On polynomial chaos expansion via gradient-enhanced ℓ1-minimization

Gradient-enhanced Uncertainty Quantification (UQ) has received recent attention, in which the derivatives of a Quantity of Interest (QoI) with respect to the uncertain parameters are utilized to improve the surrogate approximation. Polynomial chaos expansions (PCEs) are often employed in UQ, and when the QoI can be represented by a sparse PCE, ? 1 -minimization can identify the PCE coefficients with a relatively small number of samples. In this work, we investigate a gradient-enhanced ? 1 -minimization, where derivative information is computed to accelerate the identification of the PCE coefficients. For this approach, stability and convergence analysis are lacking, and thus we address these here with a probabilistic result. In particular, with an appropriate normalization, we show the inclusion of derivative information will almost-surely lead to improved conditions, e.g. related to the null-space and coherence of the measurement matrix, for a successful solution recovery. Further, we demonstrate our analysis empirically via three numerical examples: a manufactured PCE, an elliptic partial differential equation with random inputs, and a plane Poiseuille flow with random boundaries. These examples all suggest that including derivative information admits solution recovery at reduced computational cost. An approach for sparse polynomial chaos (PC) expansions including derivative information is presented.Method relies on ? 1 -minimization to solve for the PC coefficients.Formal analysis is presented to guarantee the stability and convergence of gradient-enhanced ? 1 -minimization.Three examples presented to illustrate (cost/accuracy) improvements achieved by gradient-enhanced ? 1 -minimization.

[1]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[2]  Arkadi Nemirovski,et al.  Accuracy Guarantees for ℓ1-Recovery , 2010, IEEE Trans. Inf. Theory.

[3]  Patrick Le Tallec,et al.  Numerical methods in sensitivity analysis and shape optimization , 2002, Modeling and simulation in science, engineering and technology.

[4]  Brandon A. Jones,et al.  Postmaneuver Collision Probability Estimation Using Sparse Polynomial Chaos Expansions , 2015 .

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

[6]  Ben Adcock,et al.  Infinite-dimensional $\ell^1$ minimization and function approximation from pointwise data , 2015, 1503.02352.

[7]  Emmanuel J. Candès,et al.  A Probabilistic and RIPless Theory of Compressed Sensing , 2010, IEEE Transactions on Information Theory.

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

[9]  Edward J. Haug,et al.  Design Sensitivity Analysis of Structural Systems , 1986 .

[10]  D. Xiu,et al.  STOCHASTIC COLLOCATION ALGORITHMS USING 𝓁 1 -MINIMIZATION , 2012 .

[11]  Louis B. Rall,et al.  Automatic Differentiation: Techniques and Applications , 1981, Lecture Notes in Computer Science.

[12]  E. Candès.  The restricted isometry property and its implications for compressed sensing , 2008 .

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

[14]  Xiu Yang,et al.  Reweighted ℓ1ℓ1 minimization method for stochastic elliptic differential equations , 2013, J. Comput. Phys..

[15]  Arkadi Nemirovski,et al.  On verifiable sufficient conditions for sparse signal recovery via ℓ1 minimization , 2008, Math. Program..

[16]  R. Askey,et al.  Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials , 1985 .

[17]  Niles A. Pierce,et al.  An Introduction to the Adjoint Approach to Design , 2000 .

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

[19]  Alireza Doostan,et al.  Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies , 2014, J. Comput. Phys..

[20]  Joel A. Tropp,et al.  User-Friendly Tail Bounds for Sums of Random Matrices , 2010, Found. Comput. Math..

[21]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[22]  M. Anitescu,et al.  ORTHOGONAL BASES FOR POLYNOMIAL REGRESSION WITH DERIVATIVE INFORMATION IN UNCERTAINTY QUANTIFICATION , 2010 .

[23]  Michael S. Eldred,et al.  Sparse Pseudospectral Approximation Method , 2011, 1109.2936.

[24]  Holger Rauhut,et al.  Suprema of Chaos Processes and the Restricted Isometry Property , 2012, ArXiv.

[25]  Shengtai Li,et al.  Adjoint Sensitivity Analysis for Differential-Algebraic Equations: The Adjoint DAE System and Its Numerical Solution , 2002, SIAM J. Sci. Comput..

[26]  Luis Santos,et al.  Aerodynamic shape optimization using the adjoint method , 2007 .

[27]  B. Lockwood,et al.  Gradient-based methods for uncertainty quantification in hypersonic flows , 2013 .

[28]  M. Anitescu,et al.  Polynomial Regression Approaches Using Derivative Information for Uncertainty Quantification , 2010 .

[29]  Hester Bijl,et al.  EFFICIENT UNCERTAINTY QUANTIFICATION WITH GRADIENT-ENHANCED KRIGING: APPLICATIONS IN FSI , 2012 .

[30]  Luca Weisz,et al.  Design Sensitivity Analysis Of Structural Systems , 2016 .

[31]  Albert Cohen,et al.  On the Stability and Accuracy of Least Squares Approximations , 2011, Foundations of Computational Mathematics.

[32]  M. Rudelson,et al.  On sparse reconstruction from Fourier and Gaussian measurements , 2008 .

[33]  Holger Rauhut,et al.  Sparse Legendre expansions via l1-minimization , 2012, J. Approx. Theory.

[34]  H. Rauhut,et al.  Interpolation via weighted $l_1$ minimization , 2013, 1308.0759.

[35]  Sylvain Arlot,et al.  A survey of cross-validation procedures for model selection , 2009, 0907.4728.

[36]  G. Szegő.  General properties of orthogonal polynomials , 1939 .

[37]  Ionel M. Navon,et al.  The estimation of functional uncertainty using polynomial chaos and adjoint equations , 2011 .

[38]  Dongbin Xiu,et al.  High-Order Collocation Methods for Differential Equations with Random Inputs , 2005, SIAM J. Sci. Comput..

[39]  O. L. Maître,et al.  Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics , 2010 .

[40]  Holger Rauhut,et al.  Compressive Sensing with structured random matrices , 2012 .

[41]  Alireza Doostan,et al.  Coherence motivated sampling and convergence analysis of least squares polynomial Chaos regression , 2014, 1410.1931.

[42]  R. DeVore,et al.  A Simple Proof of the Restricted Isometry Property for Random Matrices , 2008 .

[43]  R. W. Andrews,et al.  Sensitivity Analysis for Steady State Groundwater Flow Using Adjoint Operators , 1985 .

[44]  L. Mathelin,et al.  A Stochastic Collocation Algorithm for Uncertainty Analysis , 2003 .

[45]  David L Donoho,et al.  Compressed sensing , 2006, IEEE Transactions on Information Theory.

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

[47]  Fabio Nobile,et al.  Approximation of Quantities of Interest in Stochastic PDEs by the Random Discrete L2 Projection on Polynomial Spaces , 2013, SIAM J. Sci. Comput..

[48]  D. Xiu.  Numerical Methods for Stochastic Computations: A Spectral Method Approach , 2010 .

[49]  Anders Logg,et al.  Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book , 2012 .

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

[51]  George M. Siouris,et al.  Applied Optimal Control: Optimization, Estimation, and Control , 1979, IEEE Transactions on Systems, Man, and Cybernetics.