Off-axis anisotropy in multivariate functions

Transcription

1 Off-axis anisotropy in multivariate functions PAUL CONSTANTINE Assistant Professor Department of Computer Science University of Colorado, Boulder Thanks to: Jeff Hokanson (CU Boulder) Andrew Glaws (CU Boulder, NREL) SLIDES AVAILABLE UPON REQUEST DISCLAIMER: These slides are meant to complement the oral presentation. Use out of context at your own risk.

2 Ebola transmission models Diaz, Constantine, Kalmbach, Jones, and Pankavich (2018) Aerospace design Lukaczyk, Palacios, Alonso, and Constantine (2014) Integrated hydrologic models Jefferson, Gilbert, Constantine, and Maxwell (2015) Automobile design Othmer, Lukaczyk, Constantine, and Alonso (2016) Hypersonic scramjet models Constantine, Emory, Larsson, and Iaccarino (2015) Magnetohydrodynamics models Glaws, Constantine, Shadid, and Wildey (2017) Solar cell models Constantine, Zaharatos, and Campanelli (2015) f (x) Lithium ion battery model Constantine and Doostan (2017)

3 PROPERTIES: Computer model of a physical system Several independent inputs Deterministic Continuous inputs / outputs Smoothness TO DO: APPROXIMATION f(x) f(x) Z INTEGRATION f(x)dx f(x) OPTIMIZATION minimize x f(x)

4 Structure-exploiting methods STRUCTURE f(x) f 1 (x 1 )+ + f m (x m ) METHODS Sparse grids [Bungartz & Griebel (2004)], HDMR [Sobol (2003)], ANOVA [Hoeffding (1948)], QMC [Niederreiter (1992)], f(x) rx k=1 f k,1 (x 1 ) f k,m (x m ) Separation of variables [Beylkin & Mohlenkamp (2005)], Tensor-train [Oseledets (2011)], Adaptive cross approximation [Bebendorff (2011)], Proper generalized decomposition [Chinesta et al. (2011)], px f(x) k=1 a k k (x), kak 0 p Compressed sensing [Donoho (2006), Candès & Wakin (2008)],

5 Even more understanding is lost if we consider each thing we can do to data only in terms of some set of very restrictive assumptions under which that thing is best possible--- assumptions we know we CANNOT check in practice.

6 The best way to fight the curse is to reduce the dimension. But what is dimension reduction? dimensional analysis [Barrenblatt (1996)] correlation-based reduction [Jolliffe (2002)] sensitivity analysis [Saltelli et al. (2008)]

7

8 Design a jet nozzle under uncertainty (DARPA SEQUOIA project) 10-parameter engine performance model (See animation at

9 Ridge approximations f(x) g(u T x) where U T : R m! R n g : R n! R f(x1,x2) x 2 x 1 Constantine, Eftekhari, Hokanson, and Ward (2017)

10 Ridge approximations f(x) g(u T x) A subset of related literature Approximation theory: Mayer et al. (2015), Pinkus (2015), Diaconis and Shahshahani (1984), Donoho and Johnstone (1989) Compressed sensing: Fornasier et al. (2012), Cohen et al. (2012), Tyagi and Cevher (2014) Statistical regression: Friedman and Stuetzle (1981), Ichimura (1993), Hristache et al. (2001), Xia et al. (2002) Uncertainty quantification & computational science: Tipireddy and Ghanem (2014); Lei et al. (2015); Stoyanov and Webster (2015); Tripathy, Bilionis, and Gonzalez (2016); Li, Lin, and Li (2016);

11 Do these structures arise in real models? (Or, why do I care?)

12 Evidence of 1d ridge structures across science and engineering models Hypersonic scramjet models Constantine, Emory, Larsson, and Iaccarino (2015)

13 Evidence of 1d ridge structures across science and engineering models Integrated jet nozzle models Alonso, Eldred, Constantine, Duraisamy, Farhat, Iaccarino, and Jakeman (2017)

14 Evidence of 1d ridge structures across science and engineering models Integrated hydrologic models Jefferson, Gilbert, Constantine, and Maxwell (2015)

15 Evidence of 1d ridge structures across science and engineering models Drag Active Variable Aerospace vehicle geometries Lift Active Variable 1 Lukaczyk, Constantine, Palacios, and Alonso (2014) 2

16 Evidence of 1d ridge structures across science and engineering models T-cell count In-host HIV dynamical models Loudon and Pankavich (2016)

17 Evidence of 1d ridge structures across science and engineering models P max (watts) Active Variable 1 Solar cell circuit models Constantine, Zaharatos, and Campanelli (2015)

