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Title: Model reduction of dynamical systems by proper orthogonal decomposition: Error bounds and comparison of methods using snapshots from the solution and the time derivatives

Journal Article · · Journal of Computational and Applied Mathematics
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In this study, we consider two proper orthogonal decomposition (POD) methods for dimension reduction of dynamical systems. The first method (M1) uses only time snapshots of the solution, while the second method (M2) augments the snapshot set with time-derivative snapshots. The goal of the paper is to analyze and compare the approximation errors resulting from the two methods by using error bounds. We derive several new bounds of the error from POD model reduction by each of the two methods. The new error bounds involve a multiplicative factor depending on the time steps between the snapshots. For method M1 the factor depends on the second power of the time step, while for method 2 the dependence is on the fourth power of the time step, suggesting that method M2 can be more accurate for small between-snapshot intervals. However, three other factors also affect the size of the error bounds. These include (i) the norm of the second (for M1) and fourth derivatives (M2); (ii) the first neglected singular value and (iii) the spectral properties of the projection of the system’s Jacobian in the reduced space. Because of the interplay of these factors neither method is more accurate than the other in all cases. Finally, we present numerical examples demonstrating that when the number of collected snapshots is small and the first neglected singular value has a value of zero, method M2 results in a better approximation.

Research Organization:
Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
Sponsoring Organization:
USDOE
Grant/Contract Number:
LDRD 13-ERD-031; 17-ERD-026; AC52-07NA27344
OSTI ID:
1846851
Alternate ID(s):
OSTI ID: 1409976; OSTI ID: 1549927
Report Number(s):
LLNL-JRNL-720257; S0377042717304181; PII: S0377042717304181
Journal Information:
Journal of Computational and Applied Mathematics, Journal Name: Journal of Computational and Applied Mathematics Vol. 330 Journal Issue: C; ISSN 0377-0427
Publisher:
ElsevierCopyright Statement
Country of Publication:
Belgium
Language:
English
Citation Metrics:
Cited by: 13 works
Citation information provided by
Web of Science

References (26)

A Reduced-Order Method for Simulation and Control of Fluid Flows journal July 1998
New POD Error Expressions, Error Bounds, and Asymptotic Results for Reduced Order Models of Parabolic PDEs journal January 2014
The logarithmic norm. History and modern theory journal August 2006
Limited-memory adaptive snapshot selection for proper orthogonal decomposition: ADAPTIVE SNAPSHOT SELECTION FOR POD
  • Oxberry, Geoffrey M.; Kostova-Vassilevska, Tanya; Arrighi, William
  • International Journal for Numerical Methods in Engineering, Vol. 109, Issue 2 https://doi.org/10.1002/nme.5283
journal July 2016
Dimension Reduction of Dynamical Systems: Methods, Models, Applications journal August 2005
Optimal Control of a Phase-Field Model Using Proper Orthogonal Decomposition journal February 2001
The measure of a matrix as a tool to analyze computer algorithms for circuit analysis journal January 1972
Are the Snapshot Difference Quotients Needed in the Proper Orthogonal Decomposition? journal January 2014
Mixed Finite Element Formulation and Error Estimates Based on Proper Orthogonal Decomposition for the Nonstationary Navier–Stokes Equations journal January 2009
Control of laser surface hardening by a reduced-order approach using proper orthogonal decomposition journal November 2003
Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples journal February 2012
Note on the Derivatives with Respect to a Parameter of the Solutions of a System of Differential Equations journal July 1919
POD and CVT-based reduced-order modeling of Navier–Stokes flows journal December 2006
On Logarithmic Norms journal October 1975
A low-cost, goal-oriented ‘compact proper orthogonal decomposition’ basis for model reduction of static systems journal December 2010
Gradient-enhanced surrogate modeling based on proper orthogonal decomposition journal January 2013
A New Look at Proper Orthogonal Decomposition journal January 2003
Fitzhugh–Nagumo Revisited: Types of Bifurcations, Periodical Forcing and Stability Regions by a Lyapunov Functional journal March 2004
Inertial manifolds for nonlinear evolutionary equations journal June 1988
The Effect of Problem Perturbations on Nonlinear Dynamical Systems and their Reduced-Order Models journal January 2007
Galerkin Proper Orthogonal Decomposition Methods for a General Equation in Fluid Dynamics journal January 2002
Turbulence and the dynamics of coherent structures. I. Coherent structures journal January 1987
Galerkin proper orthogonal decomposition methods for parabolic problems journal November 2001
Nonlinear Model Reduction via Discrete Empirical Interpolation journal January 2010
A State Space Error Estimate for POD-DEIM Nonlinear Model Reduction journal January 2012
Error Estimation for Reduced‐Order Models of Dynamical Systems journal January 2007

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97 MATHEMATICS AND COMPUTING
Model reduction
Proper orthogonal decomposition
Error bound