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A geometric proximal gradient method for sparse least squares regression with probabilistic simplex constraint

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arXiv 2021年

作者： Xiao, Guiyun Bai, Zheng-Jian School of Mathematical Sciences Xiamen University Xiamen361005 China School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling & High Performance Scientific Computing Xiamen University Xiamen361005 China

In this paper, we consider the sparse least squares regression problem with probabilistic simplex constraint. Due to the probabilistic simplex constraint, one could not apply the l1 regularization to the considered regression model. To find a sparse solution, we reformulate the least squares regression problem as a nonconvex and nonsmooth l1 regularized minimization problem over the unit sphere. Then we propose a geometric proximal gradient method for solving the regularized problem, where the explicit expression of the global solution to every involved subproblem is obtained. The global convergence of the proposed method is established under some mild assumptions. Some numerical results are reported to illustrate the effectiveness of the proposed *** Codes 65K05, 90C25, 90C26 Copyright © 2021, The Authors. All rights reserved.

关键词： Gradient methods

ON COMPUTABILITY OF EQUILIBRIUM STATES

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arXiv 2023年

作者： Binder, Ilia He, Qiandu Li, Zhiqiang Zhang, Yiwei Department of Mathematics University of Toronto Bahen Centre 40 St. George St. TorontoONM5S 2E4 Canada School of Mathematical Sciences Peking University Beijing100871 China School of Mathematical Sciences Beijing International Center for Mathematical Research Peking University Beijing100871 China School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

Equilibrium states are natural dynamical analogs of Gibbs states in thermodynamic formalism. This paper is devoted to the study of their computability in the sense of Computable Analysis. We show that the unique equilibrium state associated to a pair of a computable, topologically exact, distance-expanding, open map T: X → X and a computable Hölder continuous potential : X → R is always computable. Furthermore, the Hausdorff dimension of the Julia set and the equilibrium state for the geometric potential of a computable hyperbolic rational map are computable. On the other hand, we introduce a mechanism to establish non-uniqueness of equilibrium states. We also present some computable dynamical systems whose equilibrium states are all *** Codes Primary: 03D78, Secondary: 37D35, 37D20, 37F15 Copyright © 2023, The Authors. All rights reserved.

关键词： Dynamical systems

Randomized regularized extended Kaczmarz algorithms for tensor recovery

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arXiv 2021年

作者： Du, Kui Sun, Xiao-Hui School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing Xiamen University Xiamen361005 China School of Mathematical Sciences Xiamen University Xiamen361005 China

Randomized regularized Kaczmarz algorithms have recently been proposed to solve tensor recovery models with consistent linear measurements. In this work, we propose a novel algorithm based on the randomized extended Kaczmarz algorithm (which converges linearly in expectation to the unique minimum norm least squares solution of a linear system) for tensor recovery models with inconsistent linear measurements. We prove the linear convergence in expectation of our algorithm. Numerical experiments on a tensor least squares problem and a sparse tensor recovery problem are given to illustrate the theoretical *** Codes 65F10, 68W20, 90C25, 15A69 Copyright © 2021, The Authors. All rights reserved.

关键词： Linear systems

ASYMPTOTIC BEHAVIOR OF THE PRINCIPAL EIGENVALUE OF A LINEAR SECOND ORDER ELLIPTIC OPERATOR WITH LARGE ADVECTION AND GENERAL BOUNDARY CONDITIONS

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arXiv 2023年

作者： Peng, Rui Zhang, Guanghui Zhou, Maolin School of Mathematical Sciences Zhejiang Normal University Zhejiang Jinhua321004 China 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 Chern Institute of Mathematics and LPMC Nankai University Tianjin300071 China

Consider the eigenvalue problem of a linear second order elliptic operator: − D∆ − 2α∇m(x) · ∇ + V (x) = λ in Ω, complemented by the Dirichlet boundary condition or the following general Robin boundary condition: (Formula Presented) is a bounded smooth domain, n(x) is the unit exterior normal to ∂Ω at x ∈ ∂Ω, D > 0 and α > 0 are, respectively, the diffusion and advection coefficients, m ∈ C2(Ω), V ∈ C(Ω), β ∈ C(∂Ω) are given functions, and β allows to be positive, sign-changing or negative. In [41], the asymptotic behavior of the principal eigenvalue of the above eigenvalue problem as D → 0 or D → ∞ was studied. In this paper, when N ≥ 2, under proper conditions on the advection function m, we establish the asymptotic behavior of the principal eigenvalue as α → ∞, and when N = 1, we obtain a complete characterization for such asymptotic behavior provided m0 changes sign at most finitely many times. Our results complement or improve those in [4, 7, 44] and also partially answer some questions raised in [4]. © 2023, CC0.

关键词： Advection

Asymptotic Behavior of the Principal Eigenvalue of a Linear Second Order Elliptic Operator with Large Advection and General Boundary Conditions

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SSRN

SSRN 2023年

In this paper, we consider asymptotic behavior of the principal eigenvalue of some second order elliptic operator with general boundary conditions. For $N=1$, we provide a complete characterization of the asymptotic behavior, assuming that $m'$ changes sign at most finitely many times. Our findings not only improve upon the previous work in \cite{BHN2005,CL2008,PZ2018}, but also partially address some of the open questions posed in \cite{BHN2005}. Furthermore, our results elucidate the novel influence of boundary conditions on such asymptotics. To achieve these results, we develop several sophisticated techniques to overcome the new difficulties posed by the problems under consideration. © 2023, The Authors. All rights reserved.

