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
Structure preserving integration and model order reduction of skew-gradient reaction-diffusion systems
TL;DR: In this paper, the authors show that the fully discrete energy of the FHN equation satisfies the mini-maximizer property of the continuous energy for the skew-gradient systems.
Journal ArticleDOI
Model order reduction for nonlinear Schrödinger equation
TL;DR: Numerical results for one and two dimensional NLS equations, coupled NLS equation with soliton solutions show that the low-dimensional approximations obtained by POD reproduce very well the characteristic dynamics of the system, such as preservation of energy and the solutions.
Journal ArticleDOI
A few techniques to improve data-driven reduced-order simulations for unsteady flows
TL;DR: In this article, a data-driven transfer function that predicts the time evolution of the reduced-order modes accurately is proposed, and the authors demonstrate a couple of useful techniques to achieve this objective: one is to preprocess time-series of reducedorder modes with a low-pass filter, e.g. a polynomial filter and B-spline, and another is to compute a data driven transfer function from multiple past time-steps, corresponding to a high-order temporal scheme.
Posted Content
On Optimal Pointwise in Time Error Bounds and Difference Quotients for the Proper Orthogonal Decomposition
TL;DR: The role played by difference quotients (DQs) in obtaining reduced order model (ROM) error bounds that are optimal with respect to both the time discretization error and the ROM discretized error is studied.
Journal ArticleDOI
A New Reduced-Order fve Algorithm Based on POD Method for Viscoelastic Equations
TL;DR: In this article, a reduced-order fully discrete FVE algorithm with fewer degrees of freedom and sufficiently high accuracy based on orthogonal decomposition (POD) is established.
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
Karl Kunisch,Stefan Volkwein +1 more