An Empirical Interpolation and Model-Variance Reduction Method for Computing Statistical Outputs of Parametrized Stochastic Partial Differential Equations

We present an empirical interpolation and model-variance reduction method for the fast and reliable computation of statistical outputs of parametrized stochastic elliptic partial differential equations. Our method consists of three main ingredients: (1) the real-time computation of reduced basis (RB) outputs approximating high-fidelity outputs computed with the hybridizable discontinuous Galerkin (HDG) discretization; (2) the empirical interpolation for an efficient offline-online decoupling of the parametric and stochastic influence; and (3) a multilevel variance reduction method that exploits the statistical correlation between the low-fidelity approximations and the high-fidelity HDG discretization to accelerate the convergence of the Monte Carlo simulations. The multilevel variance reduction method provides efficient computation of the statistical outputs by shifting most of the computational burden from the high-fidelity HDG approximation to the RB approximations. Furthermore, we develop a posteriori...

[1]  Robert M. Freund,et al.  Fabrication-Adaptive Optimization with an Application to Photonic Crystal Design , 2013, Oper. Res..

[2]  Michael B. Giles,et al.  Adjoint Recovery of Superconvergent Functionals from PDE Approximations , 2000, SIAM Rev..

[3]  H. Robbins.  A Stochastic Approximation Method , 1951 .

[4]  Leo Wai-Tsun Ng,et al.  Multifidelity Uncertainty Propagation for Optimization Under Uncertainty , 2012 .

[5]  Michael B. Giles,et al.  A model and variance reduction method for computing statistical outputs of stochastic elliptic partial differential equations , 2014, J. Comput. Phys..

[6]  S'ebastien Boyaval,et al.  A fast Monte–Carlo method with a reduced basis of control variates applied to uncertainty propagation and Bayesian estimation , 2012, 1202.0781.

[7]  Gianluigi Rozza,et al.  Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations , 2016, Numerische Mathematik.

[8]  Pierre Priouret,et al.  Adaptive Algorithms and Stochastic Approximations , 1990, Applications of Mathematics.

[9]  Stefan Heinrich,et al.  Monte Carlo Complexity of Parametric Integration , 1999, J. Complex..

[10]  O. Sigmund,et al.  Robust topology optimization accounting for spatially varying manufacturing errors , 2011 .

[11]  C. Leake.  Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method , 1994 .

[12]  N. Nguyen,et al.  An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations , 2004 .

[13]  Ahmed K. Noor,et al.  Reduced Basis Technique for Nonlinear Analysis of Structures , 1980 .

[14]  Alfio Quarteroni,et al.  Weighted Reduced Basis Method for Stochastic Optimal Control Problems with Elliptic PDE Constraint , 2014, SIAM/ASA J. Uncertain. Quantification.

[15]  P. L’Ecuyer,et al.  On the interchange of derivative and expectation for likelihood ratio derivative estimators , 1995 .

[16]  A. Shapiro.  Monte Carlo Sampling Methods , 2003 .

[17]  Andreas Griewank,et al.  Evaluating derivatives - principles and techniques of algorithmic differentiation, Second Edition , 2000, Frontiers in applied mathematics.

[18]  Trond Steihaug,et al.  Truncated-newtono algorithms for large-scale unconstrained optimization , 1983, Math. Program..

[19]  Alexander Shapiro,et al.  Asymptotic analysis of stochastic programs , 1991, Ann. Oper. Res..

[20]  Alexander Shapiro,et al.  Stochastic Approximation approach to Stochastic Programming , 2013 .

[21]  Nguyen Ngoc Cuong,et al.  Certified Real-Time Solution of Parametrized Partial Differential Equations , 2005 .

[22]  Mattias Schevenels,et al.  Topology optimization with geometric uncertainties by perturbation techniques , 2012 .

[23]  T. Santoso.  A stochastic programming approach for supply chain network design under uncertainty , 2004 .

[24]  Michael B. Giles,et al.  Multilevel Monte Carlo Path Simulation , 2008, Oper. Res..

[25]  A. Patera,et al.  Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations , 2007 .

[26]  W. Schachermayer,et al.  Multilevel quasi-Monte Carlo path simulation , 2009 .

[27]  P. L’Ecuyer,et al.  A Unified View of the IPA, SF, and LR Gradient Estimation Techniques , 1990 .

[28]  Boris Polyak,et al.  Acceleration of stochastic approximation by averaging , 1992 .

[29]  Jason H. Goodfriend,et al.  Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method , 1995 .

[30]  Gianluigi Rozza,et al.  Reduced order methods for uncertainty quantification problems , 2015 .

[31]  David P. Morton,et al.  Monte Carlo bounding techniques for determining solution quality in stochastic programs , 1999, Oper. Res. Lett..

[32]  D. Rovas,et al.  Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods , 2002 .

[33]  Karsten Urban,et al.  Reduced Basis Methods for Parameterized Partial Differential Equations with Stochastic Influences Using the Karhunen-Loève Expansion , 2013, SIAM/ASA J. Uncertain. Quantification.

[34]  David P. Morton,et al.  Assessing solution quality in stochastic programs , 2006, Algorithms for Optimization with Incomplete Information.

[35]  Y. Maday,et al.  Reduced Basis Techniques for Stochastic Problems , 2010, 1004.0357.

[36]  Elisabeth Ullmann,et al.  Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients , 2012, Numerische Mathematik.

[37]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[38]  K. A. Cliffe,et al.  Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients , 2011, Comput. Vis. Sci..

[39]  Andrea Barth,et al.  Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients , 2011, Numerische Mathematik.

[40]  Stefan Heinrich,et al.  Multilevel Monte Carlo Methods , 2001, LSSC.

[41]  Rémi Bardenet,et al.  Monte Carlo Methods , 2013, Encyclopedia of Social Network Analysis and Mining. 2nd Ed..

[42]  Alexander Shapiro,et al.  The Sample Average Approximation Method for Stochastic Discrete Optimization , 2002, SIAM J. Optim..

[43]  Colas Schretter,et al.  Monte Carlo and Quasi-Monte Carlo Methods , 2016 .