Contrast-independent partially explicit time discretizations for multiscale wave problems

Chung, Eric T.; Efendiev, Yalchin; Leung, Wing Tat; Vabishchevich, Petr N.

doi:10.48550/arxiv.2102.13198

Cited by 3 publications

(15 citation statements)

References 35 publications

(44 reference statements)

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“…In this section, we will review the partially explicit splitting scheme [12,11] which is designed to tackle the multiscale high contrast problems. The scheme is based on the multiscale finite element methods [9,13] and the stability of the scheme is independent of the contrast ratio of the problem.…”

Section: Preliminariesmentioning

confidence: 99%

“…where u H = u H,1 + u H,2 . For the time discretization, we consider the temporal splitting introduced in [12] for linear problems and in [11] for nonlinear problems. That is, we can use a partially explicit time discretization.…”

Section: Preliminariesmentioning

confidence: 99%

“…The bilinear form is then given by a(u, v) = Ω κ(x)∇u∇vdx for u, v ∈ V H and Ω is the domain of the problem. For the stability and convergence concerns of the scheme, we refer to [12,11].…”

Section: Preliminariesmentioning

confidence: 99%

“…To achieve this, one needs to include all high-contrast multiscale features in u 1 . This construction is described in [12] and uses GMsFEM and NLMC concepts developed in [9,13]. Other multiscale methods have been developed for various applications [17,19,20,5,6,7,10,16,8,9,13,15,18,23,24,1,3], which we do not mention.…”

Section: Introductionmentioning

confidence: 99%

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HEI: hybrid explicit-implicit learning for multiscale problems

Efendiev,

Leung,

Lin

et al. 2021

Preprint

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Splitting method is a powerful method to handle application problems by splitting physics, scales, domain, and so on. Many splitting algorithms have been designed for efficient temporal discretization. In this paper, our goal is to use temporal splitting concepts in designing machine learning algorithms and, at the same time, help splitting algorithms by incorporating data and speeding them up. Since the spitting solution usually has an explicit and implicit part, we will call our method hybrid explicit-implict (HEI) learning. We will consider a recently introduced multiscale splitting algorithms, where the multiscale problem is solved on a coarse grid. To approximate the dynamics, only a few degrees of freedom are solved implicitly, while others explicitly. This splitting concept allows identifying degrees of freedom that need implicit treatment. In this paper, we use this splitting concept in machine learning and propose several strategies. First, the implicit part of the solution can be learned as it is more difficult to solve, while the explicit part can be computed. This provides a speed-up and data incorporation for splitting approaches. Secondly, one can design a hybrid neural network architecture because handling explicit parts requires much fewer communications among neurons and can be done efficiently. Thirdly, one can solve the coarse grid component via PDEs or other approximation methods and construct simpler neural networks for the explicit part of the solutions. We discuss these options and implement one of them by interpreting it as a machine translation task. This interpretation of the splitting scheme successfully enables us using the Transformer since it can perform model reduction for multiple time series and learn the connection between them. We also find that the splitting scheme is a great platform to predict the coarse solution with insufficient information of the target model: the target problem is partially given and we need to solve it through a known problem which approximates the target. Our machine learning model can incorporate and encode the given information from two different problems and then solve the target problems. We conduct four numerical examples and the results show that our method is stable and accurate.

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Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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HEI: hybrid explicit-implicit learning for multiscale problems

Efendiev,

Leung,

Lin

et al. 2021

Preprint

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“…Our proposed approach provides explicit treatment for nonlinear terms (f (u) and g(u)), while we still have implicit coupling in time derivative part of the discrete system. The coupling in time derivative can be removed by designing some mass lumping (see [13]). However, the design of mass lumping for time fractional diffusion is challenging and we leave it for future studies.…”

Section: Introductionmentioning

confidence: 99%

Partially Explicit Time Discretization for Nonlinear Time Fractional Diffusion Equations

Li,

Alikhanov,

Efendiev

et al. 2021

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Nonlinear time fractional partial differential equations are widely used in modeling and simulations. In many applications, there are high contrast changes in media properties. For solving these problems, one often uses coarse spatial grid for spatial resolution. For temporal discretization, implicit methods are often used. For implicit methods, though the time step can be relatively large, the equations are difficult to compute due to the nonlinearity and the fact that one deals with large-scale systems. On the other hand, the discrete system in explicit methods are easier to compute but it requires small time steps. In this work, we propose the partially explicit scheme following earlier works on developing partially explicit methods for nonlinear diffusion equations. In this scheme, the diffusion term is treated partially explicitly and the reaction term is treated fully explicitly. With the appropriate construction of spaces and stability analysis, we find that the required time step in our proposed scheme scales as the coarse mesh size, which creates a great saving in computing. The main novelty of this work is the extension of our earlier works for diffusion equations to time fractional diffusion equations. For the case of fractional diffusion equations, the constraints on time steps are more severe and the proposed methods alleviate this since the time step in partially explicit method scales as the coarse mesh size. We present stability results. Numerical results are presented where we compare our proposed partially explicit methods with a fully implicit approach. We show that our proposed approach provides similar results, while treating many degrees of freedom in nonlinear terms explicitly.

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Multirate partially explicit scheme for multiscale flow problems

Leung¹,

Wang²

2021

Preprint

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For time-dependent problems with high-contrast multiscale coefficients, the time step size for explicit methods is affected by the magnitude of the coefficient parameter. With a suitable construction of multiscale space, one can achieve a stable temporal splitting scheme where the time step size is independent of the contrast [15]. Consider the parabolic equation with heterogeneous diffusion parameter, the flow rates vary significantly in different regions due to the high-contrast features of the diffusivity. In this work, we aim to introduce a multirate partially explicit splitting scheme to achieve efficient simulation with the desired accuracy. We first design multiscale subspaces to handle flow with different speed. For the fast flow, we obtain a low-dimensional subspace with respect to the high-diffusive component and adopt an implicit time discretization scheme. The other multiscale subspace will take care of the slow flow, and the corresponding degrees of freedom are treated explicitly. Then a multirate time stepping is introduced for the two parts. The stability of the multirate methods is analyzed for the partially explicit scheme. Moreover, we derive local error estimators corresponding to the two components of the solutions and provide an upper bound of the errors. An adaptive local temporal refinement framework is then proposed to achieve higher computational efficiency. Several numerical tests are presented to demonstrate the performance of the proposed method.

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Contrast-independent partially explicit time discretizations for multiscale wave problems

Cited by 3 publications

References 35 publications

HEI: hybrid explicit-implicit learning for multiscale problems

HEI: hybrid explicit-implicit learning for multiscale problems

Partially Explicit Time Discretization for Nonlinear Time Fractional Diffusion Equations

Multirate partially explicit scheme for multiscale flow problems

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