ACTIVE SUBSPACES in PDEs with many inputs PAUL CONSTANTINE Assistant Professor Department of Computer Science University of Colorado, Boulder activesubspaces.org! @DrPaulynomial! SLIDES AVAILABLE UPON REQUEST DISCLAIMER: These slides are meant to complement the oral presentation. Use out of context at your own risk. 

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Selected parameterized PDE literature My personal bibliography Ghanem and Spanos (1991); Babuška, Tempone, and Zouraris (2004); Babuška, Nobile, and Tempone (2007), Nobile, Tempone, and Webster (2008); Doostan and Iaccarino (2009); Doostan, Validi, and Iaccarino (2013); Doostan and Owhadi (2011); Gunzburger, Webster, and Zhang (2014); Chen, Gittelson, Jakeman, and Xiu (2015) Cohen, DeVore, and Schwab (2010, 2011); Chkifa, Cohen, and Schwab (2014); Cohen and DeVore (2015) Prud’homme, Rovas, Veroy, Machiels, Maday, Patera, Turinici (2002); Barrault, Maday, Nguyen, and Patera (2004); Rozza, Huynh, and Patera (2007); Haasdonk and Ohlberger (2008) Graham, Kuo, Nuyens, Scheichl, and Sloan (2012); Kuo and Nuyens (2016); Parametric PDEs: Sparse or Low-Rank Approximations? Bachmayr, Cohen, and Dahmen (arXiv, 2017) 

How many dimensions is high dimensions? Goal is to estimate … •  the solution map •  spatially varying statistics •  statistics of output of interest with as few function evaluations as possible. 

r · (aru) = 1 

D 1 2 The PDE The coefficients r · (aru) = 1, s 2 D u = 0, s 2 1 n · aru = 0, s 2 2 log(a(s, x)) = m X k=1 p ✓k k(s) xk Cov( s1, s2) = 2 exp ✓ ks1 s2 k1 2 ` ◆ The quantity of interest f( x ) = Z u( s , x ) d 2 •  100-term KL •  Gaussian r.v.’s Rough fields Spatial average over Neumann boundary Constantine, Dow, and Wang (2014) 

Constantine, Dow, and Wang (2014) 

Long length scale, ` = 1 log( a (s , x)) u( s , x ) Constantine, Dow, and Wang (2014) 

Short length scale, log( a (s , x)) u( s , x ) ` = 0.01 Constantine, Dow, and Wang (2014) 

f( x ) ⇢( x ) rf( x ) PDE solution’s spatial average along the Neumann boundary Standard Gaussian density Gradient computed with adjoint equations Constantine, Dow, and Wang (2014) NOT SOLUTION APPROXIMATION. NO MODEL REDUCTION IN SOLUTION SPACE. 

C = Z rf( x ) rf( x )T ⇢( x ) d x = W ⇤W T Define the active subspace Consider a function and its gradient vector, The average outer product of the gradient and its eigendecomposition, Partition the eigendecomposition, Rotate and separate the coordinates, ⇤ =  ⇤1 ⇤2 , W = ⇥ W 1 W 2 ⇤ , W 1 2 Rm⇥n x = W W T x = W 1W T 1 x + W 2W T 2 x = W 1y + W 2z active variables inactive variables f = f( x ), x 2 Rm, rf( x ) 2 Rm, ⇢ : Rm ! R + Constantine, Dow, and Wang (2014) 

i = Z w T i rf( x ) 2 ⇢( x ) d x , i = 1, . . . , m LEMMA Constantine, Dow, and Wang (2014) average, squared, directional derivative along eigenvector eigenvalue i > j | f( x + h wi) f( x ) | > | f( x + h wj) f( x ) | For small , on average, h WHAT DOES IT MEAN? 

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

Index 1 2 3 4 5 6 Eigenvalues 10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 Est BI Index 1 2 3 4 5 6 Eigenvalues 10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 Index 1 2 3 4 5 6 Eigenvalues 10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 Est BI Index 1 2 3 4 5 6 Eigenvalues 10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 Long length scale, ` = 1 Short length scale, ` = 0.01 Eigenvalues of Z rf rfT ⇢ d x 

Parameter index 0 50 100 Eigenvector components -1 -0.5 0 0.5 1 Parameter index 0 50 100 Eigenvector components -1 -0.5 0 0.5 1 First eigenvector of Z rf rfT ⇢ d x Long length scale, ` = 1 Short length scale, ` = 0.01 

-4 -2 0 2 4 Quantity of interest #10-3 0 1 2 3 4 5 -4 -2 0 2 4 Quantity of interest #10-4 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 Plotting versus Long length scale, ` = 1 Short length scale, ` = 0.01 f( x ) w T x w T x w T x 

-4 -2 0 2 4 Quantity of interest 10-10 10-5 100 105 Index 0 2 4 6 Eigenvalues 10-10 100 1010 Est BI Index 0 2 4 6 Eigenvalues 10-10 100 1010 ` = 1, 2 = 1/64 w T x 

