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POINTWISE ESTIMATES FOR THE FUNDAMENTAL SOLUTIONS OF HIGHER ORDER SCHRÖDINGER EQUATIONS IN ODD DIMENSIONS I: LOW DIMENSIONAL CASE

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arXiv 2024年

In this paper, for any odd n and any integer m ≥ 1 with n tu(x, t) = ((−∆)m + V(x))u(x, t), t ∈ , x ∈ n, where V is a real-valued potential with certain decay. Let Pac(H) denote the projection onto the absolutely continuous spectrum space of H = (−∆)m + V, and assume that H has no positive embedded eigenvalue. Our main result says that e−itHPac(H) has integral kernel K(t, x, y) satisfying (Equation presented), where the constants C, h > 0, and h can be specified by m, n and the zero energy resonances of H. A similar result for smoothing operators like (Equation presented) is also given. One of the main novelties lies in a new and unified approach to obtain the asymptotic expansions of perturbed resolvents under various types of zero energy resonances. © 2024, CC0.

关键词： Schrodinger equation

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

Gradient Descent for Symmetric Tensor Decomposition

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Annals of Applied Mathematics 2022年第4期38卷 385-413页

作者： Jian-Feng Cai Haixia Liu Yang Wang Department of Mathematics The Hong Kong University of Science and TechnologyClear Water BayKowloonHong KongChina School of Mathematics and Statistics&Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhanHubei 430074China

Symmetric tensor decomposition is of great importance in *** studies have employed a greedy approach,where the main idea is to first find a best rank-one approximation of a given tensor,and then repeat the process to the residual tensor by subtracting the rank-one *** this paper,we focus on finding a best rank-one approximation of a given orthogonally order-3 symmetric *** give a geometric landscape analysis of a nonconvex optimization for the best rank-one approximation of orthogonally symmetric *** show that any local minimizer must be a factor in this orthogonally symmetric tensor decomposition,and any other critical points are linear combinations of the ***,we propose a gradient descent algorithm with a carefully designed initialization to solve this nonconvex optimization problem,and we prove that the algorithm converges to the global minimum with high probability for orthogonal decomposable *** result,combined with the landscape analysis,reveals that the greedy algorithm will get the tensor CP low-rank *** results are provided to verify our theoretical results.

关键词： Gradient descent random initialization symmetric tensor decomposition CP decomposition linear convergence

An Unconditionally Energy-Stable and Orthonormality-Preserving Iterative Scheme for the Kohn-Sham Gradient Flow Based Model

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arXiv 2023年

作者： Wang, Xiuping Chen, Huangxin Kou, Jisheng Sun, Shuyu Saudi Arabia School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling and HighPerformance Scientific Computing Xiamen University Fujian Xiamen China Key Laboratory of Rock Mechanics and Geohazards of Zhejiang Province Shaoxing University Shaoxing China School of Mathematics and Statistics Hubei Engineering University Hubei Xiaogan China

We propose an unconditionally energy-stable, orthonormality-preserving, component-wise splitting iterative scheme for the Kohn-Sham gradient flow based model in the electronic structure calculation. We first study the scheme discretized in time but still continuous in space. The componentwise splitting iterative scheme changes one wave function at a time, similar to the Gauss-Seidel iteration for solving a linear equation system. At the time step n, the orthogonality of the wave function being updated to other wave functions is preserved by projecting the gradient of the Kohn-Sham energy onto the subspace orthogonal to all other wave functions known at the current time, while the normalization of this wave function is preserved by projecting the gradient of the Kohn-Sham energy onto the subspace orthogonal to this wave function at tn+1/2. The unconditional energy stability is nontrivial, and it comes from a subtle treatment of the two-electron integral as well as a consistent treatment of the two projections. Rigorous mathematical derivations are presented to show our proposed scheme indeed satisfies the desired properties. We then study the fully-discretized scheme, where the space is further approximated by a conforming finite element subspace. For the fully-discretized scheme, not only the preservation of orthogonality and normalization (together we called orthonormalization) can be quickly shown using the same idea as for the semi-discretized scheme, but also the highlight property of the scheme, i.e., the unconditional energy stability can be rigorously proven. The scheme allows us to use large time step sizes and deal with small systems involving only a single wave function during each iteration step. Several numerical experiments are performed to verify the theoretical analysis, where the number of iterations is indeed greatly reduced as compared to similar examples solved by the Kohn-Sham gradient flow based model in the literature. Copyright © 2023, The Aut

