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New subspace minimization conjugate gradient methods based on regularization model for unconstrained optimization

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arXiv 2020年

作者： Zhao, Ting Liu, Hongwei Liu, Zexian School of Mathematics and Statistics Xidian University Xi’an710126 China State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering computing AMSS Chinese Academy of Sciences Beijing100190 China

In this paper, two new subspace minimization conjugate gradient methods based on p−regularization models are proposed, where a special scaled norm in p−regularization model is analyzed. Different choices for special scaled norm lead to different solutions to the p−regularized subproblem. Based on the analyses of the solutions in a two-dimensional subspace, we derive new directions satisfying the sufficient descent condition. With a modified nonmonotone line search, we establish the global convergence of the proposed methods under mild assumptions. R−linear convergence of the proposed methods are also analyzed. Numerical results show that, for the CUTEr library, the proposed methods are superior to four conjugate gradient methods, which were proposed by Hager and Zhang (SIAM J Optim 16(1):170-192, 2005), Dai and Kou (SIAM J Optim 23(1):296-320, 2013), Liu and Liu (J Optim Theory Appl 180(3):879-906, 2019) and Li et al. (Comput Appl Math 38(1): 2019), respectively. Copyright © 2020, The Authors. All rights reserved.

关键词： Conjugate gradient method

CDOpt: A Python Package for a Class of Riemannian Optimization

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arXiv 2022年

作者： Xiao, Nachuan Hu, Xiaoyin Liu, Xin Toh, Kim-Chuan The Institute of Operations Research and Analytics National University of Singapore Singapore School of Computer and Computing Science Hangzhou City University Hangzhou China Academy of Edge Intelligence Hangzhou City University Gongshu Hangzhou China State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences University of Chinese Academy of Sciences China Department of Mathematics Institute of Operations Research and Analytics National University of Singapore Singapore

Optimization over an embedded submanifold defined by equality constraints c(x) = 0 has attracted much interest over the past few decades due to its wide applications in various areas, including computer vision, signal processing, numerical linear algebra, and deep learning. For solving such problems, many related optimization packages have been developed based on Riemannian optimization approaches, which rely on some basic geometrical materials of Riemannian manifolds, including Riemannian gradients, retractions, vector transports, etc. These geometrical materials can be challenging to determine in general. In fact, existing packages only accommodate a few well-known manifolds whose geometrical materials are more easily accessible. For other manifolds that are not contained in these packages, the users have to develop the geometric materials by themselves. In addition, it is not always tractable to adapt the advanced features from various state-of-the-art unconstrained optimization solvers to Riemannian optimization approaches. Here we introduce a user-friendly Python package, CDOpt (available at https: //***/ under BSD 3-clause license), for solving a class of Riemannian optimization problems. CDOpt is designed to complement existing Riemannian optimization packages by transforming Riemannian optimization problems into their unconstrained counterparts through the constraint dissolving approach. We prove that the when the penalty parameter in the constraint dissolving approach is sufficiently large, Riemannian optimization problems and their unconstrained counterparts are equivalent. Therefore, solving Riemannian optimization problems through CDOpt can directly benefit from various existing solvers and the rich expertise gained over the past few decades for unconstrained optimization. Moreover, all the computations in CDOpt related to any manifold in question are conducted through its constraints expression, hence users can easily define new manifolds in

关键词： Constrained optimization

Influence of Small-Size Contaminations on Thin Film Structural Properties

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Chinese Physics Letters 2019年第3期36卷 82-85页

作者： F.V.Grigoriev V.B.Sulimov Jinlong Zhang Xinbin Cheng Zhanshan Wang A.V.Tikhonravov Research Computing Center M.V.Lomonosov Moscow State University MOE Key Laboratory of Advanced Micro-Structured Materials Institute of Precision Optical Engineering School of Physics Science and EngineeringTongji University IFSA Collaborative Innovation Center Shanghai Jiao Tong University

An approach for studying the influence of nano-particles on the structural properties of deposited thin films is proposed. It is based on the molecular dynamic modeling of the deposition process in the presence of contaminating nano-particles. The nano-particle is assumed to be immobile and its interaction with film atoms is described by a spherically symmetric potential. The approach is applied to the investigation of properties of silicon dioxide films. Visualization tools are used to investigate the porosity associated with nano-particles. The structure of the film near the nano-particle is studied using the radial distribution function. It is found that fluctuations of film density near the nano-particles are essentially different in the cases of low-energy and high-energy deposition processes.