关键词： Advection

A Column-Wise Update Algorithm for Sparse Stochastic Matrix Factorization

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arXiv 2021年

作者： Xiao, Guiyun Bai, Zheng-Jian Ching, Wai-Ki School of Mathematical Sciences Xiamen University Xiamen361005 China School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling & High Performance Scientific Computing Xiamen University Xiamen361005 China Advanced Modeling and Applied Computing Laboratory Department of Mathematics The University of Hong Kong Pokfulam Road Hong Kong

Nonnegative matrix factorization arises widely in machine learning and data analysis. In this paper, for a given factorization of rank r, we consider the sparse stochastic matrix factorization (SSMF) of decomposing a prescribed m-by-n stochastic matrix V into a product of an m-by-r stochastic matrix W and an r-by-n stochastic matrix H, where both W and H are required to be sparse. With the prescribed sparsity level, we reformulate the SSMF as an unconstrained nonconvex-nonsmooth minimization problem and introduce a column-wise update algorithm for solving the minimization problem. We show that our algorithm converges globally. The main advantage of our algorithm is that the generated sequence converges to a special critical point of the cost function, which is nearly a global minimizer over each column vector of the W-factor and is a global minimizer over the H-factor as a whole if there is no sparsity requirement on H. Numerical experiments on both synthetic and real data sets are given to demonstrate the effectiveness of our proposed *** Codes 65K05, 90C06, 90C26 Copyright © 2021, The Authors. All rights reserved.

关键词： Gradient methods

A Well-Balanced Moving Mesh Discontinuous Galerkin Method for the Ripa Model on Triangular Meshes

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SSRN

SSRN 2022年

作者： Huang, Weizhang Li, Ruo Qiu, Jianxian Zhang, Min Department of Mathematics University of Kansas LawrenceKS66045 United States CAPT LMAM School of Mathematical Sciences Peking University Beijing100871 China School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Fujian Xiamen361005 China School of Mathematical Sciences Peking University Beijing100871 China

A well-balanced moving mesh discontinuous Galerkin (DG) method is proposed for the numerical solution of the Ripa model – a generalization of the shallow water equations that accounts for effects of water temperature variations. Thermodynamic processes are important particularly in the upper layers of the ocean where the variations of sea surface temperature play a fundamental role in climate change. The well-balance property which requires numerical schemes to preserve the lake-at-rest steady state is crucial to the simulation of perturbation waves over that steady state such as waves on a lake or tsunami waves in the deep ocean. To ensure the well-balance, positivity-preserving, and high-order properties, a DG-interpolation scheme (with or without scaling positivity-preserving limiter) and special treatments pertaining to the Ripa model are employed in the transfer of both the flow variables and bottom topography from the old mesh to the new one and in the TVB limiting process. Mesh adaptivity is realized using an MMPDE moving mesh approach and a metric tensor based on an equilibrium variable and water depth. A motivation is to adapt the mesh according to both the perturbations of the lake-at-rest steady state and the water depth distribution (bottom structure). Numerical examples in one and two dimensions are presented to demonstrate the well-balance, high-order accuracy, and positivity preserving properties of the method and its ability to capture small perturbations of the lake-at-rest steady state. © 2022, The Authors. All rights reserved.

关键词： Lakes

A well-balanced moving mesh discontinuous Galerkin method for the Ripa model on triangular meshes

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arXiv 2022年

A well-balanced moving mesh discontinuous Galerkin (DG) method is proposed for the numerical solution of the Ripa model - a generalization of the shallow water equations that accounts for effects of water temperature variations. Thermodynamic processes are important particularly in the upper layers of the ocean where the variations of sea surface temperature play a fundamental role in climate change. The well-balance property which requires numerical schemes to preserve the lake-at-rest steady state is crucial to the simulation of perturbation waves over that steady state such as waves on a lake or tsunami waves in the deep ocean. To ensure the well-balance, positivity-preserving, and high-order properties, a DG-interpolation scheme (with or without scaling positivity-preserving limiter) and special treatments pertaining to the Ripa model are employed in the transfer of both the flow variables and bottom topography from the old mesh to the new one and in the TVB limiting process. Mesh adaptivity is realized using an MMPDE moving mesh approach and a metric tensor based on an equilibrium variable and water depth. A motivation is to adapt the mesh according to both the perturbations of the lake-at-rest steady state and the water depth distribution (bottom structure). Numerical examples in one and two dimensions are presented to demonstrate the well-balance, high-order accuracy, and positivity-preserving properties of the method and its ability to capture small perturbations of the lake-at-rest steady *** Codes 65M50, 65M60, 76B15, 35Q35 Copyright © 2022, The Authors. All rights reserved.

关键词： Interpolation

ON THE HEAT FLOW OF EQUATION OF H-SURFACE

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Acta Mathematica Scientia 2019年第5期39卷 1397-1405页

作者： Guochun WU Zhong TAN Jiankai XU Fujian Province University Key Laboratory of Computational Science School of Mathematical Sciences Huaqiao University Quanzhou 362021 China School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and Scientific Computing Xiamen University Xiamen 361005 China College of Information Science and Technology Hunan Agriculture University Changsha 410128 China

We study the heat flow of equation of H-surface with non-zero Dirichlet boundary in the present article. Introducing the "stable set" M2 and "unstable set" M1, we show that there exists a unique global solution provided the initial data belong to M2 and the global solution converges to zero in H^1 exponentially as time goes to infinity. Moreover, we also prove that the local regular solution must blow up at finite time provided the initial data belong to M1.

关键词： H-surface non-zero Dirichlet boundary singularity global solutions