Concepts in Global Sensitivity Analysis IMA UQ Short Course, June 23, 2015

Transcription

1 Concepts in Global Sensitivity Analysis IMA UQ Short Course, June 23, 2015 A good reference is Global Sensitivity Analysis: The Primer. Saltelli, et al. (2008) Paul Constantine Colorado School of Mines inside.mines.edu/~pconstan WARNING: These slides are meant to complement the oral presentation in the short course. Use out of context at your own risk.

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3 Von Neumann, John, and Herman H. Goldstine. "Numerical inverting of matrices of high order." Bulletin of the American Mathematical Society (1947):

4 What kinds of science/engineering models do you care about? Do you have a simulation that you trust? What are the inputs and outputs? How would you characterize the uncertainty in the inputs? In other words, what do you know about the unknown inputs? What question are you trying to answer with your model?

5 f(x)

6 x Finite dimensional vector Independent components Centered and scaled to remove units

7 Response 1 Baseline Time Which perturbation shows the largest change? Perturbation 2 Baseline Response Perturbation 1 Baseline Response Time 2- norm difference 20.5 infinity- norm difference 2.0 difference at final 5me Time 2- norm difference 31.6 infinity- norm difference 1.8 difference at final 5me 0.0

8 f Scalar-valued Smooth No noise!

9 Sensitivity analysis seeks to identify the most important parameters. What are the most important parameters in your model? What are the least important parameters? What does it mean for a parameter to be important?

10 i (x) Derivatives measure local sensitivity. But we want something global.

11 Some Global Sensitivity Metrics 1. Morris elementary effects 2. Sobol sensitivity indices 3. Mean (squared) derivatives 4. Active subspaces

12 Morris Elementary Effects (Like bad approximations to average derivatives) x 2 p-level grid x 1 Step size h 2 2n p 1,n=1,...,p 1 Elementary effect EE ij (h) = f(x j + he i ) f(x j ) h Sensitivity indices µ i (h) = 1 N µ i (h) = 1 N NX EE ij (h) j=1 NX EE ij (h) j=1

13 Variance-based decompositions f(x) =f 0 mx + f i (x i ) + i=1 mx i=1 j>i mx f i,j (x i,x j ) + f 1,...,m (x 1,...,x m ) constant functions of one variable functions of two variables functions of 3, 4, variables function of m variables

14 Variance-based decompositions f 0 = E [f] orthogonal functions f i = E [f x i ] f 0 f i,j = E [f x i,x j ] X f i f 0 i. f 1,...,m = f(x) everything else Decomposition of variance Var [f] = X i Var [f i ]+ X i,j Var [f i,j ]+ + Var [f 1,...,m ]

15 Sobol indices First order sensitivity index Interaction effects S i = Var [f i] Var [f] S i1,...,i k = Var [f i 1,...,i k ] Var [f] Total effects S T 1 = S 1 + S 1,2 + S 1,3 + S 1,2,3 (e.g., sum everything with a 1 ) (PAUL: Mention the relationship to polynomial chaos.)

16 Mean (squared) derivatives 2 # i i Kucherenko, et al., DGSM, RESS (2008)

17 Let s play! Think of an interesting bivariate function.

18 Estimating with Monte Carlo is loud Monte Carlo Error Number of samples

19 What is it goood for? Sensitivity metrics can be hard to interpret if not zero. May provide or confirm understanding. Lots of ideas for using them as weights for anisotropic approximation schemes. Would like to use them to reduce the dimension.

20 AUDIENCE POLL How many dimensions is high dimensions?

21 APPROXIMATION f(x) f(x) Z INTEGRATION f(x) dx OPTIMIZATION minimize x f(x)

22 Dimension 10 points / dimension 1 second / evaluation s ~ 1.6 min 3 1,000 ~ 16 min 4 10,000 ~ 2.7 hours 5 100,000 ~ 1.1 days 6 1,000,000 ~ 1.6 weeks 20 1e20 3 trillion years (240x age of the universe)

23 Reduced order models Dimension 10 points / dimension 1 second / evaluation s ~ 1.6 min 3 1,000 ~ 16 min 4 10,000 ~ 2.7 hours 5 100,000 ~ 1.1 days 6 1,000,000 ~ 1.6 weeks 20 1e20 3 trillion years (240x age of the universe)

24 Better designs Dimension 10 points / dimension 1 second / evaluation s ~ 1.6 min 3 1,000 ~ 16 min 4 10,000 ~ 2.7 hours 5 100,000 ~ 1.1 days 6 1,000,000 ~ 1.6 weeks 20 1e20 3 trillion years (240x age of the universe)

