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. ZACHARY GREY PhD Student Aerospace Engineering Sciences University of Colorado Boulder Robust Design Technical Lead Engineering Capability – Design Sciences Rolls-Royce Corporation
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 or PARALLEL PROCESSING
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) BETTER DESIGNS or ADAPTIVE SAMPLING
assumes a different type of low-dimensional structure. Sparse grids [Bungartz & Griebel (2004)] High Dimensional Model Representation [Sobol (2003)] ANOVA [Hoeffding (1948)] Quasi Monte Carlo [Niederreiter (1992)] Separation of variables [Beylkin & Mohlenkamp (2005)] Tensor-train [Oseledets (2011)] Adaptive cross approximation [Bebendorff (2011)] Proper generalized decomposition [Chinesta et al. (2011)] Compressed sensing [Donoho (2006)] … How do you know which technique to use?
5 10 15 f(x) Average velocity Glaws, Constantine, Shadid, and Wildey (2017) -1 0 1 wT 1 x 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 f(x) Induce magnetic field Evidence of 1d ridge structures across science and engineering models
Constantine, Dow, and Wang (2014) mean-squared, directional derivative along eigenvector eigenvalue Eigenvalues measure low-d structure with eigenvectors: E ⇥ rf( x ) rf( x )T ⇤ = W ⇤W T i = E ⇥ ( w T i rf( x ))2 ⇤ NUMERICS (1) gradients (adjoints, finite differences) (2) averages (Monte Carlo, numerical integration) (3) eigendecomposition Active subspaces
x ) g(UT x ) 2 i f( x ) ⇡ g(UT x ) What is U? Minimize an error (loss) function: Ridge approximations Constantine, Eftekhari, Hokanson, and Ward (2017), Hokanson and Constantine (2018) NUMERICS (1) averages (Monte Carlo, numerical integration) (2) optimization (on Grassmann manifold)
to reduce the dimension! Use practical techniques that assess low-dimensional structure. Dimension reduction approaches: dimensional analysis ✗ only based on quantities’ units PCA / covariance ✗ no connection to output of interest ✗ only finds linear relationships sensitivity analysis ✗ limited to coordinate-based reduction active subspaces ✓ verifiable structure ✓ enables visualization (1 and 2 dimensions) ✓ directions provide sensitivity analysis (what about nonlinear reduction?)
different disciplines shape optimization “infinite-dimensional” inputs, space of shapes image-valued / time-series inputs same as above transient models important directions change over time improved algorithms lower cost, better accuracy, greater reliability high-impact applications change the world!
well-defined problem (e.g., design optimization)? Do you have a reliable computational model with inputs and outputs of interest? Can you run your model number-of-parameters-squared times in a reasonable amount of time? If there is no strong evidence of exploitable low-dimensional structure, then do what you would otherwise do. OPTION 1: Work through publicly available material and examples on your own. OPTION 2: Work with Zach and me to understand how the low-dimensional structure, if present, can accelerate your workflow. ($$$)
model doesn’t fit your setup? (no gradients, multiple outputs, correlated inputs, …) PAUL CONSTANTINE Assistant Professor University of Colorado Boulder activesubspaces.org! @DrPaulynomial! QUESTIONS? Active Subspaces SIAM (2015)