Reduced Basis Methods: Success, Limitations and Future Challenges

Ohlberger, Mario; Rave, Stephan

doi:10.48550/arxiv.1511.02021

Cited by 22 publications

(28 citation statements)

References 29 publications

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“…The fact that these quantities decay slower for transport-dominant problems is consistent with the slow decay of the Kolmogorov N -width for transport problems, see e.g. [20,35]. As a result, the performance of our method and many other linear ROMs suffer.…”

Section: N0supporting

confidence: 68%

A reduced basis method for radiative transfer equation

Peng

Chen

Cheng

et al. 2021

Preprint

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Linear kinetic transport equations play a critical role in optical tomography, radiative transfer and neutron transport. The fundamental difficulty hampering their efficient and accurate numerical resolution lies in the high dimensionality of the physical and velocity/angular variables and the fact that the problem is multiscale in nature. Leveraging the existence of a hidden low-rank structure hinted by the diffusive limit, in this work, we design and test the angularspace reduced order model for the linear radiative transfer equation, the first such effort based on the celebrated reduced basis method (RBM).Our method is built upon a high-fidelity solver employing the discrete ordinates method in the angular space, an asymptotic preserving upwind discontinuous Galerkin method for the physical space, and an efficient synthetic accelerated source iteration for the resulting linear system. Addressing the challenge of the parameter values (or angular directions) being coupled through an integration operator, the first novel ingredient of our method is an iterative procedure where the macroscopic density is constructed from the RBM snapshots, treated explicitly and allowing a transport sweep, and then updated afterwards. A greedy algorithm can then proceed to adaptively select the representative samples in the angular space and form a surrogate solution space. The second novelty is a least-squares density reconstruction strategy, at each of the relevant physical locations, enabling the robust and accurate integration over an arbitrarily unstructured set of angular samples toward the macroscopic density. Numerical experiments indicate that our method is highly effective for computational cost reduction in a variety of regimes.

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Section: N0supporting

confidence: 68%

A reduced basis method for radiative transfer equation

Peng

Chen

Cheng

et al. 2021

Preprint

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“…where d (S h ; S n ) is the maximum distance between S n and S h . Theories show that Kolmogorov n-width decreases with the increase of n, but the decay rate depends on the problem [29]. For the linear trial subspace S n generated by POD, the Kolmogorov n-width defined above decays quickly for diffusion-dominant problems.…”

Section: Projection Based Rommentioning

confidence: 99%

“…However, Kolmogorov n-width decays slowly for convection-dominant problems. We must increase the order of ROM largely to obtain the required accuracy [29]. Next, we introduce a convolutional autoencoder based nonlinear ROM.…”

Section: Projection Based Rommentioning

confidence: 99%

Solving time-dependent parametric PDEs by multiclass classification-based reduced order model

Chen¹,

Jiang²,

Shu³

2021

Preprint

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In this paper, we propose a network model, the multiclass classification-based ROM (MC-ROM), for solving time-dependent parametric partial differential equations (PPDEs). This work is inspired by the observation of applying the deep learning-based reduced order model (DL-ROM) [1] to solve diffusion-dominant PPDEs. We find that the DL-ROM has a good approximation for some special model parameters, but it cannot approximate the drastic changes of the solution as time evolves. Based on this fact, we classify the dataset according to the magnitude of the solutions, and construct corresponding subnets dependent on different types of data. Then we train a classifier to integrate different subnets together to obtain the MC-ROM. When subsets have the same architecture, we can use transfer learning technology to accelerate the offline training. Numerical experiments show that the MC-ROM improves the generalization ability of the DL-ROM both for diffusion-and convection-dominant problems, and maintains the DL-ROM's advantage of good approximation ability. We also compare the approximation accuracy and computational efficiency of the proper orthogonal decomposition (POD) which is not suitable for convection-dominant problems. For diffusion-dominant problems, the MC-ROM can save about 100 times online computational cost than the POD with a slightly better approximation in the reduced space of the same dimension.

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“…Galerkin POD is unable to capture the dynamics of the projection coefficients for the entire testing set. In fact, classical POD techniques struggle to resolve advection-dominated problems, and must be augmented by various stabilization techniques in order to be effective [11,25,26,27]. On the other hand, the neural network is able to maintain accuracy over the time interval.…”

Section: Two-dimensional Casementioning

confidence: 99%

Reduced Basis Approximations of Parameterized Dynamical Partial Differential Equations via Neural Networks

Peter¹,

Beckwith²,

Cyr³

et al. 2021

Preprint

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Projection-based reduced order models are effective at approximating parameter-dependent differential equations that are parametrically separable. When parametric separability is not satisfied, which occurs in both linear and nonlinear problems, projection-based methods fail to adequately reduce the computational complexity. Devising alternative reduced order models is crucial for obtaining efficient and accurate approximations to expensive high-fidelity models. In this work, we develop a time-stepping procedure for dynamical parameter-dependent problems, in which a neural-network is trained to propagate the coefficients of a reduced basis expansion. This results in an online stage with a computational cost independent of the size of the underlying problem. We demonstrate our method on several parabolic partial differential equations, including a problem that is not parametrically separable.

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Reduced Basis Methods: Success, Limitations and Future Challenges

Cited by 22 publications

References 29 publications

A reduced basis method for radiative transfer equation

A reduced basis method for radiative transfer equation

Solving time-dependent parametric PDEs by multiclass classification-based reduced order model

Reduced Basis Approximations of Parameterized Dynamical Partial Differential Equations via Neural Networks

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