A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations
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TLDR
A new definition of PGD is introduced, called Minimax PGD, which can be interpreted as a Petrov–Galerkin model reduction technique, where test and trial reduced basis functions are related by an adjoint problem, and improves convergence properties of separated representations with respect to a chosen metric.About:
This article is published in Computer Methods in Applied Mechanics and Engineering.The article was published on 2010-04-15 and is currently open access. It has received 298 citations till now. The article focuses on the topics: Partial differential equation & Separation of variables.read more
Citations
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Journal ArticleDOI
A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems
TL;DR: Model reduction aims to reduce the computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior as mentioned in this paper. But model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books.
Journal ArticleDOI
A Short Review on Model Order Reduction Based on Proper Generalized Decomposition
TL;DR: This paper revisits a new model reduction methodology based on the use of separated representations, the so called Proper Generalized Decomposition—PGD, which allows to treat efficiently models defined in degenerated domains as well as the multidimensional models arising from multiddimensional physics or from the standard ones when some sources of variability are introduced in the model as extra-coordinates.
MonographDOI
Low-rank methods for high-dimensional approximation and model order reduction
TL;DR: The problem of best approximation in subsets of low-rank tensors is analyzed and its connection with the problem of optimal model reduction in low-dimensional reduced spaces is discussed.
Journal ArticleDOI
Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions
TL;DR: The demonstrated convergence is comparable to or even better than the one of the DMRG algorithm, and the proposed algorithms are also efficient for non-SPD systems, for example, those arising from the chemical master equation describing the gene regulatory model at the mesoscopic scale.
Journal ArticleDOI
Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems
TL;DR: An efficient algorithm is proposed for the a priori construction of separated representations of square integrable vector-valued functions defined on a high-dimensional probability space, which are the solutions of systems of stochastic algebraic equations.
References
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Book
Convex analysis and variational problems
Ivar Ekeland,Roger Téman +1 more
TL;DR: In this article, the authors consider non-convex variational problems with a priori estimate in convex programming and show that they can be solved by the minimax theorem.
Journal ArticleDOI
The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows
TL;DR: The Navier-Stokes equations are well-known to be a good model for turbulence as discussed by the authors, and the results of well over a century of increasingly sophisticated experiments are available at our disposal.
Journal ArticleDOI
Galerkin Proper Orthogonal Decomposition Methods for a General Equation in Fluid Dynamics
Karl Kunisch,Stefan Volkwein +1 more
TL;DR: Error estimates for Galerkin proper orthogonal decomposition (POD) methods for nonlinear parabolic systems arising in fluid dynamics are proved and the backward Euler scheme is considered.