Active subspace poster

Poster presented during Purdue ME prospective graduate students visitation in 2016.

Transcript

1. Gaussian processes with built-in dimensionality reduction Rohit Tripathy, Ilias Bilionis, Marcial Gonzalez School of Mechanical Engineering, Purdue University 1. High dimensional uncertainty quantification is a hard problem. Why? Curse of dimensionality. 2. Traditional approach – Monte Carlo (inefficient). 3. Surrogate models (GP, Neural net, SVM etc) – do not scale well with high input dimension. 4. Look for special structure in inputs (active subspace (AS)) . 1. Use Eigen decomposition of an empirical “covariance” matrix computed using gradients of the quantity of interest (QoI). 2. Problems? i. Need gradients. Ii. Observational Noise. Iii. Code may be expensive-to-evaluate black box function. Methodology Our approach 1. Probabilistic framework (deal with observational noise and quantify epistemic uncertainty). 2. Novel Gaussian process regression – projection matrix as covariance kernel hyperparameter (bypasses need for gradients). 3. Bayesian Information Criterion (BIC) – Automatic selection of AS dimensionality. Classic Approach to AS discovery Introduction Demonstration of the methodology Future Work Verification and validation of the methodology 1. Learn covariance kernel hyperparameters via an alternating two-step log likelihood maximization procedure with Projection matrix constrained to be in the Stiefel manifold. 2. Compute BIC score for each AS dimensionality. Synthetic example Elliptic PDE 1. 10 dimensional Input. 2. Quadratic link function. 1 dimensional active subspace. 2 dimensional active subspace. Long correlation length Short correlation length 1. 100 dimensional input. 2. 2 correlation lengths. 3. 1-d active subspace in both cases. Granular crystals (a) (b) 1. Propagation of soliton through 1-D chain of 47 granular particles. 2. Properties of soliton obtained through solution of a high dimensional non-linear dynamical system. 3. Uncertainty in particle radii, Young’s moduli and striker velocity (95 uncertain parameters). 4. Study amplitude, time of flight and half width for 2 selected particles. Joint dist. Particle 20 amplitude and time of flight. Marginal distribution particle 20 time of flight. Active subspace particle 20 amplitude. Active subspace particle 20 time of flight. 1. Fully Bayesian approach – specification of priors for all covariance kernel hyperparameters. 2. Derivation of MCMC schemes - selection of proposals that force projection matrix to remain on Stiefel manifold. NOTE • Article submitted to Journal of Computational Physics (arXiv link: http://arxiv.org/abs/1602.04550) • Code: https://github.com/PredictiveScienceLab/py-aspgp