ACTIVE SUBSPACES Emerging Ideas for Dimension Reduction in Parameter Studies PAUL CONSTANTINE Ben L. Fryrear Assistant Professor Applied Mathematics & Statistics Colorado School of Mines activesubspaces.org! @DrPaulynomial! DISCLAIMER: These slides are meant to complement the oral presentation. Use out of context at your own risk. 

Paul Diaz MS 2016 Zach Grey PhD 2019 Kerrek Stinson BS 2016 Andrew Glaws PhD 2018 

Gianluca Iaccarino Mechanical Engineering Juan Alonso Aeronautics and Astronautics Qiqi Wang Aeronautics and Astronautics Youssef Marzouk Aeronautics and Astronautics Reed Maxwell Hyrdological Science & Engineering Michael Wakin Electrical Engineering Stephen Pankavich Applied Mathematics Tan Bui-Thanh Aerospace Engineering David Gleich Computer Science Michael Eldred Optimization & Uncertainty Quantification John Shadid Computational Mathematics 

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To help scientists with complex, highly parameterized models answer their science questions WHY? 

x inputs parameters knobs data independent variables f calculations and computations outputs predictions quantities of interest dependent variables f( x ) 

f( x ) Uncertainty quantification for hypersonic scramjets (with G. Iaccarino, J. Larsson, M. Emory) Sensitivity analysis in hydrological models (with R. Maxwell, J. Jefferson, J. Gilbert) Shape optimization in aerospace vehicles (with J. Alonso, T. Lukaczyk) 

f( x ) Sensitivity analysis in solar cell models (with B. Zaharatos, M. Campanelli) Sensitivity analysis in HIV modeling (with T. Loudon, S. Pankavich) Sensitivity analysis in Ebola transmission models (with P. Diaz, S. Pankavich) Calibration of an atmospheric reentry vehicle model (with P. Congedo, T. Magin) 

f( x ) What is the range of predictions? Which inputs are most important? Which inputs make the model agree with data? 

Discover and exploit low-dimensional structure in the model HOW? 

How many dimensions is high dimensions? 

Number of parameters (the dimension) Number of model runs (at 10 points per dimension) Time for parameter study (at 1 second per run) 1 10 10 sec 2 100 ~ 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) REDUCED-ORDER MODELS BETTER DESIGNS DIMENSION REDUCTION 

WHAT? 

https://youtu.be/mJvKzjT6lmY 

WHAT IS THE ACTIVE SUBSPACE? Z rf( x ) rf( x )T ⇢( x ) d x = W ⇤W T The dominant eigenspace of a matrix derived from the model’s gradient gradient of the model output averaged over the input space eigenvalue decomposition 

WHAT DOES THE ACTIVE SUBSPACE DO? Enables low-dimensional models for high-dimensional parameter studies f( x ) ⇡ g(W T 1 x ) first few eigenvectors function of a few variables 

Seven parameters characterizing the operating conditions What is the range of pressures at the channel exit? P. Constantine, M. Emory, J. Larsson, and G. Iaccarino Exploiting active subspaces to quantify uncertainty in the numerical simulation of the HyShot II scramjet Journal of Computational Physics (2015) 

https://youtu.be/frWa-P__kms 

$27 VISIT THE SIAM BOOKSTORE! Coupon code: BKSL16 

Algorithmic challenges “Will this work on my model?” CHALLENGES 

How do active subspaces relate to [insert method]? What kinds of models does this work on? How new is all this? PAUL CONSTANTINE Ben L. Fryrear Assistant Professor Applied Mathematics & Statistics Colorado School of Mines activesubspaces.org! @DrPaulynomial! QUESTIONS?