Approximated Lax pairs for the reduced order integration of nonlinear evolution equations
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TLDR
A reduced-order model algorithm, called ALP, is proposed to solve nonlinear evolution partial differential equations, based on approximations of generalized Lax pairs, which is well-suited to solving problems with progressive front or wave propagation.About:
This article is published in Journal of Computational Physics.The article was published on 2014-05-01 and is currently open access. It has received 66 citations till now. The article focuses on the topics: Lax equivalence theorem & Lax pair.read more
Citations
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Journal ArticleDOI
Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders
Kookjin Lee,Kevin Carlberg +1 more
TL;DR: The ability of the method to significantly outperform even the optimal linear-subspace ROM on benchmark advection-dominated problems is demonstrated, thereby demonstrating the method's ability to overcome the intrinsic $n$-width limitations of linear subspaces.
Reduced Basis Methods: Success, Limitations and Future Challenges
Mario Ohlberger,Stephan Rave +1 more
TL;DR: This contribution discusses what is known about the convergence properties of these methods: when they succeed and when they are bound to fail, and highlights some recent approaches employing nonlinear approximation techniques which aim to overcome the current limitations of reduced basis methods.
Posted Content
A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs
TL;DR: Numerical results indicate that DL-ROMs whose dimension is equal to the intrinsic dimensionality of the PDE solutions manifold are able to efficiently approximate the solution of parametrized PDEs, especially in cases for which a huge number of POD modes would have been necessary to achieve the same degree of accuracy.
Book ChapterDOI
Model Order Reduction for Problems with large Convection Effects
TL;DR: A simple approach towards this direction, preliminary simulations support this approach and the set of solutions needs to be transformed/twisted so that the combination of the proper twist and the appropriate linear combination recovers an accurate approximation.
References
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Book
Solitons and the Inverse Scattering Transform
Mark J. Ablowitz,Harvey Segur +1 more
TL;DR: In this paper, the authors developed the theory of the inverse scattering transform (IST) for ocean wave evolution, which can be solved exactly by the soliton solution of the Korteweg-deVries equation.
Book
Integrals of Nonlinear Equations of Evolution and Solitary Waves
TL;DR: In this article, a general principle for associating nonlinear equations evolutions with linear operators so that the eigenvalues of the linear operator integrals of the nonlinear equation can be found is presented, where the main tool used is the first remarkable series of integrals discovered by Kruskal and Zabusky.
Book
Solitons: An Introduction
P. G. Drazin,Robin Johnson +1 more
TL;DR: In this article, the authors introduce the Inverse Scattering Transform (IST) and its application in the theory of solitons and its applications to nonlinear systems that arise in the physical sciences.
Journal ArticleDOI
Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations
TL;DR: (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equations are considered.