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Group Projected Subspace Pursuit for Block Sparse Signal Reconstruction: Convergence Analysis and Applications

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arXiv 2024年

作者： He, Roy Y. Liu, Haixia Liu, Hao Department of Mathematics City University of Hong Kong Kowloon Tong Hong Kong School of Mathematics and Statistics Institute of Interdisciplinary Research for Mathematics and Applied Science Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan China Department of Mathematics Hong Kong Baptist University Kowloon Tong Hong Kong

In this paper, we present a convergence analysis of the Group Projected Subspace Pursuit (GPSP) algorithm proposed by He et al. [HKL+23] (Group Projected subspace pursuit for IDENTification of variable coefficient differential equations (GP-IDENT), Journal of Computational Physics, 494, 112526) and extend its application to general tasks of block sparse signal recovery. We prove that when the sampling matrix satisfies the Block Restricted Isometry Property (BRIP) with a sufficiently small Block Restricted Isometry Constant (BRIC), GPSP exactly recovers the true block sparse signals. When the observations are noisy, this convergence property of GPSP remains valid if the magnitude of the true signal is sufficiently large. GPSP selects the features by subspace projection criterion (SPC) for candidate inclusion and response magnitude criterion (RMC) for candidate exclusion. We compare these criteria with counterparts of other state-of-the-art greedy algorithms. Our theoretical analysis and numerical ablation studies reveal that SPC is critical to the superior performances of GPSP, and that RMC can enhance the robustness of feature identification when observations contain noises. We test and compare GPSP with other methods in diverse settings, including heterogeneous random block matrices, inexact observations, face recognition, and PDE identification. We find that GPSP outperforms the other algorithms in most cases for various levels of block sparsity and block sizes, justifying its effectiveness for general applications. © 2024, CC BY.

关键词： Face recognition

A ternary phase-field model for two-phase flows in complex geometries

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arXiv 2023年

作者： Zhan, Chengjie Chai, Zhenhua Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

In this work, a ternary phase-field model for two-phase flows in complex geometries is proposed. In this model, one of the three components in the classical ternary Cahn-Hilliard model is considered as the solid phase, and only one Cahn-Hilliard equation with degenerate mobility needs to be solved due to the condition of volume conservation, which is consistent with the standard phase-field model with a single-scalar variable for two-phase flows. To depict different wetting properties at the complex fluid-solid boundaries, the spreading parameters in ternary phase-field model are determined based on the Young's law, in which the liquid-solid surface tension coefficient is assumed to be a linear function of gas-liquid surface tension coefficient and related to the contact angle and the minimum curvature of the solid surface. In addition, to achieve a high viscosity in the solid phase and preserve the velocity boundary conditions on the solid surface, the phase-field variable of the solid phase is also used to derive the modified Navier-Stokes equations. To test the present model, we further develop a consistent and conservative Hermite-moment based lattice Boltzmann method where an adjustable scale factor is introduced to improve the numerical stability, and conduct the numerical simulations of several benchmark problems. The results illustrate that present model has the good capability in the study of the two-phase flows in complex geometries. © 2023, CC BY.

关键词： Two phase flow

Precise asymptotics on the Birkhoff sums for dynamical systems

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arXiv 2020年

作者： Fang, Lulu Takahasi, Hiroki Zhang, Yiwei School of Science Nanjing University of Science and Technology Nanjing210094 China Department of Mathematics Keio University Yokohama223-8522 Japan School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Sciences and Technology Wuhan430074 China

We establish two precise asymptotic results on the Birkhoff sums for dynamical systems. These results are parallel to that on the arithmetic sums of independent and identically distributed random variables previously obtained by Hsu and Robbins, Erdős, Heyde. We apply our results to the Gauss map and obtain new precise asymptotics in the theorem of Lévy on the regular continued fraction expansion of irrational numbers in (0, 1).MSC Codes 37A44, 37A50, 60F10 © 2020, CC BY.

