Compressive sensing adaptation for polynomial chaos expansions

Abstract Basis adaptation in Homogeneous Chaos spaces rely on a suitable rotation of the underlying Gaussian germ. Several rotations have been proposed in the literature resulting in adaptations with different convergence properties. In this paper we present a new adaptation mechanism that builds on compressive sensing algorithms, resulting in a reduced polynomial chaos approximation with optimal sparsity. The developed adaptation algorithm consists of a two-step optimization procedure that computes the optimal coefficients and the input projection matrix of a low dimensional chaos expansion with respect to an optimally rotated basis. We demonstrate the attractive features of our algorithm through several numerical examples including the application on Large-Eddy Simulation (LES) calculations of turbulent combustion in a HIFiRE scramjet engine.

[1]  N. Cutland,et al.  On homogeneous chaos , 1991, Mathematical Proceedings of the Cambridge Philosophical Society.

[2]  Roger Ghanem,et al.  Stochastic convergence acceleration through basis enrichment of polynomial chaos expansions , 2008 .

[3]  Paul G. Constantine,et al.  Data-Driven Polynomial Ridge Approximation Using Variable Projection , 2017, SIAM J. Sci. Comput..

[4]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..

[5]  Roger Ghanem,et al.  Ingredients for a general purpose stochastic finite elements implementation , 1999 .

[6]  Habib N. Najm,et al.  Stochastic spectral methods for efficient Bayesian solution of inverse problems , 2005, J. Comput. Phys..

[7]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[8]  Neal E. Hass,et al.  HIFiRE Direct-Connect Rig (HDCR) Phase I Scramjet Test Results from the NASA Langley Arc-Heated Scramjet Test Facility , 2011 .

[9]  Stephen J. Wright.  Coordinate descent algorithms , 2015, Mathematical Programming.

[10]  Peter E. Thornton,et al.  DIMENSIONALITY REDUCTION FOR COMPLEX MODELS VIA BAYESIAN COMPRESSIVE SENSING , 2014 .

[11]  Nathan A. Baker,et al.  Enhancing sparsity of Hermite polynomial expansions by iterative rotations , 2015, J. Comput. Phys..

[12]  BabuskaIvo,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2007 .

[13]  Xun Huan,et al.  Global Sensitivity Analysis and Estimation of Model Error, Toward Uncertainty Quantification in Scramjet Computations , 2017, 1707.09478.

[14]  Lloyd G. Wilson,et al.  Opposed Jet Burner Extinction Limits: Simple Mixed Hydrocarbon Scramjet Fuels vs Air , 2007 .

[15]  Jiwen Liu,et al.  Combustor Operability and Performance Verification for HIFiRE Flight 2 , 2011 .

[16]  Roger Ghanem,et al.  Stochastic model reduction for chaos representations , 2007 .

[17]  Houman Owhadi,et al.  A non-adapted sparse approximation of PDEs with stochastic inputs , 2010, J. Comput. Phys..

[18]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[19]  Habib N. Najm,et al.  Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics , 2009 .

[20]  Roger G. Ghanem,et al.  Homogeneous chaos basis adaptation for design optimization under uncertainty: Application to the oil well placement problem , 2017, Artificial Intelligence for Engineering Design, Analysis and Manufacturing.

[21]  Zachary P. Vane,et al.  Large Eddy Simulation of the HIFiRE Direct Connect Rig Scramjet Combustor. , 2017 .

[22]  H. Najm,et al.  A stochastic projection method for fluid flow II.: random process , 2002 .

[23]  Douglas J. Dolvin,et al.  Hypersonic International Flight Research and Experimentation (HIFiRE) Fundamental Sciences and Technology Development Strategy , 2008 .

[24]  Roger G. Ghanem,et al.  Reduced Wiener Chaos representation of random fields via basis adaptation and projection , 2016, J. Comput. Phys..

[25]  Mark Gruber,et al.  HIFIRE Flight 2 Overview and Status Update 2011 , 2011 .

[26]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[27]  Roger Ghanem,et al.  Stochastic Finite Element Analysis for Multiphase Flow in Heterogeneous Porous Media , 1998 .

[28]  Richard Bellman,et al.  Adaptive Control Processes: A Guided Tour , 1961, The Mathematical Gazette.

[29]  Karl Pearson F.R.S..  LIII. On lines and planes of closest fit to systems of points in space , 1901 .

[30]  Ilias Bilionis,et al.  Gaussian processes with built-in dimensionality reduction: Applications in high-dimensional uncertainty propagation , 2016, 1602.04550.

[31]  Xun Huan,et al.  Global Sensitivity Analysis and Quantification of Model Error in Scramjet Computations , 2017 .

[32]  Douglas J. Dolvin,et al.  Hypersonic International Flight Research and Experimentation Technology Development and Flight Certification Strategy , 2009 .

[33]  H. H. Rachford,et al.  On the numerical solution of heat conduction problems in two and three space variables , 1956 .

