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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Posted Content
Two-Grid based Adaptive Proper Orthogonal Decomposition Algorithm for Time Dependent Partial Differential Equations
TL;DR: This article proposes a two-grid based adaptive proper orthogonal decomposition (POD) method to solve the time dependent partial differential equations with the Kolmogorov flow and the ABC flow and shows that the method is more efficient than the existing POD methods.
DissertationDOI
Direct Methods for PDE-Constrained Optimization Using Derivative-Extended POD Reduced-Order Models
TL;DR: This thesis proposes methods for a suitable enhancement of the ROMs for the optimization purpose which are based on the inclusion of derivative information in the POD and DEIM subspaces, and develops methods based on model order reduction (MOR) for the solution of optimization problems constrained by time-dependent partial differential equations (PDEs).
Journal ArticleDOI
Randomized quasi-optimal local approximation spaces in time
TL;DR: By solving the PDE locally in time with random initial conditions, the proposed method can outperform existing methods like the proper orthogonal decomposition even in a sequential setting and is well capable of approximating advection-dominated problems.
Posted Content
Error estimates for a POD method for solving viscous G-equations in incompressible cellular flows
Haotian Gu,Jack Xin,Zhiwen Zhang +2 more
TL;DR: An efficient model reduction method for computing regular solutions of viscous G-equations in incompressible steady and time-periodic cellular flows based on the Galerkin proper orthogonal decomposition (POD) method is developed.
Journal ArticleDOI
Dynamical Behaviors Analysis of the Rotor Model with Coupling Faults and Applications of the TPOD Method
TL;DR: In this paper, a transient proper orthogonal decomposition (TPOD) method is applied for order reduction in the rotor-bearing system with the coupling faults in order to obtain the optimal order reduction model based on the POM energy.
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.
Related Papers (5)
Galerkin Proper Orthogonal Decomposition Methods for a General Equation in Fluid Dynamics
Karl Kunisch,Stefan Volkwein +1 more