关键词： Influence Small-Size Properties

Quantum algorithms for computing observables of nonlinear partial differential equations

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arXiv 2022年

作者： Jin, Shi Liu, Nana Institute of Natural Sciences Shanghai Jiao Tong University Shanghai 200240 China School of Mathematical Sciences Shanghai Jiao Tong University Shanghai 200240 China Ministry of Education Key Laboratory in Scientific and Engineering Computing Shanghai Jiao Tong University Shanghai 200240 China University of Michigan Shanghai Jiao Tong University Joint Institute Shanghai 200240 China

We construct quantum algorithms to compute physical observables of nonlinear PDEs with M initial data. Based on an exact mapping between nonlinear and linear PDEs using the level set method, these new quantum algorithms for nonlinear Hamilton-Jacobi and scalar hyperbolic PDEs can be performed with a computational cost that is independent of M, for arbitrary nonlinearity. Depending on the details of the initial data, it can also display up to exponential advantage in both the dimension of the PDE and the error in computing its observables. For general nonlinear PDEs, quantum advantage with respect to M is possible in the large M limit. © 2022, CC BY.

关键词： Quantum theory

LIOUVILLE-TYPE THEOREMS FOR STEADY SOLUTIONS TO THE NAVIER-STOKES SYSTEM IN A SLAB

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arXiv 2022年

作者： Bang, Jeaheang Gui, Changfeng Wang, Yun Xie, Chunjing Department of Mathematics The University of Texas San Antonio United States School of Mathematical Sciences Center for dynamical systems and differential equations Soochow University Suzhou China School of mathematical Sciences Institute of Natural Sciences Ministry of Education Key Laboratory of Scientific and Engineering Computing CMA-Shanghai Shanghai Jiao Tong University 800 Dongchuan Road Shanghai China

Liouville-type theorems for the steady incompressible Navier-Stokes system are investigated for solutions in a three-dimensional slab with either no-slip boundary conditions or periodic boundary conditions. When the no-slip boundary conditions are prescribed, we prove that any bounded solution is trivial if it is axisymmetric or rur is bounded, and that general three-dimensional solutions must be Poiseuille flows when the velocity is not big in L∞ space. When the periodic boundary conditions are imposed on the slab boundaries, we prove that the bounded solutions must be constant vectors if either the swirl or radial velocity is independent of the angular variable, or rur decays to zero as r tends to infinity. The proofs are based on the fundamental structure of the equations and energy estimates. The key technique is to establish a Saint-Venant type estimate that characterizes the growth of Dirichlet integral of nontrivial solutions. Copyright © 2022, The Authors. All rights reserved.

关键词： Navier Stokes equations

Convergence analysis of one-point large deviations rate functions of numerical discretizations for stochastic wave equations with small noise

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arXiv 2022年

作者： Jin, Diancong Hong, Jialin Sheng, Derui 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 Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

In this work, we present the convergence analysis of one-point large deviations rate functions (LDRFs) of the spatial finite difference method (FDM) for stochastic wave equations with small noise, which is essentially about the asymptotical limit of minimization problems and not a trivial task for the nonlinear cases. In order to overcome the difficulty that objective functions for the original equation and the spatial FDM have different effective domains, we propose a new technical route for analyzing the pointwise convergence of the one-point LDRFs of the spatial FDM, based on the Γ -convergence of objective functions. Based on the new technical route, the intractable convergence analysis of one-point LDRFs boils down to the qualitative analysis of skeleton equations of the original equation and its numerical discretizations. Copyright © 2022, The Authors. All rights reserved.

