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
On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far‐field boundary treatment
TL;DR: In this article, a reduced order model based on the proper orthogonal decomposition (POD)/Galerkin projection method is proposed as an alternative discretization of the linearized compressible Euler equations.
Journal ArticleDOI
Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition
Michael Hinze,Stefan Volkwein +1 more
TL;DR: This paper applies the discrete technique developed by Hinze to POD discretizations of abstract linear–quadratic optimal control problems with control constraints and proves error estimates for the corresponding discrete controls.
Journal ArticleDOI
SUPG reduced order models for convection-dominated convection–diffusion–reaction equations
TL;DR: A Streamline-Upwind Petrov–Galerkin (SUPG) reduced order model (ROM) based on proper orthogonal decomposition (POD) is investigated theoretically and numerically for convection-dominated convection–diffusion–reaction problems.
Journal ArticleDOI
On closures for reduced order models—A spectrum of first-principle to machine-learned avenues
Shady E. Ahmed,Suraj Pawar,Omer San,Adil Rasheed,Traian Iliescu,Bernd R. Noack,Bernd R. Noack +6 more
TL;DR: In this article, the effect of the discarded reduced order modes in under-resolved simulations is modeled using data-driven proper orthogonal decomposition (POD) modeling.
Journal ArticleDOI
Mixed Finite Element Formulation and Error Estimates Based on Proper Orthogonal Decomposition for the Nonstationary Navier-Stokes Equations
TL;DR: It is shown by considering numerical simulation results obtained for the illustrating example of cavity flows that the error between POD solution of reduced MFE formulation and the reference solution is consistent with theoretical results and validates the feasibility and efficiency of the POD method.
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