Adaptive h -refinement for reduced-order models: ADAPTIVE h -refinement for reduced-order models

Carlberg, Kevin T.

doi:10.1002/nme.4800

Adaptive h -refinement for reduced-order models: ADAPTIVE h -refinement for reduced-order models

Journal Article · Tue Nov 04 23:00:00 EST 2014 · International Journal for Numerical Methods in Engineering

DOI:https://doi.org/10.1002/nme.4800· OSTI ID:1141744

Carlberg, Kevin T. ^[1]

Sandia National Lab. (SNL-CA), Livermore, CA (United States)

Our work presents a method to adaptively refine reduced-order models a posteriori without requiring additional full-order-model solves. The technique is analogous to mesh-adaptive h-refinement: it enriches the reduced-basis space online by ‘splitting’ a given basis vector into several vectors with disjoint support. The splitting scheme is defined by a tree structure constructed offline via recursive k-means clustering of the state variables using snapshot data. This method identifies the vectors to split online using a dual-weighted-residual approach that aims to reduce error in an output quantity of interest. The resulting method generates a hierarchy of subspaces online without requiring large-scale operations or full-order-model solves. Furthermore, it enables the reduced-order model to satisfy any prescribed error tolerance regardless of its original fidelity, as a completely refined reduced-order model is mathematically equivalent to the original full-order model. Experiments on a parameterized inviscid Burgers equation highlight the ability of the method to capture phenomena (e.g., moving shocks) not contained in the span of the original reduced basis.

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Laboratories (SNL-CA), Livermore, CA (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: AC04-94AL85000

OSTI ID:: 1141744

Report Number(s):: SAND--2014-2732J; 507023

Journal Information:: International Journal for Numerical Methods in Engineering, Journal Name: International Journal for Numerical Methods in Engineering Journal Issue: 5 Vol. 102; ISSN 0029-5981

Publisher:: WileyCopyright Statement

Country of Publication:: United States

Language:: English

References (28)

