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Phase-field modeling of dendritic growth with gas bubbles in the solidification of binary alloys

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arXiv 2024年

作者： Zhan, Chengjie Chai, Zhenhua Sun, Dongke Shi, Baochang Geng, Shaoning Jiang, Ping 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 School of Mechanical Engineering Southeast University Nanjing211189 China The State Key Laboratory of Digital Manufacturing Equipment and Technology School of Mechanical Science and Engineering Huazhong University of Science and Technology Wuhan430074 China

In this work, a phase-field model is developed for the dendritic growth with gas bubbles in the solidification of binary alloys. In this model, a total free energy for the complex gas-liquid-dendrite system is proposed through considering the interactions of gas bubbles, liquid melt and solid dendrites, and it can reduce to the energy for gas-liquid flows in the region far from the solid phase, while degenerate to the energy for thermosolutal dendritic growth when the gas bubble disappears. The governing equations are usually obtained by minimizing the total free energy, but here some modifications are made to improve the capacity of the conservative phase-field equation for gas bubbles and convection-diffusion equation for solute transfer. Additionally, through the asymptotic analysis of the thin-interface limit, the present general phase-field model for alloy solidification can match the corresponding free boundary problem, and it is identical to the commonly used models under a specific choice of model parameters. Furthermore, to describe the fluid flow, the incompressible Navier-Stokes equations are adopted in the entire domain including gas, liquid, and solid regions, where the fluid-structure interaction is considered by a simple diffuse-interface method. To test the present phase-field model, the lattice Boltzmann method is used to study several problems of gas-liquid flows, dendritic growth as well as the solidification in presence of gas bubbles, and a good performance of the present model for such complex problems is observed. © 2024, CC BY-NC-ND.

关键词： Solidification

Learning-rate-free Momentum SGD with Reshuffling Converges in Nonsmooth Nonconvex Optimization

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arXiv 2024年

作者： Hu, Xiaoyin Xiao, Nachuan Liu, Xin Toh, Kim-Chuan School of Computer and Computing Science Hangzhou City University Hangzhou310015 China Institute of Operational Research and Analytics National University of Singapore Singapore 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 Singapore119076 Singapore

In this paper, we propose a generalized framework for developing learning-rate-free momentum stochastic gradient descent (SGD) methods in the minimization of nonsmooth nonconvex functions, especially in training nonsmooth neural networks. Our framework adaptively generates learning rates based on the historical data of stochastic subgradients and iterates. Under mild conditions, we prove that our proposed framework enjoys global convergence to the stationary points of the objective function in the sense of the conservative field, hence providing convergence guarantees for training nonsmooth neural networks. Based on our proposed framework, we propose a novel learning-rate-free momentum SGD method (LFM). Preliminary numerical experiments reveal that LFM performs comparably to the state-of-the-art learning-rate-free methods (which have not been shown theoretically to be convergence) across well-known neural network training benchmarks. Copyright © 2024, The Authors. All rights reserved.

关键词： Momentum

Proximal Subgradient Norm Minimization of Ista and Fista

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SSRN

SSRN 2022年

作者： Li, Bowen Shi, Bin Yuan, Ya-Xiang State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing Beijing100190 China University of Chinese Academy of Sciences Beijing Beijing100049 China

For first-order smooth optimization, the research on the acceleration phenomenon has a long-time history. Until recently, the mechanism leading to acceleration was not successfully uncovered by the gradient correction term and its equivalent implicit-velocity form. Furthermore, based on the high-resolution differential equation framework with the corresponding emerging techniques, phase-space representation and Lyapunov function, the squared gradient norm of Nesterov's accelerated gradient descent (NAG) method at an inverse cubic rate is discovered. However, this result cannot be directly generalized to composite optimization widely used in practice, e.g., the linear inverse problem with sparse representation. In this paper, we meticulously observe a pivotal inequality used in composite optimization about the step size s and the Lipschitz constant L and find that it can be improved tighter. We apply the tighter inequality discovered in the well-constructed Lyapunov function and then obtain the proximal subgradient norm minimization by the phase-space representation, regardless of gradient-correction or implicit-velocity. Furthermore, we demonstrate that the squared proximal subgradient norm for the class of iterative shrinkage-thresholding algorithms (ISTA) converges at an inverse square rate, and the squared proximal subgradient norm for the class of faster iterative shrinkage-thresholding algorithms (FISTA) is accelerated to convergence at an inverse cubic rate. © 2022, The Authors. All rights reserved.

