Journal ArticleDOI
Galerkin proper orthogonal decomposition methods for parabolic problems
Karl Kunisch,Stefan Volkwein +1 more
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In this article, error bounds for Galerkin proper orthogonal decomposition (POD) methods for linear and certain non-linear parabolic systems are proved and the resulting error bounds depend on the number of POD basis functions and on the time discretization.Abstract:
In this work error estimates for Galerkin proper orthogonal decomposition (POD) methods for linear and certain non-linear parabolic systems are proved. The resulting error bounds depend on the number of POD basis functions and on the time discretization. Numerical examples are included.read more
Citations
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Journal ArticleDOI
A reduced-order extrapolated technique about the unknown coefficient vectors of solutions in the finite element method for hyperbolic type equation
Zhendong Luo,Wenrui Jiang +1 more
TL;DR: In this paper, a reduced-order extrapolated finite element (ROEFE) format for the hyperbolic type equation (HTE) was proposed, and the existence and stability as well as error estimates of the ROEFE solutions were demonstrated by matrix analysis, leading to an elegant theoretical development.
Journal ArticleDOI
An analysis of galerkin proper orthogonal decomposition for subdiffusion
Bangti Jin,Zhi Zhou +1 more
TL;DR: In this paper, a Galerkin-L1-POD scheme for the subdiffusion model with a Caputo fractional derivative of order α ∈ (0, 1) in time was developed.
Journal ArticleDOI
Isogeometric analysis and proper orthogonal decomposition for the acoustic wave equation
TL;DR: A novel discretization technique-Isogeometric Analysis (IGA) is used in combination with proper orthogonal decomposition (POD) for model order reduction of the time parameterized acoustic wave equations.
Journal ArticleDOI
Closure Learning for Nonlinear Model Reduction Using Deep Residual Neural Network
TL;DR: In this article, a deep residual neural network (ResNet) closure learning framework for nonlinear systems is proposed, where data is used only to complement classical physical modeling (i.e., only in the closure modeling component), not to completely replace it.
Journal ArticleDOI
A multiscale large time increment/FAS algorithm with time-space model reduction for frictional contact problems
Anthony Giacoma,Anthony Giacoma,David Dureisseix,Anthony Gravouil,Anthony Gravouil,Michel Rochette +5 more
TL;DR: In this article, a multiscale strategy using model reduction for frictional contact computation is presented, which is based on a combination between the LATIN (LArge Time Increment) method and the FAS multigrid solver.
References
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Book
Infinite-Dimensional Dynamical Systems in Mechanics and Physics
TL;DR: In this article, the authors give bounds on the number of degrees of freedom and the dimension of attractors of some physical systems, including inertial manifolds and slow manifolds.
Book
Turbulence, Coherent Structures, Dynamical Systems and Symmetry
TL;DR: In this article, the authors present a review of rigor properties of low-dimensional models and their applications in the field of fluid mechanics. But they do not consider the effects of random perturbation on models.
Book
Galerkin Finite Element Methods for Parabolic Problems
TL;DR: The standard Galerkin method is based on more general approximations of the elliptic problem as discussed by the authors, and is used to solve problems in algebraic systems at the time level.
Journal ArticleDOI
Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition
Karl Kunisch,Stefan Volkwein +1 more
TL;DR: POD is utilized to solve open-loop and closed-loop optimal control problems for the Burgers equation to comparison of POD-based algorithms with numerical results obtained from finite-element discretization of the optimality system.
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Galerkin Proper Orthogonal Decomposition Methods for a General Equation in Fluid Dynamics
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