25 Dimension reduction Dimension 10 points / dimension 1 second / evaluation s ~ 1.6 min 3 1,000 ~ 16 min 4 10,000 ~ 2.7 hours 5 100,000 ~ 1.1 days 6 1,000,000 ~ 1.6 weeks 20 1e20 3 trillion years (240x age of the universe)

26 f(x 1,x 2 ) = exp(0.7x x 2 ) direction of flat direction of change

27 $27 Coupon code: BKSL15 bookstore.siam.org/sl02/

28 DEFINE the active subspace. Consider a function and its gradient vector, f = f(x), x 2 R m, rf(x) 2 R m, : R m! R + The average outer product of the gradient and its eigendecomposition, Z C = (r x f)(r x f) T dx = W W T Partition the eigendecomposition, = apple 1 2, W = W 1 W 2, W 1 2 R m n Rotate and separate the coordinates, active variables x = WW T x = W 1 W T 1 x + W 2 W T 2 x = W 1 y + W 2 z inactive variables

29 Z r x f r x f T dx VS. Z xx T dx

30 The eigenvectors indicate perturbations that change the function more, on average. LEMMA 1: i = Z (r x f) T w i 2 dx, i =1,...,m LEMMA 2: Z Z (r y f) T (r y f) dx = n (r z f) T (r z f) dx = n m

31 DISCOVER the active subspace with random sampling. Draw samples: x j Compute: f j = f(x j ) and r x f j = r x f(x j ) Approximate with Monte Carlo C Equivalent to SVD of samples of the gradient. 1 N NX r x f j r x fj T = Ŵ ˆ Ŵ T j=1 1 T p rx f 1 r x f N = Ŵ pˆ ˆV N Called an active subspace method in T. Russi s 2010 Ph.D. thesis, Uncertainty Quantification with Experimental Data in Complex System Models

32 Let s be abundantly clear about the problem we are trying to solve. Low-rank approximation of the collection of gradients: q 1 T p rx f 1 r x f N Ŵ 1 ˆ 1 ˆV 1 N Low-dimensional linear approximation of the gradient: rf(x) Ŵ 1 a(x) Approximate a function of many variables by a function of a few linear combinations of the variables: f(x) g T Ŵ 1 x

33 What is the approximation error? f(x) g What is the effect of the approximate eigenvectors? Ŵ T x 1 How do you construct g?

34 [ Show them the animation! ]

35 EXPLOIT active subspaces for response surfaces with conditional averaging. Define the conditional expectation: Z g(y) = f(w 1 y + W 2 z) (z y) dz, f(x) g(w T 1 x) THEOREM: Z f(x) 1 2 g(w T 2 1 x) dx apple C P ( n m ) 1 2 Define the Monte Carlo approximation: ĝ(y) = 1 NX f(w 1 y + W 2 z i ), N THEOREM: Z f(x) i=1 1 2 ĝ(w T 2 1 x) dx z i (z y) apple C P 1+N 1 2 ( n m ) 1 2

36 EXPLOIT active subspaces for response surfaces with conditional averaging. Define the subspace error: " = dist(w 1, Ŵ 1) THEOREM: Z f(x) g(ŵ T x) dx Eigenvalues for inactive variables apple C P " ( n ) 1 2 +( n m ) 1 2 Subspace error Eigenvalues for active variables

37 THE BIG IDEA 1. Choose points in the domain of g. 2. Estimate conditional averages at each point. 3. Construct the approximation in n < m dimensions.

38 There s an active subspace in this parameterized PDE. Eigenvalues Est BI Index Two-d Poisson with 100-term Karhunen-Loeve coefficients 1 D 2 Subspace Distance BI Est Subspace Dimension r (aru) =1,x2D DIMENSION REDUCTION: 100 to 1 u =0,x2 1 n aru =0,x2 2

39 DIMENSION REDUCTION: 100 to 1 There s an active subspace in this parameterized PDE #10-3 Two-d Poisson with 100-term Karhunen-Loeve coefficients Quantity of Interest D Active variable 2 3 r (aru) =1,x2D u =0,x2 1 n aru =0,x2 2

40 Active subspaces can be sensitivity metrics. Components of first eigenvector β=0.01 β=1 Short correlation length Long correlation length Index

41 Questions? How do the active subspaces relate to the coordinate-based sensitivity metrics? How does this relate to PCA/POD? How many gradient samples do I need? How new is all this? Paul Constantine Colorado School of Mines