Reduced-order modelling of parameter-dependent, linear and nonlinear dynamic partial differential equation models

In this paper, we develop reduced-order models for dynamic, parameter-dependent, linear and nonlinear partial differential equations using proper orthogonal decomposition (POD). The main challenges are to accurately and efficiently approximate the POD bases for new parameter values and, in the case of nonlinear problems, to efficiently handle the nonlinear terms. We use a Bayesian nonlinear regression approach to learn the snapshots of the solutions and the nonlinearities for new parameter values. Computational efficiency is ensured by using manifold learning to perform the emulation in a low-dimensional space. The accuracy of the method is demonstrated on a linear and a nonlinear example, with comparisons with a global basis approach.

[1]  Heng Tao Shen,et al.  Principal Component Analysis , 2009, Encyclopedia of Biometrics.

[2]  Charbel Farhat,et al.  Stabilization of projection‐based reduced‐order models , 2012 .

[3]  Sonja Kuhnt,et al.  Design and analysis of computer experiments , 2010 .

[4]  Karen Willcox,et al.  Parametric reduced-order models for probabilistic analysis of unsteady aerodynamic applications , 2007 .

[5]  Bernard Haasdonk,et al.  Reduced Basis Approximation for Nonlinear Parametrized Evolution Equations based on Empirical Operator Interpolation , 2012, SIAM J. Sci. Comput..

[6]  Charbel Farhat,et al.  An Online Method for Interpolating Linear Parametric Reduced-Order Models , 2011, SIAM J. Sci. Comput..

[7]  C. Farhat,et al.  Interpolation Method for Adapting Reduced-Order Models and Application to Aeroelasticity , 2008 .

[8]  A. R. Mitchell,et al.  Product Approximation for Non-linear Problems in the Finite Element Method , 1981 .

[9]  Karen Willcox,et al.  Parameter and State Model Reduction for Large-Scale Statistical Inverse Problems , 2010, SIAM J. Sci. Comput..

[10]  R. Murray,et al.  Model reduction for compressible flows using POD and Galerkin projection , 2004 .

[11]  Adrian Sandu,et al.  Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations , 2014, International Journal for Numerical Methods in Fluids.

[12]  D. Higdon,et al.  Computer Model Calibration Using High-Dimensional Output , 2008 .

[13]  I. Sobol.  Uniformly distributed sequences with an additional uniform property , 1976 .

[14]  Robert Cedergren,et al.  Guided tour , 1990, Nature.

[15]  Muruhan Rathinam,et al.  A New Look at Proper Orthogonal Decomposition , 2003, SIAM J. Numer. Anal..

[16]  Stefan Volkwein,et al.  Galerkin Proper Orthogonal Decomposition Methods for a General Equation in Fluid Dynamics , 2002, SIAM J. Numer. Anal..

[17]  Siep Weiland,et al.  Missing Point Estimation in Models Described by Proper Orthogonal Decomposition , 2004, IEEE Transactions on Automatic Control.

[18]  Charbel Farhat,et al.  Reduced-order fluid/structure modeling of a complete aircraft configuration , 2006 .

[19]  Bernhard Schölkopf,et al.  Nonlinear Component Analysis as a Kernel Eigenvalue Problem , 1998, Neural Computation.

[20]  Wolfgang Dahmen,et al.  Convergence Rates for Greedy Algorithms in Reduced Basis Methods , 2010, SIAM J. Math. Anal..

[21]  N. Nguyen,et al.  EFFICIENT REDUCED-BASIS TREATMENT OF NONAFFINE AND NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS , 2007 .

[22]  Benjamin Peherstorfer,et al.  Localized Discrete Empirical Interpolation Method , 2014, SIAM J. Sci. Comput..

[23]  Mario Ohlberger,et al.  Pr ep rin t Data-Driven Combined State and Parameter Reduction for Extreme-Scale Inverse Problems , 2014 .

[24]  Alexander Hay,et al.  On the use of sensitivity analysis in model reduction to predict flows for varying inflow conditions , 2012 .

[25]  Prasanth B. Nair,et al.  Reduced dimensional Gaussian process emulators of parametrized partial differential equations based on Isomap , 2015, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[26]  M. Baumann.  Nonlinear Model Order Reduction using POD/DEIM for Optimal Control of Burgers' Equation , 2013 .

[27]  P. Holmes,et al.  The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows , 1993 .

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

[29]  Danny C. Sorensen,et al.  Nonlinear Model Reduction via Discrete Empirical Interpolation , 2010, SIAM J. Sci. Comput..

[30]  Boris Lohmann,et al.  Parametric Model Order Reduction by Matrix Interpolation , 2010, Autom..

[31]  Stian Kristoffersen.  The Empirical Interpolation Method , 2013 .

[32]  Mario Ohlberger,et al.  Data-driven combined state and parameter reduction for inverse problems , 2014, Advances in Computational Mathematics.

[33]  Z. Bai.  Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems , 2002 .

[34]  L. Sirovich.  TURBULENCE AND THE DYNAMICS OF COHERENT STRUCTURES PART I : COHERENT STRUCTURES , 2016 .

[35]  Christopher J. C. Burges,et al.  Dimension Reduction: A Guided Tour , 2010, Found. Trends Mach. Learn..

[36]  Matthew F. Barone,et al.  Stabilization of projection-based reduced order models for linear time-invariant systems via optimization-based eigenvalue reassignment , 2014 .

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

[38]  Karen Willcox,et al.  Model Reduction for Large-Scale Systems with High-Dimensional Parametric Input Space , 2008, SIAM J. Sci. Comput..

[39]  Andrew J. Newman,et al.  Model Reduction via the Karhunen-Loeve Expansion Part I: An Exposition , 1996 .

[40]  K. Willcox,et al.  Interpolation among reduced‐order matrices to obtain parameterized models for design, optimization and probabilistic analysis , 2009 .

[41]  Karen Willcox,et al.  Goal-oriented, model-constrained optimization for reduction of large-scale systems , 2007, J. Comput. Phys..

[42]  Karen Willcox,et al.  A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems , 2015, SIAM Rev..

[43]  A.A. Shah,et al.  Manifold learning for the emulation of spatial fields from computational models , 2016, J. Comput. Phys..

[44]  Mark N. Glauser,et al.  Towards practical flow sensing and control via POD and LSE based low-dimensional tools , 2004 .

[45]  Harold J. Kushner,et al.  Stochastic processes in information and dynamical systems , 1972 .

[46]  L. Sirovich.  Turbulence and the dynamics of coherent structures. I. Coherent structures , 1987 .

[47]  Allen Tannenbaum,et al.  Statistical shape analysis using kernel PCA , 2006, Electronic Imaging.

[48]  Charbel Farhat,et al.  A method for interpolating on manifolds structural dynamics reduced‐order models , 2009 .

[49]  Charbel Farhat,et al.  The GNAT method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows , 2012, J. Comput. Phys..

[50]  Lucia Faravelli,et al.  Model Order Reduction in Nonlinear Systems , 2016 .