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
Model order reduction for seismic waveform modelling: inspiration from normal modes
TL;DR: In this paper , the eigenmodes of the seismic wave equation are used to span this lower dimensional space and free body oscillations and a form of Petrov-Galerkin projection can be applied in regional scale problems.
Journal ArticleDOI
On the influence of the nonlinear term in the numerical approximation of Incompressible Flows by means of proper orthogonal decomposition methods
TL;DR: It is proved that an additional error term appears in this case, compared with the case in which the same discretization of the nonlinear term is applied for both the FOM and the POD methods.
Journal ArticleDOI
On the influence of the nonlinear term in the numerical approximation of Incompressible Flows by means of proper orthogonal decomposition methods
TL;DR: In this article , proper orthogonal decomposition (POD) methods to approximate the incompressible Navier-Stokes equations were considered and an additional error term appeared in this case, compared with the case in which the same discretization of the nonlinear term is applied for both the FOM and the POD methods.
Journal ArticleDOI
Structure preserving reduced order modeling for gradient systems
TL;DR: A reduced order model (ROM) is developed which preserves the gradient dissipative structure and demonstrates that the POD-DEIM reduced order solutions preserve the energy dissipation over time and at the steady state.
Journal ArticleDOI
POD-ROMs for incompressible flows including snapshots of the temporal derivative of the full order solution
TL;DR: This paper studies the importance of including snapshots that approach the velocity time derivative in the numerical approximation of the incompressible Navier-Stokes equations by means of proper orthogonal decomposition (POD) methods and proves that pointwise in time error bounds can be proved.
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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