Computational multiscale methods for quasi-gas dynamic equations

Chetverushkin, Boris; Chung, Eric; Efendiev, Yalchin; Pun, Sai-Mang; Zhang, Zecheng

doi:10.48550/arxiv.2009.00068

Cited by 3 publications

(11 citation statements)

References 41 publications

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“…Proposition 3.1 (Lemma 4.5 in [11]). Assume that {χ ms j,k } Nc j=1 is a set of bilinear partition of unity.…”

Section: Fully Discretizationmentioning

confidence: 99%

“…We show that the fully discretization is stable in time. To this aim, we first recall the inverse inequality for the multiscale space, which is proved in [11].…”

Section: Fully Discretizationmentioning

confidence: 99%

“…It also relies on the constant C inv which comes from the inverse inequality (see Lemma 4.5 in [11]). From the proof of that lemma, the constant C inv is independent of the ratio of the contrast values in the permeability.…”

Section: Fully Discretizationmentioning

confidence: 99%

“…The second property makes it possible to solve the problem as an equation evolving in z direction; we hence can discretize (∂ z v) by applying classical difference scheme. Consequently, we obtain a quasi-gas-dynamic (QGD) equation which have been thoroughly studied in literature [11,12,13].…”

Section: Introductionmentioning

confidence: 99%

“…If we discretize z evolution using some classical difference scheme, each z level is a QGD model with heterogeneous. This model can be solved in the framework of CEM-GMsFEM [11]; however the coarse scale basis evaluation is time consuming; in particular, this process will be repeated for each level of z. We hence proposed the proper orthogonal decomposition (POD) technique [39] which is target to find low dimensional subspace such that the error of the orthogonal projection is minimized in the sense of the norm induced by the inner product of the original space.…”

Section: Introductionmentioning

confidence: 99%

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Computational multiscale methods for parabolic wave approximations in heterogeneous media

Chung,

Efendiev,

Pun

et al. 2021

Preprint

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In this paper, we develop a computational multiscale to solve the parabolic wave approximation with heterogeneous and variable media. Parabolic wave approximation is a technique to approximate the full wave equation. One benefit of the method is that: one wave propagation direction can be taken as an evolution direction, and we then can discretize it using a classical scheme like Backward Euler. Consequently, we obtain a set of quasi-gas-dynamic (QGD) models with different heterogeneous permeability fields. Then, we employ constraint energy minimization generalized multiscale finite element method (CEM-GMsFEM) to perform spatial discretization for the problem. The resulting system can be solved by combining the central difference in time evolution. Due to the variable media, we apply the technique of proper orthogonal decomposition (POD) to further the dimension of the problem and solve the corresponding model problem in the POD space instead of in the whole multiscale space spanned by all possible multiscale basis functions. We prove the stability of the full discretization scheme and give the convergence analysis of the proposed approximation scheme. Numerical results verify the effectiveness of the proposed method.

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“…Proposition 3.1 (Lemma 4.5 in [11]). Assume that {χ ms j,k } Nc j=1 is a set of bilinear partition of unity.…”

Section: Fully Discretizationmentioning

confidence: 99%

“…We show that the fully discretization is stable in time. To this aim, we first recall the inverse inequality for the multiscale space, which is proved in [11].…”

Section: Fully Discretizationmentioning

confidence: 99%

Section: Fully Discretizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Computational multiscale methods for parabolic wave approximations in heterogeneous media

Chung,

Efendiev,

Pun

et al. 2021

Preprint

Self Cite

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Multi-agent Reinforcement Learning Accelerated MCMC on Multiscale Inversion Problem

Chung,

Efendiev,

Leung

et al. 2020

Preprint

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In this work, we propose a multi-agent actor-critic reinforcement learning (RL) algorithm to accelerate the multi-level Monte Carlo Markov Chain (MCMC) sampling algorithms. The policies (actors) of the agents are used to generate the proposal in the MCMC steps; and the critic, which is centralized, is in charge of estimating the long term reward. We verify our proposed algorithm by solving an inverse problem with multiple scales. There are several difficulties in the implementation of this problem by using traditional MCMC sampling. Firstly, the computation of the posterior distribution involves evaluating the forward solver, which is very time consuming for a problem with heterogeneous. We hence propose to use the multi-level algorithm. More precisely, we use the generalized multiscale finite element method (GMsFEM) as the forward solver in evaluating a posterior distribution in the multilevel rejection procedure. Secondly, it is hard to find a function which can generate samplings which are meaningful. To solve this issue, we learn an RL policy as the proposal generator. Our experiments show that the proposed method significantly improves the sampling process.

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Multi-variance replica exchange stochastic gradient MCMC for inverse and forward Bayesian physics-informed neural network

Lin,

Wang,

Zhang

2021

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Physics-informed neural network (PINN) has been successfully applied in solving a variety of nonlinear non-convex forward and inverse problems. However, the training is challenging because of the non-convex loss functions and the multiple optima in the Bayesian inverse problem. In this work, we propose a multi-variance replica exchange stochastic gradient Langevin diffusion method to tackle the challenge of the multiple local optima in the optimization and the challenge of the multiple modal posterior distribution in the inverse problem. Replica exchange methods are capable of escaping from the local traps and accelerating the convergence; two chains with different temperatures are designed where the low temperature chain aims for the local convergence, and the target of the high temperature chain is to travel globally and explore the whole loss function entropy landscape. However, it may not be efficient to solve mathematical inversion problems by using the vanilla replica method directly since the method doubles the computational cost in evaluating the forward solvers (likelihood functions) in the two chains. To address this issue, we propose to make different assumptions on the energy function estimation and this facilities one to use solvers of different fidelities in the likelihood function evaluation. More precisely, one can use a solver with low fidelity in the high temperature chain while use a solver with high fidelity in the low temperature chain. Our proposed method significantly lowers the computational cost in the high temperature chain, meanwhile preserves the accuracy and converges very fast. We give an unbiased estimate of the swapping rate and give an estimation of the discretization error of the scheme. To verify our idea, we design and solve four inverse problems which have multiple modes. The proposed method is also employed to train the Bayesian PINN to solve the forward and inverse problems; faster and more accurate convergence has been observed when compared to the stochastic gradient Langevin diffusion (SGLD) method and vanila replica exchange methods.

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Computational multiscale methods for quasi-gas dynamic equations

Cited by 3 publications

References 41 publications

Computational multiscale methods for parabolic wave approximations in heterogeneous media

Computational multiscale methods for parabolic wave approximations in heterogeneous media

Multi-agent Reinforcement Learning Accelerated MCMC on Multiscale Inversion Problem

Multi-variance replica exchange stochastic gradient MCMC for inverse and forward Bayesian physics-informed neural network

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