关键词： Lyapunov functions

On the Hyperparameters in Stochastic Gradient Descent with Momentum

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arXiv 2021年

作者： Shi, Bin State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

Following the same routine as [SSJ20], we continue to present the theoretical analysis for stochastic gradient descent with momentum (SGD with momentum) in this paper. Differently, for SGD with momentum, we demonstrate that the two hyperparameters together, the learning rate and the momentum coefficient, play a significant role in the linear convergence rate in nonconvex optimizations. Our analysis is based on using a hyperparameters-dependent stochastic differential equation (hp-dependent SDE) that serves as a continuous surrogate for SGD with momentum. Similarly, we establish the linear convergence for the continuous-time formulation of SGD with momentum and obtain an explicit expression for the optimal linear rate by analyzing the spectrum of the Kramers-Fokker-Planck operator. By comparison, we demonstrate how the optimal linear rate of convergence and the final gap for SGD only about the learning rate varies with the momentum coefficient increasing from zero to one when the momentum is introduced. Then, we propose a mathematical interpretation of why, in practice, SGD with momentum converges faster and is more robust in the learning rate than standard stochastic gradient descent (SGD). Finally, we show the Nesterov momentum under the presence of noise has no essential difference from the traditional momentum. © 2021, CC BY-NC-SA.

关键词： Momentum

The Sampling Method for Optimal Precursors of ENSO Events

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arXiv 2023年

作者： Shi, Bin Ma, Junjie State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China School of Mathematics North University of China Taiyuan030051 China

El Niño-Southern Oscillation (ENSO) is one of the significant climate phenomena, which appears periodically in the tropic Pacific. The intermediate coupled ocean-atmosphere Zebiak-Cane (ZC) model is the first and classical one designed to numerically forecast the ENSO events. Traditionally, the conditional nonlinear optimal perturbation (CNOP) approach has been used to capture optimal precursors in practice. In this paper, based on state-of-the-art statistical machine learning techniques,1 we investigate the sampling algorithm proposed in [Shi and Sun, 2023] to obtain optimal precursors via the CNOP approach in the ZC model. For the ZC model, or more generally, the numerical models with dimension O(104−105), the numerical performance, regardless of the statically spatial patterns and the dynamical nonlinear time evolution behaviors as well as the corresponding quantities and indices, shows the high efficiency of the sampling method by comparison with the traditional adjoint method. The sampling algorithm does not only reduce the gradient (first-order information) to the objective function value (zeroth-order information) but also avoids the use of the adjoint model, which is hard to develop in the coupled ocean-atmosphere models and the parameterization models. In addition, based on the key characteristic that the samples are independently and identically distributed, we can implement the sampling algorithm by parallel computation to shorten the computation time. Meanwhile, we also show in the numerical experiments that the important features of optimal precursors can be still captured even when the number of samples is reduced sharply. © 2023, CC BY-NC-SA.