[34]  Johan Larsson,et al.  Exploiting active subspaces to quantify uncertainty in the numerical simulation of the HyShot II scramjet , 2014, J. Comput. Phys..

[35]  Datta V. Gaitonde,et al.  Exploratory Simulations of the HIFiRE 2 Scramjet Flowpath , 2012 .

[36]  Karthik Duraisamy,et al.  Uncertainty quantification and error estimation in scramjet simulation , 2011 .

[37]  Habib N. Najm,et al.  Bayesian estimation of Karhunen-Loève expansions; A random subspace approach , 2016, J. Comput. Phys..

[38]  Lawrence Carin,et al.  Bayesian Compressive Sensing , 2008, IEEE Transactions on Signal Processing.

[39]  Michael Elad,et al.  Stable recovery of sparse overcomplete representations in the presence of noise , 2006, IEEE Transactions on Information Theory.

[40]  E. M. Wright,et al.  Adaptive Control Processes: A Guided Tour , 1961, The Mathematical Gazette.

[41]  Joseph C. Oefelein,et al.  Large eddy simulation of turbulent combustion processes in propulsion and power systems , 2006 .

[42]  P. Tsilifis,et al.  Gradient-informed basis adaptation for Legendre Chaos expansions , 2016, 1611.02754.

[43]  Roger Ghanem,et al.  Scales of fluctuation and the propagation of uncertainty in random porous media , 1998 .

[44]  J. Oefelein.  Simulation and analysis of turbulent multiphase combustion processes at high pressures , 1997 .

[45]  P. Spalart.  Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach , 1997 .

[46]  Khachik Sargsyan,et al.  Enhancing ℓ1-minimization estimates of polynomial chaos expansions using basis selection , 2014, J. Comput. Phys..

[47]  Qiqi Wang,et al.  Erratum: Active Subspace Methods in Theory and Practice: Applications to Kriging Surfaces , 2013, SIAM J. Sci. Comput..

[48]  Roger G. Ghanem,et al.  Basis adaptation in homogeneous chaos spaces , 2014, J. Comput. Phys..

[49]  Carl E. Rasmussen,et al.  Gaussian processes for machine learning , 2005, Adaptive computation and machine learning.

[50]  Fabio Nobile,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2007, SIAM Rev..

[51]  D. Xiu,et al.  Modeling uncertainty in flow simulations via generalized polynomial chaos , 2003 .

[52]  R. Ghanem,et al.  Bayesian adaptation of chaos representations using variational inference and sampling on geodesics , 2018, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[53]  Alireza Doostan,et al.  A weighted l1-minimization approach for sparse polynomial chaos expansions , 2013, J. Comput. Phys..

[54]  Gianluca Iaccarino,et al.  A least-squares approximation of partial differential equations with high-dimensional random inputs , 2009, J. Comput. Phys..

[55]  Saltelli Andrea,et al.  Global Sensitivity Analysis: The Primer , 2008 .

[56]  Roger G. Ghanem,et al.  Efficient Bayesian Experimentation Using an Expected Information Gain Lower Bound , 2015, SIAM/ASA J. Uncertain. Quantification.

[57]  P. Tsilifis,et al.  Reduced-dimensionality Legendre Chaos expansions via basis adaptation on 1d active subspaces , 2016 .

[58]  Jiwen Liu,et al.  Hydrocarbon-Fueled Scramjet Combustor Flowpath Development for Mach 6-8 HIFire Flight Experiments (Preprint) , 2008 .

[59]  Wotao Yin,et al.  A feasible method for optimization with orthogonality constraints , 2013, Math. Program..

[60]  W. T. Martin,et al.  The Orthogonal Development of Non-Linear Functionals in Series of Fourier-Hermite Functionals , 1947 .

[61]  Fabio Nobile,et al.  A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data , 2008, SIAM J. Numer. Anal..

[62]  S. Frick,et al.  Compressed Sensing , 2014, Computer Vision, A Reference Guide.

[63]  Bruno Sudret,et al.  Adaptive sparse polynomial chaos expansion based on least angle regression , 2011, J. Comput. Phys..

[64]  Xun Huan,et al.  Compressive Sensing with Cross-Validation and Stop-Sampling for Sparse Polynomial Chaos Expansions , 2017, SIAM/ASA J. Uncertain. Quantification.

[65]  H. Najm,et al.  Uncertainty quantification in reacting-flow simulations through non-intrusive spectral projection , 2003 .

[66]  John Red-Horse,et al.  Propagation of probabilistic uncertainty in complex physical systems using a stochastic finite element approach , 1999 .

[67]  J.-C. Pesquet,et al.  A Douglas–Rachford Splitting Approach to Nonsmooth Convex Variational Signal Recovery , 2007, IEEE Journal of Selected Topics in Signal Processing.

[68]  Roger Ghanem,et al.  Measure transformation and efficient quadrature in reduced‐dimensional stochastic modeling of coupled problems , 2011, 1112.4772.