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 parameterized-background data-weak approach to variational data assimilation: formulation, analysis, and application to acoustics
TL;DR: In this article, a parameterized-background data-weak (PBDW) formulation of the variational data assimilation (state estimation) problem for systems modeled by partial differential equations is presented.
Journal ArticleDOI
Multi-scale optimization for process systems engineering
TL;DR: A rigorous multi-scale optimization framework is developed that substitutes RMs for complex original detailed models (ODMs) and guarantees convergence to the original optimization problem and leads to three related NLP algorithms for RM-based optimization.
Journal ArticleDOI
Optimal snapshot location for computing POD basis functions
Karl Kunisch,Stefan Volkwein +1 more
TL;DR: Numerical examples illustrate that the proposed criterion is sensitive with respect to the choice of the time instances and further they demonstrate the feasibility of the method in determining optimal snapshot locations for concrete diffusion equations.
Journal ArticleDOI
An optimized Crank–Nicolson finite difference extrapolating model for the fractional-order parabolic-type sine-Gordon equation
Yanjie Zhou,Zhendong Luo +1 more
TL;DR: In this paper, a proper orthogonal decomposition (POD) is proposed to reduce the order of the classical Crank-Nicolson finite difference (CCNFD) model for the fractional-order parabolic-type sine-Gordon equations (FOPTSGEs).
Journal ArticleDOI
Variational multiscale proper orthogonal decomposition: Navier‐stokes equations
Traian Iliescu,Zhu Wang +1 more
TL;DR: In this paper, a variational multiscale proper orthogonal decomposition (POD) reduced-order model for turbulent incompressible Navier-Stokes equations is presented.
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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