Contrast-Independent, Partially-Explicit Time Discretizations for Nonlinear Multiscale Problems

Chung, Eric T.; Efendiev, Yalchin; Leung, Wing Tat; Li, Wenyuan

doi:10.3390/math9233000

Cited by 10 publications

(5 citation statements)

References 39 publications

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“…In this paper, we use the partially explicit splitting scheme introduced in [18]. We split the space V H to be the direct sum of two subspaces V H,1 and V H,2 , i.e.…”

Section: Partially Explicit Schemementioning

confidence: 99%

“…We refer to [28,18] for the details of convergence assumptions and stability analysis of the above scheme.…”

Section: Partially Explicit Schemementioning

confidence: 99%

“…For example, porous media flow and transport occur over many space and time scales. Many multiscale and splitting algorithms have been proposed to solve these problems [28,18,41]. In this paper, we would like to combine machine learning and splitting/multiscale approaches to predict the dynamics of complex forward problems.…”

Section: Introductionmentioning

confidence: 99%

“…We couple them and obtain the final solution after we solve the smaller-scale problems. Recently, a partially explicit splitting method [28,18,24] is proposed for linear and nonlinear multiscale problems. In this approach, we treat some solution parts implicitly while treating the other parts explicitly.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Hybrid explicit-implicit learning for multiscale problems with time dependent source

Efendiev¹,

Leung²,

Li³

et al. 2022

Preprint

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The splitting method is a powerful method for solving partial differential equations. Various splitting methods have been designed to separate different physics, nonlinearities, and so on. Recently, a new splitting approach has been proposed where some degrees of freedom are handled implicitly while other degrees of freedom are handled explicitly. As a result, the scheme contains two equations, one implicit and the other explicit. The stability of this approach has been studied. It was shown that the time step scales as the coarse spatial mesh size, which can provide a significant computational advantage. However, the implicit solution part can be expensive, especially for nonlinear problems. In this paper, we introduce modified partial machine learning algorithms to replace the implicit solution part of the algorithm. These algorithms are first introduced in 'HEI: Hybrid Explicit-Implicit Learning For Multiscale Problems', where a homogeneous source term is considered along with the Transformer, which is a neural network that can predict future dynamics. In this paper, we consider time-dependent source terms which is a generalization of the previous work. Moreover, we use the whole history of the solution to train the network. As the implicit part of the equations is more complicated to solve, we design a neural network to predict it based on training. Furthermore, we compute the explicit part of the solution using our splitting strategy. In addition, we use Proper Orthogonal Decomposition based model reduction in machine learning. The machine learning algorithms provide computational saving without sacrificing accuracy. We present three numerical examples which show that our machine learning scheme is stable and accurate.

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“…In this paper, we use the partially explicit splitting scheme introduced in [18]. We split the space V H to be the direct sum of two subspaces V H,1 and V H,2 , i.e.…”

Section: Partially Explicit Schemementioning

confidence: 99%

“…We refer to [28,18] for the details of convergence assumptions and stability analysis of the above scheme.…”

Section: Partially Explicit Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Hybrid explicit-implicit learning for multiscale problems with time dependent source

Efendiev¹,

Leung²,

Li³

et al. 2022

Preprint

Self Cite

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show abstract

“…The time step depends on the contrast. In [4], a partially explicit approach was first proposed for heterogeneous parabolic equations. The main idea of this approach is to handle some degrees of freedom implicitly, while the rest explicitly.…”

Section: Introductionmentioning

confidence: 99%

Nonlocal transport equations in multiscale media. Modeling, dememorization, and discretizations

Efendiev¹,

Leung²,

Li³

et al. 2022

Preprint

Self Cite

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In this paper, we consider a class of convection-diffusion equations with memory effects. These equations arise as a result of homogenization or upscaling of linear transport equations in heterogeneous media and play an important role in many applications. First, we present a dememorization technique for these equations. We show that the convection-diffusion equations with memory effects can be written as a system of standard convection diffusion reaction equations. This allows removing the memory term and simplifying the computations. We consider a relation between dememorized equations and micro-scale equations, which do not contain memory terms. We note that dememorized equations differ from microscale equations and constitute a macroscopic model. Next, we consider both implicit and partially explicit methods. The latter is introduced for problems in multiscale media with high-contrast properties. Because of high-contrast, explicit methods are restrictive and require time steps that are very small (scales as the inverse of the contrast). We show that, by appropriately decomposing the space, we can treat only a few degrees of freedom implicitly and the remaining degrees of freedom explicitly. We present a stability analysis. Numerical results are presented that confirm our theoretical findings about partially explicit schemes applied to dememorized systems of equations.

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Partial learning using partially explicit discretization for multicontinuum/multiscale problems with limited observation: Language interactions simulation

Ammosov

Stepanov

Tyrylgin

et al. 2023

Journal of Computational and Applied Mathematics

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Contrast-Independent, Partially-Explicit Time Discretizations for Nonlinear Multiscale Problems

Cited by 10 publications

References 39 publications

Hybrid explicit-implicit learning for multiscale problems with time dependent source

Hybrid explicit-implicit learning for multiscale problems with time dependent source

Nonlocal transport equations in multiscale media. Modeling, dememorization, and discretizations

Partial learning using partially explicit discretization for multicontinuum/multiscale problems with limited observation: Language interactions simulation

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