The post-processing approach in the finite element method—part 1: Calculation of displacements, stresses and other higher derivatives of the displacements Babuška, I.; Miller, A. International Journal for Numerical Methods in Engineering, Vol. 20, Issue 6 https://doi.org/10.1002/nme.1620200610	journal	June 1984
A method for interpolating on manifolds structural dynamics reduced-order models Amsallem, David; Cortial, Julien; Carlberg, Kevin International Journal for Numerical Methods in Engineering, Vol. 80, Issue 9 https://doi.org/10.1002/nme.2681	journal	November 2009
Nonlinear model order reduction based on local reduced-order bases: NONLINEAR MODEL REDUCTION BASED ON LOCAL REDUCED-ORDER BASES Amsallem, David; Zahr, Matthew J.; Farhat, Charbel International Journal for Numerical Methods in Engineering, Vol. 92, Issue 10 https://doi.org/10.1002/nme.4371	journal	June 2012
Adjoint Error Estimation and Grid Adaptation for Functional Outputs: Application to Quasi-One-Dimensional Flow Venditti, David A.; Darmofal, David L. Journal of Computational Physics, Vol. 164, Issue 1 https://doi.org/10.1006/jcph.2000.6600	journal	October 2000
Grid Adaptation for Functional Outputs: Application to Two-Dimensional Inviscid Flows Venditti, David A.; Darmofal, David L. Journal of Computational Physics, Vol. 176, Issue 1 https://doi.org/10.1006/jcph.2001.6967	journal	February 2002
Adaptive Reduced Basis Methods for Nonlinear Convection–Diffusion Equations Drohmann, Martin; Haasdonk, Bernard; Ohlberger, Mario Finite Volumes for Complex Applications VI Problems & Perspectives https://doi.org/10.1007/978-3-642-20671-9_39	book	January 2011
Adaptive error control for multigrid finite element Becker, R.; Johnson, C.; Rannacher, R. Computing, Vol. 55, Issue 4 https://doi.org/10.1007/BF02238483	journal	December 1995
A multi-point reduced-order modeling approach of transient structural dynamics with application to robust design optimization Weickum, G.; Eldred, M. S.; Maute, K. Structural and Multidisciplinary Optimization, Vol. 38, Issue 6 https://doi.org/10.1007/s00158-008-0309-5	journal	September 2008
Efficient model reduction in non-linear dynamics using the Karhunen-Lo�ve expansion and dual-weighted-residual methods Meyer, M.; Matthies, H. G. Computational Mechanics, Vol. 31, Issue 1-2 https://doi.org/10.1007/s00466-002-0404-1	journal	May 2003
Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data Gunzburger, Max D.; Peterson, Janet S.; Shadid, John N. Computer Methods in Applied Mechanics and Engineering, Vol. 196, Issue 4-6 https://doi.org/10.1016/j.cma.2006.08.004	journal	January 2007
An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations Barrault, Maxime; Maday, Yvon; Nguyen, Ngoc Cuong Comptes Rendus Mathematique, Vol. 339, Issue 9 https://doi.org/10.1016/j.crma.2004.08.006	journal	November 2004
A priori hyperreduction method: an adaptive approach Ryckelynck, D. Journal of Computational Physics, Vol. 202, Issue 1 https://doi.org/10.1016/j.jcp.2004.07.015	journal	January 2005
Stable Galerkin reduced order models for linearized compressible flow Barone, Matthew F.; Kalashnikova, Irina; Segalman, Daniel J. Journal of Computational Physics, Vol. 228, Issue 6 https://doi.org/10.1016/j.jcp.2008.11.015	journal	April 2009
The GNAT method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows Carlberg, Kevin; Farhat, Charbel; Cortial, Julien Journal of Computational Physics, Vol. 242 https://doi.org/10.1016/j.jcp.2013.02.028	journal	June 2013
Reduced basis method for finite volume approximations of parametrized linear evolution equations Haasdonk, Bernard; Ohlberger, Mario ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 42, Issue 2 https://doi.org/10.1051/m2an:2008001	journal	March 2008
A training set and multiple bases generation approach for parameterized model reduction based on adaptive grids in parameter space Haasdonk, Bernard; Dihlmann, Markus; Ohlberger, Mario Mathematical and Computer Modelling of Dynamical Systems, Vol. 17, Issue 4 https://doi.org/10.1080/13873954.2011.547674	journal	August 2011
Missing Point Estimation in Models Described by Proper Orthogonal Decomposition Astrid, Patricia; Weiland, Siep; Willcox, Karen IEEE Transactions on Automatic Control, Vol. 53, Issue 10 https://doi.org/10.1109/TAC.2008.2006102	journal	November 2008
Least squares quantization in PCM Lloyd, S. IEEE Transactions on Information Theory, Vol. 28, Issue 2 https://doi.org/10.1109/TIT.1982.1056489	journal	March 1982
A Posteriori Error Bounds and Global Error Control for Approximation of Ordinary Differential Equations Estep, Donald SIAM Journal on Numerical Analysis, Vol. 32, Issue 1 https://doi.org/10.1137/0732001	journal	February 1995
Nonlinear Model Reduction via Discrete Empirical Interpolation Chaturantabut, Saifon; Sorensen, Danny C. SIAM Journal on Scientific Computing, Vol. 32, Issue 5 https://doi.org/10.1137/090766498	journal	January 2010
An "$hp$" Certified Reduced Basis Method for Parametrized Elliptic Partial Differential Equations Eftang, Jens L.; Patera, Anthony T.; Rønquist, Einar M. SIAM Journal on Scientific Computing, Vol. 32, Issue 6 https://doi.org/10.1137/090780122	journal	January 2010
Reduced Basis Approximation for Nonlinear Parametrized Evolution Equations based on Empirical Operator Interpolation Drohmann, Martin; Haasdonk, Bernard; Ohlberger, Mario SIAM Journal on Scientific Computing, Vol. 34, Issue 2 https://doi.org/10.1137/10081157X	journal	January 2012
Localized Discrete Empirical Interpolation Method Peherstorfer, Benjamin; Butnaru, Daniel; Willcox, Karen SIAM Journal on Scientific Computing, Vol. 36, Issue 1 https://doi.org/10.1137/130924408	journal	January 2014
Adjoint Recovery of Superconvergent Functionals from PDE Approximations Pierce, Niles A.; Giles, Michael B. SIAM Review, Vol. 42, Issue 2 https://doi.org/10.1137/S0036144598349423	journal	January 2000
Analysis of Inexact Trust-Region SQP Algorithms Heinkenschloss, Matthias; Vicente, Luis N. SIAM Journal on Optimization, Vol. 12, Issue 2 https://doi.org/10.1137/S1052623499361543	journal	January 2002
Adjoint-Based, Three-Dimensional Error Prediction and Grid Adaptation Park, Michael Andrew AIAA Journal, Vol. 42, Issue 9 https://doi.org/10.2514/1.10051	journal	September 2004
Interpolation Method for Adapting Reduced-Order Models and Application to Aeroelasticity Amsallem, David; Farhat, Charbel AIAA Journal, Vol. 46, Issue 7 https://doi.org/10.2514/1.35374	journal	July 2008
An Adaptive Simplex Cut-Cell Method for Discontinuous Galerkin Discretizations of the Navier-Stokes Equations Fidkowski, Krzysztof; Darmofal, David 18th AIAA Computational Fluid Dynamics Conference https://doi.org/10.2514/6.2007-3941	conference	June 2012

