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A CTIVE S UBSPACES AND S UFFICIENT D IMENSION R EDUCTION IN 3DR ESISTIVE M AGNETOHYDRODYNAMICS A NDREW G LAWS 1 ,P AUL C ONSTANTINE 1 ,T IM W ILDEY 2 , AND J OHN S HADID 2 1 Colorado School of Mines (contact:[email protected]), 2 Sandia National Laboratories A CTIVE S UBSPACES Assumptions: • Deterministic model with m scalar inputs f (x) ∈ R, x ∈ R m • Input parameters drawn according to a density function ρ(x) • Model output is differentiable with ∇f (x) ∈ R m Goal: • Determine directions in parameter space along which f (x) changes the most, on average Method: C = Z (∇f )(∇f ) T ρdx Interpretation: • Importance of basis vectors relates directly to eigenvalues λ i = Z (∇f ) T w i 2 ρdx • Dimension of reduced subspace not explicitly defined • Obtain a low-dimensional approximation to model f (x) ≈ g (W T 1 x), W T 1 x ∈ R n , n<m S UFFICIENT D IMENSION R EDUCTION Assumptions: • m scalar predictors to a scalar response y = f (x)+ , x ∈ R m ,y ∈ R Goal: • Determine subspace of predictor space where conditional distribution of response is unchanged Method: • Sliced inverse regression (SIR): C = Z E(x|y )E(x|y ) T dy • Sliced average variance estimate (SAVE): C = Z (I - cov (x|y )) 2 dy • Principal Hessian directions (PHD): C = E ( ∇ 2 E (y |x) ) Interpretation: • Reduction defined by subspace and not by any specific basis • Subspace maintains conditional distribution of response y |x ∼ y |η T 1 x, ··· , η T n x H ARTMANN P ROBLEM P arameter Description Range μ viscosity [0.05, 0.2] ρ density [1, 5] ∇P 0 volumetric applied pressure drop [0.5, 3] η resistivity [0.5, 3] B 0 applied magnetic field [0.1, 1] Active subspace analysis Sufficient dimension reduction (using SIR) Subspace convergence Subspace error = || ˆ W T 1 W 2 || 2 MHD G ENERATOR P arameter Description Range μ viscosity [0.001, 0.01] ∇P 0 volumetric applied pressure drop [0.1, 0.5] η resistivity [0.1, 10] ρ density [0.1, 10] Active subspace analysis Sliced inverse regression Sliced average variance estimate Principal Hessian directions C ONCLUSIONS For the simulations examined, we observed similar dimension reduction between ac- tive subspaces and the SDR methods. However, in studying the subspace conver- gence of the various techniques, we saw that active subspaces best approximates its true dimension reduction space when only a small number of samples is available (M< 100). Furthermore, active subspaces had much smaller dependence on the sampling than the SDR methods as shown through the bootstrap errors. R EFERENCES 1 Constantine, P. G. Active Subspaces: Emerging Ideas in Dimension Reduction for Parameter Studies. SIAM, Philadelphia (2015) 2 Cook, R. D. Regression Graphics: Ideas for Studying Regressions through Graphics. Wiley, New York(1998) 3 Cook, R. D. and Weisberg, S. “Sliced Inverse Regression for Dimensional Reduction: Comment.” Journal of the American Statistical Association. 86: 328-332. (1991) 4 Li, K. C. “On Principal Hessian Directions for Data Visualization and Dimension Reduction: Another Application of Stein’s Lemma.” Journal of the American Statistical Association. 87: 1025-1039. (1992) 5 Li, K. C. “Sliced Inverse Regression for Dimensional Reduction.” Journal of the American Statistical Association. 86: 316-342. (1991)
Transcript of 3DR Mandrewglaws.com/wp-content/uploads/2018/05/Active... · 1Constantine, P. G. Active Subspaces:...
ACTIVE SUBSPACES AND SUFFICIENT DIMENSION REDUCTIONIN 3D RESISTIVE MAGNETOHYDRODYNAMICS
ANDREW GLAWS1, PAUL CONSTANTINE1, TIM WILDEY2, AND JOHN SHADID2
1Colorado School of Mines (contact:[email protected]), 2Sandia National Laboratories
ACTIVE SUBSPACES
Assumptions:• Deterministic model with m scalar inputs
f(x) ∈ R, x ∈ Rm
• Input parameters drawn according to a density function ρ(x)• Model output is differentiable with∇f(x) ∈ Rm
Goal:• Determine directions in parameter space along which f(x)
changes the most, on average
Method:C =
∫(∇f) (∇f)T ρ dx
Interpretation:• Importance of basis vectors relates directly to eigenvalues
λi =
∫ ((∇f)T wi
)2ρ dx
• Dimension of reduced subspace not explicitly defined• Obtain a low-dimensional approximation to model
f(x) ≈ g(WT1 x), WT
1 x ∈ Rn, n < m
SUFFICIENT DIMENSION REDUCTION
Assumptions:• m scalar predictors to a scalar response
y = f(x) + ε, x ∈ Rm, y ∈ R
Goal:• Determine subspace of predictor space where conditional
distribution of response is unchanged
Method:• Sliced inverse regression (SIR):
C =
∫E(x|y)E(x|y)T dy
• Sliced average variance estimate (SAVE):
C =
∫(I − cov (x|y))2 dy
• Principal Hessian directions (PHD):
C = E(∇2E (y|x)
)Interpretation:• Reduction defined by subspace and not by any specific basis• Subspace maintains conditional distribution of response
y|x ∼ y|ηT1 x, · · · ,ηT
nx
HARTMANN PROBLEM
Parameter Description Rangeµ viscosity [0.05, 0.2]ρ density [1, 5]∇P0 volumetric applied pressure drop [0.5, 3]η resistivity [0.5, 3]B0 applied magnetic field [0.1, 1]
Active subspace analysis
Sufficient dimension reduction (using SIR)
Subspace convergence
Subspace error = ||WT1 W2||2
MHD GENERATOR
Parameter Description Rangeµ viscosity [0.001, 0.01]∇P0 volumetric applied pressure drop [0.1, 0.5]η resistivity [0.1, 10]ρ density [0.1, 10]
Active subspace analysis
Sliced inverse regression
Sliced average variance estimate
Principal Hessian directions
CONCLUSIONS
For the simulations examined, we observed similar dimension reduction between ac-tive subspaces and the SDR methods. However, in studying the subspace conver-gence of the various techniques, we saw that active subspaces best approximates itstrue dimension reduction space when only a small number of samples is available(M < 100). Furthermore, active subspaces had much smaller dependence on thesampling than the SDR methods as shown through the bootstrap errors.
REFERENCES1 Constantine, P. G. Active Subspaces: Emerging Ideas in Dimension Reduction for Parameter Studies. SIAM, Philadelphia (2015)2 Cook, R. D. Regression Graphics: Ideas for Studying Regressions through Graphics. Wiley, New York (1998)3 Cook, R. D. and Weisberg, S. “Sliced Inverse Regression for Dimensional Reduction: Comment.” Journal of the American
Statistical Association. 86: 328-332. (1991)4 Li, K. C. “On Principal Hessian Directions for Data Visualization and Dimension Reduction: Another Application of Stein’s
Lemma.” Journal of the American Statistical Association. 87: 1025-1039. (1992)5 Li, K. C. “Sliced Inverse Regression for Dimensional Reduction.” Journal of the American Statistical Association. 86: 316-342.
(1991)
