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 mfe formulation based on pod method for parabolic equations
Zhendong Luo,Lei Li,Ping Sun +2 more
TL;DR: In this article, a reduced-order MFE formulation based on proper orthogonal decomposition (POD) was proposed for solving parabolic equations with lower dimensions and sufficiently high accuracy.
Journal ArticleDOI
Reduced order modeling of random linear dynamical systems based on a new a posteriori error bound
Md. Nurtaj Hossain,Debraj Ghosh +1 more
TL;DR: In this article, an adaptive reduced order model based on a posteriori error bound is proposed for a linear random dynamical system, and two adaptive methods are proposed for building robust reduced order models, one for local and another for global.
Journal ArticleDOI
Two-Grid Based Adaptive Proper Orthogonal Decomposition Method 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 proposes an error indicator for the numerical solution obtained in the fine grid.
Dissertation
Reduced Order Models, Forward and Inverse Problems in Cardiac Electrophysiology
TL;DR: This PhD thesis is dedicated to the investigation of the forward and the inverse problem of cardiac electrophysiology and a new reduced order algorithm is proposed, based on the ALP and the Discrete Empirical Interpolation methods.
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