关键词： Wave functions

The macroscopic finite-difference scheme and modified equations of the general propagation multiple-relaxation-time lattice Boltzmann model

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arXiv 2023年

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

In this paper, we first present the general propagation multiple-relaxation-time lattice Boltzmann (GPMRT-LB) model and obtain the corresponding macroscopic finite-difference (GPMFD) scheme on conservative moments. Then based on the Maxwell iteration method, we conduct the analysis on the truncation errors and modified equations (MEs) of the GPMRT-LB model and GPMFD scheme at both diffusive and acoustic scalings. For the nonlinear anisotropic convection-diffusion equation (NACDE) and Navier-Stokes equations (NSEs), we also derive the first- and second-order MEs of the GPMRT-LB model and GPMFD scheme. In particular, for the one-dimensional convection-diffusion equation (CDE) with the constant velocity and diffusion coefficient, we can develop a fourth-order GPMRT-LB (F-GPMRT-LB) model and the corresponding fourth-order GPMFD (F-GPMFD) scheme at the diffusive scaling. Finally, two benchmark problems, Gauss hill problem and Poiseuille flow in two-dimensional space, are used to test the GPMRT-LB model and GPMFD scheme, and it is found that the numerical results are not only in good agreement with corresponding analytical solutions, but also have a second-order convergence rate in space. Additionally, a numerical study on one-dimensional CDE also demonstrates that the F-GPMRT-LB model and F-GPMFD scheme can achieve a fourth-order accuracy in space, which is consistent with our theoretical analysis. Copyright © 2023, The Authors. All rights reserved.

关键词： Navier Stokes equations

DISPERSIVE ESTIMATES FOR THE SCHRÖDINGER EQUATION WITH FINITE RANK PERTURBATIONS

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arXiv 2022年

作者： Cheng, Han Huang, Shanlin Zheng, Quan School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China School of Mathematics and Statistics Huazhong University of Science and Technology Hubei Wuhan430074 China

In this paper, we investigate dispersive estimates for the time evolution of Hamiltonians (Equation presented) where each j satisfies certain smoothness and decay conditions. We show that, under a spectral assumption, there exists a constant C = C(N, d, 1,..., N) > 0 such that (Equation presented), for t > 0. As far as we are aware, this seems to provide the first study of L1 − L∞ estimates for finite rank perturbations of the Laplacian in any dimension. We first deal with rank one perturbations (N = 1). Then we turn to the general case. The new idea in our approach is to establish the Aronszajn-Krein type formula for finite rank perturbations. This allows us to reduce the analysis to the rank one case and solve the problem in a unified manner. Moreover, we show that in some specific situations, the constant C(N, d, 1,..., N) grows polynomially in N. Finally, as an application, we are able to extend the results to N = ∞ and deal with some trace class *** Codes 35J10, 81Q15, 42B37 Copyright © 2022, The Authors. All rights reserved.

关键词：

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

A diffuse-interface lattice Boltzmann method for the dissolution through nonlinear heterogeneous reaction

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Physics of Fluids 2025年第5期37卷

作者： Yang, Xiaofeng Zhan, Chengjie Liu, Xi Chai, Zhenhua 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 State Key Laboratory of Intelligent Manufacturing Equipment and Technology Huazhong University of Science and Technology Wuhan430074 China

In this work, a diffuse-interface lattice Boltzmann method (DI-LBM) is proposed for the dissolution through nonlinear heterogeneous reaction. In this method, the sharp boundary between the fluid and solid phases is replaced by a diffuse interface with a nonzero thickness. To describe the changes of physical variables from the fluid region to the solid region, a smooth function is introduced, and simultaneously, the nonlinear Robin boundary condition imposed on the fluid-solid interface is reformulated as a source term into the modified convection-diffusion equation. We then test the DI-LBM through several benchmark problems, and find that the numerical results are in agreement with the analytical solutions and available data. Finally, the present DI-LBM is adopted to study the dissolution in a two-dimensional porous medium, and the results show that with the decrease in reaction order, the reactant in the central region with a large fluid velocity reacts faster and forms a conical pathway. © 2025 Author(s).

关键词： Kinetic theory of gases

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