关键词： Numerical methods

Joint design of hybrid beamforming and reflection coefficients in RIS-aided mmWave MIMO systems

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arXiv 2022年

作者： Li, Renwang Guo, Bei Tao, Meixia Liu, Ya-Feng Yu, Wei Department of Electronic Engineering Shanghai Jiao Tong University Shanghai China State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Department of Electrical and Engineering University of Toronto TorontoONM5S 3G4 Canada

This paper considers a reconfigurable intelligent surface (RIS)-aided millimeter wave (mmWave) downlink communication system where hybrid analog-digital beamforming is employed at the base station (BS). We formulate a power minimization problem by jointly optimizing hybrid beamforming at the BS and the response matrix at the RIS, under the signalto-interference-plus-noise ratio (SINR) constraints at all users. The problem is highly challenging to solve due to the nonconvex SINR constraints as well as the unit-modulus phase shift constraints for both the RIS reflection coefficients and the analog beamformer. A two-layer penalty-based algorithm is proposed to decouple variables in SINR constraints, and manifold optimization is adopted to handle the non-convex unitmodulus *** also propose a low-complexity sequential optimization method, which optimizes the RIS reflection coefficients, the analog beamformer, and the digital beamformer sequentially without iteration. Furthermore, the relationship between the power minimization problem and the max-min fairness (MMF) problem is discussed. Simulation results show that the proposed penalty-based algorithm outperforms the stateof-the-art semidefinite relaxation (SDR)-based algorithm. Results also demonstrate that the RIS plays an important role in the power reduction. Copyright © 2022, The Authors. All rights reserved.

关键词： Millimeter waves

An Inexact Preconditioned Zeroth-order Proximal Method for Composite Optimization

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arXiv 2024年

作者： Liu, Shanglin Wang, Lei Xiao, Nachuan Liu, Xin State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing101408 China Department of Statistics Pennsylvania State University University Park16802 United States Institute of Operational Research and Analytics National University of Singapore Singapore117597 Singapore

In this paper, we consider the composite optimization problem, where the objective function integrates a continuously differentiable loss function with a nonsmooth regularization term. Moreover, only the function values for the differentiable part of the objective function are available. To efficiently solve this composite optimization problem, we propose a preconditioned zeroth-order proximal gradient method in which the gradients and preconditioners are estimated by finite-difference schemes based on the function values at the same trial points. We establish the global convergence and worst-case complexity for our proposed method. Numerical experiments exhibit the superiority of our developed method. Copyright © 2024, The Authors. All rights reserved.

关键词： Gradient methods

Preface-Special Issue on Mathematical Optimization:Past,Present and Future

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Journal of the Operations Research Society of China 2020年第2期8卷 197-198页

作者： Zai-Wen Wen Ya-Xiang Yuan Beijing International Center for Mathematical Research Peking UniversityBeijing 100871China State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China

Mathematical optimization is one of the foundations of fields such as operations research,computational mathematics,scientific and engineering computing,machine learning,and data *** tasks usually include formulating appropriate mathematical models to describe related practical problems,designing suitable numerical methods to find optimal solutions,and exploring the theoretical properties of the models and *** evolution of modern computer architecture and the popularity of big/complex/smart data had a significant impact on mathematical *** success of large-scale optimization in machine learning and signal processing certainly provides a very exciting paradigm on“Modeling+algorithms+computing power”.

关键词： optimization. Mathematical optimization

Stochastic Gauss-Newton Algorithms for Online PCA

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arXiv 2022年

作者： Zhou, Siyun Liu, Xin Xu, Liwei School of Mathematical Sciences University of Electronic Science and Technology of China 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

In this paper, we propose a stochastic Gauss-Newton (SGN) algorithm to study the online principal component analysis (OPCA) problem, which is formulated by using the symmetric low-rank product (SLRP) model for dominant eigenspace calculation. Compared with existing OPCA solvers, SGN is of improved robustness with respect to the varying input data and algorithm parameters. In addition, turning to an evaluation of data stream based on approximated objective functions, we develop a new adaptive stepsize strategy for SGN (AdaSGN) which requires no priori knowledge of the input data, and numerically illustrate its comparable performance with SGN adopting the manaully-tuned diminishing stepsize. Without assuming the eigengap to be positive, we also establish the global and optimal convergence rate of SGN with the specified stepsize using the diffusion approximation *** Codes 65F15, 68W27, 60H10, 60J70 Copyright © 2022, The Authors. All rights reserved.

关键词： Principal component analysis