` = 1, 2 = 1/16 -4 -2 0 2 4 Quantity of interest 10-10 10-5 100 105 Index 0 2 4 6 Eigenvalues 10-10 100 1010 Est BI Index 0 2 4 6 Eigenvalues 10-10 100 1010 w T x 

-4 -2 0 2 4 Quantity of interest 10-10 10-5 100 105 Index 0 2 4 6 Eigenvalues 10-10 100 1010 Est BI Index 0 2 4 6 Eigenvalues 10-10 100 1010 ` = 1, 2 = 1/4 w T x 

-4 -2 0 2 4 Quantity of interest 10-10 10-5 100 105 Index 0 2 4 6 Eigenvalues 10-10 100 1010 Est BI Index 0 2 4 6 Eigenvalues 10-10 100 1010 ` = 1, 2 = 1 w T x 

-4 -2 0 2 4 Quantity of interest 10-10 10-5 100 105 Index 0 2 4 6 Eigenvalues 10-10 100 1010 Est BI Index 0 2 4 6 Eigenvalues 10-10 100 1010 ` = 1, 2 = 4 w T x 

-4 -2 0 2 4 Quantity of interest 10-10 10-5 100 105 Index 0 2 4 6 Eigenvalues 10-10 100 1010 Est BI Index 0 2 4 6 Eigenvalues 10-10 100 1010 ` = 1, 2 = 16 w T x 

` = 1, 2 = 64 -4 -2 0 2 4 Quantity of interest 10-10 10-5 100 105 Index 0 2 4 6 Eigenvalues 10-10 100 1010 Est BI Index 0 2 4 6 Eigenvalues 10-10 100 1010 w T x 

` = 0.01, 2 = 1/64 -4 -2 0 2 4 Quantity of interest #10-3 0 0.5 1 1.5 Index 1 2 3 4 5 6 Eigenvalues 10-16 10-14 10-12 10-10 10-8 10-6 Est BI Index 1 2 3 4 5 6 Eigenvalues 10-16 10-14 10-12 10-10 10-8 10-6 w T x 

` = 0.01, 2 = 1/16 -4 -2 0 2 4 Quantity of interest #10-3 0 0.5 1 1.5 Index 1 2 3 4 5 6 Eigenvalues 10-16 10-14 10-12 10-10 10-8 10-6 Est BI Index 1 2 3 4 5 6 Eigenvalues 10-16 10-14 10-12 10-10 10-8 10-6 w T x 

-4 -2 0 2 4 Quantity of interest #10-3 0 0.5 1 1.5 Index 1 2 3 4 5 6 Eigenvalues 10-16 10-14 10-12 10-10 10-8 10-6 Est BI Index 1 2 3 4 5 6 Eigenvalues 10-16 10-14 10-12 10-10 10-8 10-6 ` = 0.01, 2 = 1/4 w T x 

-4 -2 0 2 4 Quantity of interest #10-3 0 0.5 1 1.5 Index 1 2 3 4 5 6 Eigenvalues 10-16 10-14 10-12 10-10 10-8 10-6 Est BI Index 1 2 3 4 5 6 Eigenvalues 10-16 10-14 10-12 10-10 10-8 10-6 ` = 0.01, 2 = 1 w T x 

-4 -2 0 2 4 Quantity of interest #10-3 0 0.5 1 1.5 Index 1 2 3 4 5 6 Eigenvalues 10-16 10-14 10-12 10-10 10-8 10-6 Est BI Index 1 2 3 4 5 6 Eigenvalues 10-16 10-14 10-12 10-10 10-8 10-6 ` = 0.01, 2 = 4 w T x 

-4 -2 0 2 4 Quantity of interest #10-3 0 0.5 1 1.5 Index 1 2 3 4 5 6 Eigenvalues 10-16 10-14 10-12 10-10 10-8 10-6 Est BI Index 1 2 3 4 5 6 Eigenvalues 10-16 10-14 10-12 10-10 10-8 10-6 ` = 0.01, 2 = 16 w T x 

-4 -2 0 2 4 Quantity of interest #10-3 0 0.5 1 1.5 Index 1 2 3 4 5 6 Eigenvalues 10-16 10-14 10-12 10-10 10-8 10-6 Est BI Index 1 2 3 4 5 6 Eigenvalues 10-16 10-14 10-12 10-10 10-8 10-6 ` = 0.01, 2 = 64 w T x 

<2 10-2 100 102 ^ 61 =^ 62 101 102 103 104 105 <2 10-2 100 102 ^ 61 =^ 62 100 101 102 103 104 105 Long length scale, ` = 1 Short length scale, ` = 0.01 Plotting versus CONJECTURE: decreases linearly with log( 1/ 2) log( 2 ) log( 2 ) log( 1/ 2) 

Some numerical evidence that the low- dimensional structure in some common parameterized PDEs looks like active subspaces. TODO: Prove it! PAUL CONSTANTINE Assistant Professor University of Colorado, Boulder activesubspaces.org! @DrPaulynomial! TAKE HOMES Active Subspaces SIAM (2015)