ACTIVE SUBSPACES AND SUFFICIENT DIMENSION REDUCTIONIN 3D RESISTIVE MAGNETOHYDRODYNAMICS

ANDREW GLAWS1, PAUL CONSTANTINE1, TIM WILDEY2, AND JOHN SHADID2

1Colorado School of Mines (contact:aglaws@mines.edu), 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)