关键词： Dynamical systems

A third-order discrete unified gas kinetic scheme for continuum and rarefied flows: Low-speed isothermal case

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arXiv 2017年

作者： Wu, Chen Shu, Chang Shi, Baochang Chen, Zhen State Key Laboratory of Coal Combustion Huazhong University of Science and Technology Wuhan430074 China Department of Mechanical Engineering National University of Singapore 10 Kent Ridge Crescent Singapore119260 Singapore School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

An efficient third-order discrete unified gas kinetic scheme (DUGKS) with efficiency is presented in this work for simulating continuum and rarefied flows. By employing two-stage time-stepping scheme and the high-order DUGKS flux reconstruction strategy, third-order of accuracy in both time and space can be achieved in the present method. It is also analytically proven that the second-order DUGKS is a special case of the present method. Compared with the high-order lattice Boltzmann equation LBE based methods, the present method is capable to deal with the rarefied flows by adopting the Newton-Cotes quadrature to approximate the integrals of moments. Instead of being constrained by the second-order (or lower-order) of accuracy in time splitting scheme as in the conventional high-order Runge-Kutta (RK) based kinetic methods, the present method solves the original BE, which overcomes the limitation in time accuracy. Typical benchmark tests are carried out for comprehensive evaluation of the present method. It is observed in the tests that the present method is advantageous over the original DUGKS in accuracy and capturing delicate flow structures. Moreover, the efficiency of the present third-order method is also shown in simulating rarefied flows. Copyright © 2017, The Authors. All rights reserved.

关键词： Boltzmann equation

OBSERVABILITY INEQUALITY, LOG-TYPE HAUSDORFF CONTENT AND HEAT EQUATIONS

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arXiv 2024年

作者： Huang, Shanlin Wang, Gengsheng Wang, Ming School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Center for Applied Mathematics Tianjin University Tianjin300072 China School of Mathematics and Statistics HNP-LAMA Central South University Hunan Changsha410083 China

This paper studies observability inequalities for heat equations on both bounded domains and the whole space Rd. The observation sets are measured by log-type Hausdorff contents, which are induced by certain log-type gauge functions closely related to the heat kernel. On a bounded domain, we derive the observability inequality for observation sets of positive log-type Hausdorff content. Notably, the aforementioned inequality holds not only for all sets with Hausdorff dimension s for any s ∈ (d−1, d], but also for certain sets of Hausdorff dimension d − 1. On the whole space Rd, we establish the observability inequality for observation sets that are thick at the scale of the log-type Hausdorff content. Furthermore, we prove that for the 1-dimensional heat equation on an interval, the Hausdorff content we have chosen is an optimal scale for the observability inequality. To obtain these observability inequalities, we use the adapted Lebeau-Robiano strategy from [14]. For this purpose, we prove the following results at scale of the log-type Hausdorff content, the former being derived from the latter: We establish a spectral inequality/a Logvinenko-Sereda uncertainty principle;we set up a quantitative propagation of smallness of analytic functions;we build up a Remez’ inequality;and more fundamentally, we provide an upper bound for the log-type Hausdorff content of a set where a monic polynomial is small, based on an estimate in Lubinsky [35], which is ultimately traced back to the classical Cartan Lemma. In addition, we set up a capacity-based slicing lemma (related to the log-type gauge functions) and establish a quantitative relationship between Hausdorff contents and capacities. These tools are crucial in the studies of the aforementioned propagation of smallness in high-dimensional situations. Copyright © 2024, The Authors. All rights reserved.

关键词： Set theory

The optimal displacement of immiscible two-phase fluids in a pore doublet

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arXiv 2022年

作者： Shan, Fang Chai, Zhenhua Shi, Baochang Zhao, Meng School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

The displacement of multiphase fluid flow in a pore doublet is a fundamental problem, and is also of importance in understanding of the transport mechanisms of multiphase flows in the porous media. During the displacement of immiscible two-phase fluids in the pore doublet, the transport process is not only influenced by the capillary and viscous forces, but also affected by the channel geometry. In this paper, we first present a mathematical model to describe the two-phase fluid displacement in the pore doublet where the effects of capillary force, viscous force and the geometric structure are included. Then we derive an analytical solution of the model for the first time, and find that the displacement process is dominated by the capillary number, the viscosity ratio and the radius ratio. Furthermore, we define the optimal displacement that the wetting fluids in two daughter channels break through the branches simultaneously (both of them have the same breakthrough time), and also obtain the critical capillary number corresponding to the optimal displacement, which is related to the radius ratio of two daughter channels and viscosity ratio of two immiscible fluids. Finally, it is worthy noting that the present analytical results on the displacement in the pore doublet can be used to explain and understand the phenomenon of preferential imbibition or preferential flow in porous media. © 2022, CC BY.