18 Evidence of 1d ridge structures across science and engineering models Stagnation heat flux q st ŵq T x Stagnation pressure p st Atmospheric reentry vehicle model ŵp T x Cortesi, Constantine, Magin, and Congedo (hal, 2017)

19 Evidence of 1d ridge structures across science and engineering models 15 Average velocity f(x) 10 5 Induce magnetic f(x) field w1 T x w1 T x Magnetohydrodynamics generator model Glaws, Constantine, Shadid, and Wildey (2017)

20 Voltage [V] Evidence of 1d ridge structures across science and engineering models 3.70 Capacity [mah cm 2 ] wt x wt x Constantine and Doostan (2017) Lithium ion battery model

21 Evidence of 1d ridge structures across science and engineering models Automobile geometries Othmer, Lukaczyk, Constantine, and Alonso (2016)

22 Evidence of 1d ridge structures across science and engineering models Quantity of interest Long length scale 5 # r (aru) =1, s 2 D u =0, s 2 1 n aru =0, s Input field Solution Quantity of interest Short length scale # Short corr Long corr Constantine, Dow, and Wang (2014)

23 f(x) Jupyter notebooks: github.com/paulcon/as-data-sets

24 Ridge approximations What is the approximation error? What is U? f(x) g(u T x) What is g? Constantine, Eftekhari, Hokanson, and Ward (2017)

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Ridge approximations f(x) g(u T x) What is g? Use the conditional average: µ(y) = Z conditional density f(uy + V z) (z y) dz subspace coordinates complement subspace and coordinates µ(u T x) is the best approximation [Pinkus (2015)] <latexit sha1_base64="vubd4ije4p461ekohg46beucixo=">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</latexit> L 2 Constantine, Dow, and Wang (2014); Constantine, Eftekhari, Hokanson, and Ward (2017)

26 Define the active subspace The function, its gradient vector, and a given weight function: 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 = rf(x) rf(x) T (x) dx = W W T Some relevant literature Statistical regression: Samarov (1993), Hristache et al. (2001) Partition the eigendecomposition, Machine applelearning: 1 Rotate = and separate the, coordinates, W = Mukerjee, Wu, and Xiao (2010); Fukumizu and Leng (2014) active W 1 W 2, variables W 1 2 R m n inactive variables Detection and estimation 2 theory: van Trees (2001) x = WW T x = W 1 W T 1 x + W 2 W T 2 x = W 1 y + W 2 z Constantine, Dow, and Wang (2014)

27 Define the active subspace The function, its gradient vector, and a given weight function: 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 = rf(x) rf(x) T (x) dx = W W T Eigenvalues measure ridge structure with eigenvectors: eigenvalue i = Z w T i rf(x) 2 (x) dx, i =1,...,m average, squared, directional derivative along eigenvector Constantine, Dow, and Wang (2014)

28 Eigenvalues control the approximation error conditional average Poincaré constant f(x) µ(w T 1 x) L 2 ( ) apple C ( n m ) 1 2 active subspace eigenvalues associated with inactive subspace Constantine, Dow, and Wang (2014)

29 Estimate the active subspace with Monte Carlo (1) Draw samples: x j (x) (2) Compute: f j = f(x j ) and rf j = rf(x j ) (3) Approximate with Monte Carlo, and compute eigendecomposition C 1 N NX rf j rfj T = Ŵ ˆ Ŵ T j=1 Equivalent to SVD of samples of the gradient 1 p N rf1 rf N = Ŵ pˆ ˆV T Called an active subspace method in T. Russi s 2010 Ph.D. thesis, Uncertainty Quantification with Experimental Data in Complex System Models Constantine, Dow, and Wang (2014), Constantine and Gleich (2015, arxiv)

30 Remember the problem to solve Low-rank approximation of the collection of gradients: q 1 T p rf1 rf N Ŵ 1 ˆ 1 ˆV 1 N Low-dimensional linear approximation of the gradient: span (Ŵ 1) { rf(x) :x 2 supp (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

31 How many gradient samples? number of samples bound on gradient L 2 1 N = 2 k "2 log(m) eigenvalue error (w.h.p.) =) k ˆk apple k " dimension number of samples bound on gradient L 2 N = 1" 2 log(m) =) dist(w 1, Ŵ 1) apple 4 1 " n n+1 dimension subspace error (w.h.p.) Constantine and Gleich (2015) via Gittens and Tropp (2011), Stewart (1973)

32 In practice, bootstrap 10 4 True Est BI Eigenvalues Subspace Error True Est BI Index Subspace Dimension Eigenvalue estimates and subspace error estimates with bootstrap intervals from quadratic function of 10 variables Constantine and Gleich (2015, arxiv)

