Journal ArticleDOI
Galerkin proper orthogonal decomposition methods for parabolic problems
Karl Kunisch,Stefan Volkwein +1 more
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TLDR
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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Model order reduction of nonlinear dynamic systems using multiple projection bases and optimized state-space sampling
TL;DR: The concept of transforming the nonlinear representation into a composite structure of well defined basic functions from multiple projection bases is used, which offers a new perspective to the problem of model order reduction of nonlinear systems and the tracking or preservation of physical parameters in the final compact model.
Journal ArticleDOI
A reduced-order LSMFE formulation based on POD method and implementation of algorithm for parabolic equations
TL;DR: In this article, a reduced-order least-squares mixed finite element (LSMFE) formulation for two-dimensional parabolic equations with real practical applied background is presented.
Dissertation
POD-Galerkin Modeling for Incompressible Flows with Stochastic Boundary Conditions
TL;DR: It is demonstrated that the reduced-order solutions of the considered problems converge toward the underlying snapshots when the dimension of the POD basis is increased, leading to a significant speed-up of the overall computational process compared to a standard finite element model.
Journal ArticleDOI
Analysis of a space–time continuous Galerkin method for convection-dominated Sobolev equations
Zhihui Zhao,Hong Li,Zhendong Luo +2 more
TL;DR: It is proved the existence and uniqueness of the approximate solution and the optimal convergence rates in L ∞ ( H 1 ) norm which do not require any restriction assumptions on the space and time mesh size.
Journal ArticleDOI
Crank–Nicolson Finite Element Scheme and Modified Reduced-Order Scheme for Fractional Sobolev Equation
Jincun Liu,Hong Li,Yang Liu +2 more
TL;DR: In this article, a Crank-Nicolson finite difference/finite element method was used to obtain the numerical solution for a time fractional Sobolev equation, where the classical finite element finite difference method was replaced by a finite element 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.
Related Papers (5)
Galerkin Proper Orthogonal Decomposition Methods for a General Equation in Fluid Dynamics
Karl Kunisch,Stefan Volkwein +1 more