Energy preserving model order reduction of the nonlinear Schrödinger equation

Karasözen, Bülent; Uzunca, Murat

doi:10.1007/s10444-018-9593-9

Cited by 25 publications

(22 citation statements)

References 64 publications

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“…based on H 1 . In this work we are interested in tackling the issues of accuracy and stability through the promising approach of structurepreserving model reduction, in which reduced-order models are developed in such a way that invariants and/or symmetries of the full model are kept [2,3,14,27,35]. An example is a ROM that inherits the symplectic form of a Hamiltonian system, leading to a ROM that is applicable to stable long-time integration [35].…”

Section: Introductionmentioning

confidence: 99%

Non-linearly stable reduced-order models for incompressible flow with energy-conserving finite volume methods

Sanderse

2020

Journal of Computational Physics

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A novel reduced-order model (ROM) formulation for incompressible flows is presented with the key property that it exhibits non-linearly stability, independent of the mesh (of the full order model), the time step, the viscosity, and the number of modes. The two essential elements to non-linear stability are: (1) first discretise the full order model, and then project the discretised equations, and (2) use spatial and temporal discretisation schemes for the full order model that are globally energy-conserving (in the limit of vanishing viscosity). For this purpose, as full order model a staggered-grid finite volume method in conjunction with an implicit Runge-Kutta method is employed. In addition, a constrained singular value decomposition is employed which enforces global momentum conservation. The resulting 'velocity-only' ROM is thus globally conserving mass, momentum and kinetic energy. For non-homogeneous boundary conditions, a (onetime) Poisson equation is solved that accounts for the boundary contribution. The stability of the proposed ROM is demonstrated in several test cases. Furthermore, it is shown that explicit Runge-Kutta methods can be used as a practical alternative to implicit time integration at a slight loss in energy conservation.

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Section: Introductionmentioning

confidence: 99%

Non-linearly stable reduced-order models for incompressible flow with energy-conserving finite volume methods

Sanderse

2020

Journal of Computational Physics

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“…Alla and Kutz proposed the use of randomized methods for computing the SVD of snapshot matrices and for dynamic mode decomposition. Buhr and Smetana and Karasözen and Uzunca also used randomized SVD methods for model reduction. However, no detailed comparison has been made to other low‐rank approximation algorithms, such as the incremental or the iterative SVD.…”

Section: Introductionmentioning

confidence: 99%

Randomized low‐rank approximation methods for projection‐based model order reduction of large nonlinear dynamical problems

Bach

Ceglia

Song

et al. 2019

Numerical Meth Engineering

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Projection-based nonlinear model order reduction (MOR) methods typically make use of a reduced basis V ∈ R m×k to approximate high-dimensional quantities. However, the most popular methods for computing V, eg, through a singular value decomposition of an m × n snapshot matrix, have asymptotic time complexities of (min(mn 2 , m 2 n)) and do not scale well as m and n increase. This is problematic for large dynamical problems with many snapshots, eg, in case of explicit integration. In this work, we propose the use of randomized methods for reduced basis computation and nonlinear MOR, which have an asymptotic complexity of only (mnk) or (mn log(k)). We evaluate the suitability of randomized algorithms for nonlinear MOR and compare them to other strategies that have been proposed to mitigate the demanding computing times incurred by large nonlinear models. We analyze the computational complexities of traditional, iterative, incremental, and randomized algorithms and compare the computing times and accuracies for numerical examples. The results indicate that randomized methods exhibit an extremely high level of accuracy in practice, while generally being faster than any other analyzed approach. We conclude that randomized methods are highly suitable for the reduction of large nonlinear problems. KEYWORDSexplicit FEM, low-rank approximation, nonlinear dynamics, nonlinear model order reduction, randomized numerical linear algebra, randomized SVD Int J Numer Methods Eng. 2019;118:209-241.wileyonlinelibrary.com/journal/nme

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“…In this paper, we have constructed two different ROMs for the SWEs, in Hamiltonian form and in f ‐plane. Our main contribution is twofold: ROMs are constructed that preserve the reduced skew‐symmetry in the Poisson matrix of the SWEs with the energy‐preserving time integrator AVF method as in Gong et al and Karasözen and Uzunca 30,31 . We show that the reduced discrete energy (Hamiltonian) and other conserved quantities like enstrophy, mass, and vorticity are well‐preserved in the long‐term using POD and DEIM. Semidiscrete form of the SWE in the f ‐plane results in an ODE with quadratic nonlinearities, which is solved in time by linearly implicit Kahan's method.…”

Section: Introductionmentioning

confidence: 96%

Structure preserving model order reduction of shallow water equations

Karasözen

Yıldız

Uzunca

2020

Math Methods in App Sciences

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In this paper, we present two different approaches for constructing reducedorder models (ROMs) for the two-dimensional shallow water equation (SWE). The first one is based on the noncanonical Hamiltonian/Poisson form of the SWE. After integration in time by the fully implicit average vector field method, ROMs are constructed with proper orthogonal decomposition(POD)/discrete empirical interpolation method that preserves the Hamiltonian structure. In the second approach, the SWE as a partial differential equation with quadratic nonlinearity is integrated in time by the linearly implicit Kahan's method, and ROMs are constructed with the tensorial POD that preserves the linear-quadratic structure of the SWE. We show that in both approaches, the invariants of the SWE such as the energy, enstrophy, mass and circulation are preserved over a long period of time, leading to stable solutions. We conclude by demonstrating the accuracy and the computational efficiency of the reduced solutions by a numerical test problem.

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Energy preserving model order reduction of the nonlinear Schrödinger equation

Cited by 25 publications

References 64 publications

Non-linearly stable reduced-order models for incompressible flow with energy-conserving finite volume methods

Non-linearly stable reduced-order models for incompressible flow with energy-conserving finite volume methods

Randomized low‐rank approximation methods for projection‐based model order reduction of large nonlinear dynamical problems

Structure preserving model order reduction of shallow water equations

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