman 1935], Maxwell-Garnett approximation (MGA)[Maxwell 1873], Differential effective medium approximation (DEM)

[Bruggeman 1935]. . .

Morphology-describing features:

Lineal path function[Lu, Torquato 1992], (Convex) Blob area, Blob extent, Distance transformations. . . Left: Convex area (blue), max. extent (red), pixel-cross (green). Right: distance transform

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Sparsity priors

Strategy: Include as many features ϕj as possible, employ sparsity prior for feature selection Laplace prior (LASSO): p(θc,i) ∝ exp {−√γ |θc,i|} , ARD prior: p(θc,i) ∝ ∞ 1 τi N(θc,i|0, τi)dτi = 1 |θc,i|

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Which features are activated?

The higher the contrast, the more geometry matters

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Learned effective property λc

Note that pc(λc|λf, θc) = N(log λc|Φθc, Σ = diag(σ2)), and we plot λcpc = Φθc + 1

2σ2 SIAM CSE 17 — Mar 1st, 2017 Reduced order modeling of SPDE’s 19/26

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How many training samples do we need?

Few data is needed, errors converge quickly The finer the coarse mesh, the better the predictions The finer the coarse mesh, the less data is needed But: the finer the coarse mesh, the more expensive the training/predictions

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Predictions

Figure: Histogrammatic predictive distribution of temperature at lower right corner, ¯ p(Uf,lr|λf, θ)

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Predictions

Figure: Predictions on different test data samples for Nel,c = 8 × 8, φhi = 0.2, l = 0.078 and c = λhi

λlo = 10. Colored: U f, blue: U f¯ p,

grey: ±σ.

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictions

Figure: Test sample 3 from different angles

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Predictive uncertainty

Figure: Optimal variances σ∗2 of pc (l.) and optimal variances s of pcf.

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Scaling of the algorithm

Training: Quantity N Scaling #Data O(N) dim(λf) ? dim(U f) O(N) dim(λc), dim(Uc) O(N3) dim(θc) O(N3) Predictions: Quantity N Scaling #Data O(1) dim(λf) ? dim(U f) O(N) dim(λc), dim(Uc) O(N3) dim(θc) O(N)

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Summary & Outlook

Summary

Replace FOM by cheaper, but less accurate ROM Learn probabilistic output-output, but also input/input mappings between fine and coarse solver Predict by sampling λc, solving coarse model, sampling U f Potentially find interpretable features for effective material properties

Outlook

Anisotropic λc Account for correlations among λc,k’s Adaptive coarse mesh refinement

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