33 Effect of estimated eigenvectors? Recall the subspace error: " = dist(w 1, Ŵ 1) / O(eigval. error) n n+1 f(x) µ(ŵ T 1 x) apple C L 2 ( ) Eigenvalues for inactive variables " ( n ) 1 2 +( n m ) 1 2 Subspace error Eigenvalues for active variables Constantine, Dow, and Wang (2014)

34 Is the active subspace optimal? (No.)

35 <latexit 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An example where it doesn t work f(x 1,x 2 )=5x 1 + sin(10 x 2 ) Z C = rf rf T apple 25 0 = active subspace U = [0; 1] inactive subspace U = [1; 0] Constantine, Eftekhari, Hokanson, and Ward (2017)

36 Ridge approximations What is U? f(x) g(u T x) Define the error function: R(U) = 1 2 Z (f(x) best approximation µ(u T x)) 2 (x) dx Minimize the error: minimize U R(U) subject to U 2 G(n, m) Grassmann manifold Constantine, Eftekhari, Hokanson, and Ward (2017)

37 The active subspace is nearly stationary Assume (1) Lipschitz continuous function (2) Gaussian density function gradient on the Grassmann manifold k rr(w 1 )k F Lipschitz constant apple L dimensions 2m (m n) 2 ( n m ) 1 2 active subspace Frobenius norm eigenvalues associated with inactive subspace Constantine, Eftekhari, Hokanson, and Ward (2017)

38 Estimate the optimal subspace with discrete least squares (1) Choose points: x j (x) (2) Compute: f j = f(x j ) (3) Minimize the misfit minimize g2p p (R n ) U2G(n,m) NX j=1 2 f j g(u T x j ) Minimize over polynomials and subspaces Constantine, Eftekhari, Hokanson, and Ward (2017), Hokanson and Constantine (2018)

39 Two contenders for the least squares problem minimize g2p p (R n ) U2G(n,m) NX j=1 2 f j g(u T x j ) Variable projection Use pseudoinverse of Vandermonde matrix to express optimal polynomial coefficients Compute the derivative of the pseudoinverse of the Vandermonde matrix [Golub & Pereyra (1973)] on the Grassmann manifold [Edelman et al. (1998)] Alternating minimization Given subspace, fit polynomial Given polynomial coefficients, minimize over subspace Repeat Run Newton on loss function Constantine, Eftekhari, Hokanson, and Ward (2017), Hokanson and Constantine (2018)

40 Hokanson and Constantine (2018)

41 r (aru) =1, s 2 D u =0, s 2 1 n aru =0, s 2 2 Input field Solution Hokanson and Constantine (2018)

42 Details under the hood How do you choose points? How do you choose the subspace dimension? What is your polynomial basis for the ridge approximation? Grassmann gradient of pseudo-inverse is complicated but has some nice simplifications Open questions Can you use other bases (frames, RBFs, )? Is the continuous optimization problem well-posed? How does the discrete optimization relate to the continuous optimization? Constantine, Eftekhari, Hokanson, and Ward (2017), Hokanson and Constantine (2018)

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NEW STUFF :: The Lipschitz matrix For lower triangular L 2 R m m and R m, consider F(L, ) = n f : R m! R f(x) f(y) applekl T (x y)k 2 o Problem: Given f<latexit sha1_base64="j/8ynkd3l0kj6tdgu4+kkpwj58k=">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</latexit>, find Algorithm: x j (x) f j = f(x j ) (1) Choose points and compute (2) Solve symmetric minimize trace (M) M subject to (f i f j ) 2 apple (x i x j ) T M(x i x j ) (3) Square root factorization M = LL T Hokanson and Constantine (in prep.)

44 SUMMARY :: Why I like ridge structure (1) Exploitable + for dimension reduction, not just cheap surrogate (2) Insights + which variables are important (3) Discoverable / checkable + eigenvalues + non-residual metrics: E[Var[f U T x ]] + plots in 1 and 2d

45 TAKE HOMES The best way to fight the curse of dimensionality is to reduce the dimension! There are many notions of important subspaces; they arise in several applications. Important subspaces are discoverable and exploitable for answering science questions.

46 QUESTIONS? Are there other options for important directions? What is the trade-off between discovering the lowdimensional structure vs. solving the original problem? Why are these structures so pervasive? Active Subspaces SIAM (2015) What if my model doesn t fit your setup? (no gradients, multiple outputs, correlated inputs, ) PAUL CONSTANTINE Assistant Professor University of Colorado Boulder