关键词： Porous materials

A comparative study of two Allen-Cahn models for immiscible N-phase flows by using a consistent and conservative lattice Boltzmann method

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arXiv 2023年

作者： Zhan, Chengjie Liu, Xi Chai, Zhenhua Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

In this work, we conduct a detailed comparison between two second-order conservative Allen-Cahn (AC) models [Model A: Zheng et al., Phys. Rev. E 101, 0433202 (2020) and Model B: Mirjalili and Mani, (2023)] for the immiscible N-phase flows. Mathematically, these two AC equations can be proved to be equivalent under some approximate conditions. However, the effects of these approximations are unclear from the theoretical point of view, and would be considered numerically. To this end, we propose a consistent and conservative lattice Boltzmann method for the AC models for N-phase flows, and present some numerical comparisons of accuracy and stability between these two AC models. The results show that both two AC models have good performances in accuracy, but the Model B is more stable for the realistic complex N-phase flows, although there is an adjustable parameter in the Model A. © 2023, CC BY-NC-ND.

关键词： Kinetic theory

A unified fourth-order Bhatnagar-Gross-Krook lattice Boltzmann model for high-dimensional linear hyperbolic equations

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arXiv 2024年

作者： Chen, Ying Chai, Zhenhua Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

In this work, we first develop a unified Bhatnagar-Gross-Krook lattice Boltzmann (BGK-LB) model for the d(d ≥ 1)-dimensional linear hyperbolic equation (L-HE), where the natural moments and the DdQ(2d2 + 1) [(2d2 + 1) discrete velocities in d-dimensional space] lattice structure are considered. Subsequently, at the acoustic scaling, we conduct an accuracy analysis on the developed BGK-LB model by the direct Taylor expansion (DTE) method, and present the second- and third-order moments of the equilibrium distribution functions (EDFs) to ensure that the BGK-LB model can be fourth-order consistent with the L-HE. And on this basis, when considering the Dirichlet boundary condition, the fourth-order full-way and half-way boundary schemes are proposed to approximate the unknown distribution functions to ensure that the BGK-LB model can be overall fourth-order accurate. Thereafter, based on the kinetic entropy theory, we derive the conditions that the fourth-order moments of the EDFs should satisfy to ensure the microscopic entropy stability of the BGK-LB model. In addition, with the aid of the von Neumann stability analysis, we also discuss the L2 stability of the BGK-LB model and numerically plot the stability regions. In particular, from a numerical perspective, we find that the region of microscopic entropy stability is identical to that of L2 stability. Finally, we carry out some numerical experiments to test the accuracy and stability of the BGK-LB model, and the numerical results are in agreement with our theoretical analysis. In addition, we compare the developed full-way and half-way boundary schemes for the Dirichlet boundary condition, which shows that the full-way boundary scheme is more stable. Copyright © 2024, The Authors. All rights reserved.

关键词： Entropy

A thermodynamically consistent and conservative diffuse-interface model for gas-liquid-solid multiphase flows

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arXiv 2023年

In this work, a thermodynamically consistent and conservative diffuse-interface model for gas-liquid-solid multiphase flows is proposed. In this model, a novel free energy for the gas-liquid-solid multiphase flows is established according to a ternary phase-field model, and it not only contains the standard bulk and interface free energies for two-phase flows, but also includes some additional terms to reflect the penalty in the solid phase and the wettability on the solid surface. Furthermore, a smooth indicator function of the solid phase is also introduced in the consistent Navier-Stokes equations to achieve a high viscosity in the solid phase and preserve the velocity boundary conditions on the solid surface. Based on the proposed diffuse-interface model, the fluid interface dynamics, the fluid-structure interaction, and the wetting property of the solid surface can be described simply and efficiently. Additionally, the total energy is also proved to be dissipative for the two-phase flows in the stationary geometries. To test the present diffuse-interface model, we develop a consistent and conservative lattice Boltzmann method and conduct some simulations. The numerical results also confirm the energy dissipation and good capability of the proposed diffuse-interface model in the study of two-phase flows in complex geometries and gas-liquid-particle multiphase flows. © 2023, CC BY.

关键词： Energy dissipation