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Transported snapshot model order reduction approach for parametric, steady-state fluid flows containing parameter-dependent shocks: Model order reduction for fluid flows containing shocks Nair, Nirmal J.; Balajewicz, Maciej International Journal for Numerical Methods in Engineering, Vol. 117, Issue 12 https://doi.org/10.1002/nme.5998	journal	December 2018
Randomized low‐rank approximation methods for projection‐based model order reduction of large nonlinear dynamical problems Bach, C.; Ceglia, D.; Song, L. International Journal for Numerical Methods in Engineering, Vol. 118, Issue 4 https://doi.org/10.1002/nme.6009	journal	January 2019
Fixed‐precision randomized low‐rank approximation methods for nonlinear model order reduction of large systems Bach, C.; Duddeck, F.; Song, L. International Journal for Numerical Methods in Engineering, Vol. 119, Issue 8 https://doi.org/10.1002/nme.6068	journal	April 2019
Enhancing piecewise‐defined surrogate response surfaces with adjoints on sets of unstructured samples to solve stochastic inverse problems Mattis, Steven A.; Butler, Troy International Journal for Numerical Methods in Engineering, Vol. 119, Issue 10 https://doi.org/10.1002/nme.6078	journal	May 2019
Adaptive multi‐index collocation for uncertainty quantification and sensitivity analysis Jakeman, John D.; Eldred, Michael S.; Geraci, Gianluca International Journal for Numerical Methods in Engineering, Vol. 121, Issue 6 https://doi.org/10.1002/nme.6268	journal	November 2019
Adaptive multi‐index collocation for uncertainty quantification and sensitivity analysis Jakeman, John D.; Eldred, Michael S.; Geraci, Gianluca International Journal for Numerical Methods in Engineering, Vol. 121, Issue 19 https://doi.org/10.1002/nme.6450	journal	August 2020
POD model order reduction with space-adapted snapshots for incompressible flows Gräßle, Carmen; Hinze, Michael; Lang, Jens Advances in Computational Mathematics, Vol. 45, Issue 5-6 https://doi.org/10.1007/s10444-019-09716-7	journal	September 2019
Limited-memory adaptive snapshot selection for proper orthogonal decomposition Oxberry, Geoffrey; Kostova-Vassilevska, Tanya; Arrighi, Bill https://doi.org/10.2172/1224940	report	April 2015
Adaptive multi-index collocation for uncertainty quantification and sensitivity analysis Jakeman, John Davis; Eldred, Michael S.; Geraci, G. https://doi.org/10.2172/1574406	report	November 2019

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