关键词： Finite difference method

Optimization on the symplectic Stiefel manifold: SR decomposition-based retraction and applications

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arXiv 2022年

作者： Gao, Bin Son, Nguyen Thanh Stykel, Tatjana State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Institut für Mathematik Centre for Advanced Analytics and Predictive Sciences Universität Augsburg Universitätsstraße 12a Augsburg86159 Germany Thai Nguyen University of Sciences Thai Nguyen24118 Viet Nam

Numerous problems in optics, quantum physics, stability analysis, and control of dynamical systems can be brought to an optimization problem with matrix variable subjected to the symplecticity constraint. As this constraint nicely forms a so-called symplectic Stiefel manifold, Riemannian optimization is preferred, because one can borrow ideas from unconstrained optimization methods after preparing necessary geometric tools. Retraction is arguably the most important one which decides the way iterates are updated given a search direction. Two retractions have been constructed so far: one relies on the Cayley transform and the other is designed using quasi-geodesic curves. In this paper, we propose a new retraction which is based on an SR matrix decomposition. We prove that its domain contains the open unit ball which is essential in proving the global convergence of the associated gradient-based optimization algorithm. Moreover, we consider three applications—symplectic target matrix problem, symplectic eigenvalue computation, and symplectic model reduction of Hamiltonian systems—with various examples. The extensive numerical comparisons reveal the strengths of the proposed optimization *** Codes 15A23, 32C25, 65F15, 65F99, 65K05, 65P10, 90C30 © 2022, CC BY.

关键词： Dynamical systems

A Phase-Field-Based Multiple-Relaxation-Time Lattice Boltzmann Method for Incompressible Multiphase Flows with Density and Viscosity Contrasts

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SSRN

SSRN 2022年

作者： Yuan, Xiaolei Wu, Yao Zhang, Chunhua Chai, Zhenhua Shi, Baochang College of Mathematics and Information Science Hebei University Baoding071002 China Institute of Interdisciplinary Research for Mathematics and Applied Science 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 Department of Mechanics and Aerospace Engineering Southern University of Science and Technology Guangdong Shenzhen518055 China

In this work, we present a unified framework of phase-field-based multiple-relaxation-time lattice Boltzmann (MRT-LB) method for incompressible multiphase flows with density and viscosity contrasts where a block-lower-triangular relaxation matrix is introduced. The present framework includes the classic MRT-LB model and central-moments-based LB model (CLBM, a particular version of the MRT-LB model). The governing equations of incompressible multiphase flows can be reproduced precisely through the direct Taylor expansion method at the second-order of expansion parameters. In this work, the pressure $p$ is re-estimated, and we prove that the present pressure expression is suitable for classic MRT model, CLBM and various lattice structures. In addition, theoretical analysis and modeling are both carried out in the velocity space. As a result, the present model has a similar structure to the traditional Bhatnagar-Gross-Krook (BGK) model and is very easy to analyze and implement. We then test the present model (mainly CLBM) by five classic physical problems. The results show that the present model is found to be accurate and has better numerical stability over the BGK model, and has the ability to simulate multiphase flow problems with large density ratio ($1000$). © 2022, The Authors. All rights reserved.

关键词： Multiphase flow

A parallel scalable domain decomposition preconditioner for elastic crack simulation using XFEM

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arXiv 2021年

作者： Tian, Wei Jingjing, Huang Chen, Rongliang Jiang, Yi Shenzhen Key Laboratory for Exascale Engineering and Scientific Computing Shenzhen518055 China Shenzhen Institutes of Advanced Technology Chinese Academy of Sciences Shenzhen518055 China Department of Rehabilitation Medicine Shenzhen University General Hospital Shenzhen518055 China

In this paper, a parallel overlapping domain decomposition preconditioner is proposed to solve the linear system of equations arising from the extended finite element discretization of elastic crack problems. The algorithm partitions the computational mesh into two types of subdomains: the regular subdomains and the crack tip subdomains based on the observation that the crack tips have a significant impact on the convergence of the iterative method while the impact of the crack lines is not that different from those of regular mesh points. The tip subdomains consist of mesh points at crack tips and all neighboring points where the branch enrichment functions are applied. The regular subdomains consist of all other mesh points, including those on the crack lines. To overcome the mismatch between the number of subdomains and the number of processor cores, the proposed method is divided into two steps: solve the crack tip problem and then the regular subdomain problem during each iteration. The proposed method was used to develop a parallel XFEM package which is able to test different types of iterative methods. To achieve good parallel efficiency, additional methods were introduced to reduce communication and to maintain the load balance between processors. Numerical experiments indicate that the proposed method significantly reduces the number of iterations and the total computation time compared to the classical methods. In addition, the method scales up to 8192 processor cores with over 70% parallel efficiency to solve problems with more than 2 × 108 degrees of freedom. Copyright © 2021, The Authors. All rights reserved.

关键词： Crack tips