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Dissolving Constraints for Riemannian Optimization

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arXiv 2022年

作者： Xiao, Nachuan Liu, Xin Toh, Kim-Chuan The Institute of Operations 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 consider optimization problems over closed embedded submanifolds of Rn, which are defined by the constraints c(x) = 0. We propose a class of constraint dissolving approaches for these Riemannian optimization problems. In these proposed approaches, solving a Riemannian optimization problem is transferred into the unconstrained minimization of a constraint dissolving function named CDF. Different from existing exact penalty functions, the exact gradient and Hessian of CDF are easy to compute. We study the theoretical properties of CDF and prove that the original problem and CDF have the same first-order and second-order stationary points, local minimizers, and Lojasiewicz exponents in a neighborhood of the feasible region. Remarkably, the convergence properties of our proposed constraint dissolving approaches can be directly inherited from the existing rich results in unconstrained optimization. Therefore, the proposed constraint dissolving approaches build up short cuts from unconstrained optimization to Riemannian optimization. Several illustrative examples further demonstrate the potential of our proposed constraint dissolving approaches. Copyright © 2022, The Authors. All rights reserved.

关键词： Optimization

A force-based gradient descent method for ab initio atomic structure relaxation

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arXiv 2022年

作者： Hu, Yukuan Gao, Xingyu Zhao, Yafan Liu, Xin Song, Haifeng State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China University of Chinese Academy of Sciences Beijing100049 China Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Beijing100088 China CAEP Software Center for High Performance Numerical Simulation Beijing100088 China

Force-based algorithms for ab initio atomic structure relaxation, such as conjugate gradient methods, usually get stuck in the line minimization processes along search directions, where expensive ab initio calculations are triggered frequently to test trial positions before locating the next iterate. We present a force-based gradient descent method, WANBB, that circumvents the deficiency. At each iteration, WANBB enters the line minimization process with a trial stepsize capturing the local curvature of the energy surface. The exit is controlled by an unrestrictive criterion that tends to accept early trials. These two ingredients streamline the line minimization process in WANBB. The numerical simulations on nearly 80 systems with good universality demonstrate the considerable compression of WANBB on the cost for the unaccepted trials compared with conjugate gradient methods. We also observe across the board significant and universal speedups as well as the superior robustness of WANBB over several widely used methods. The latter point is theoretically established. The implementation of WANBB is pretty simple, in that no a priori physical knowledge is required and only two parameters are present without tuning. © 2022, CC BY.

关键词： Conjugate gradient method

Exploring Question-Specific Rewards for Generating Deep Questions 28

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Exploring Question-Specific Rewards for Generating Deep Ques...

28th International Conference on Computational Linguistics, COLING 2020

作者： Xie, Yuxi Pan, Liangming Wang, Dongzhe Kan, Min-Yen Feng, Yansong Wangxuan Institute of Computer Technology Peking University China NUS Graduate School for Integrative Sciences and Engineering ZhuiYi Technology Singapore School of Computing National University of Singapore Singapore The MOE Key Laboratory of Computational Linguistics Peking University China

ISBN: (纸本)9781952148279

Recent question generation (QG) approaches often utilize the sequence-to-sequence framework (Seq2Seq) to optimize the log-likelihood of ground-truth questions using teacher forcing. However, this training objective is inconsistent with actual question quality, which is often reflected by certain global properties such as whether the question can be answered by the document. As such, we directly optimize for QG-specific objectives via reinforcement learning to improve question quality. We design three different rewards that target to improve the fluency, relevance, and answerability of generated questions. We conduct both automatic and human evaluations in addition to a thorough analysis to explore the effect of each QG-specific reward. We find that optimizing question-specific rewards generally leads to better performance in automatic evaluation metrics. However, only the rewards that correlate well with human judgement (e.g., relevance) lead to real improvement in question quality. Optimizing for the others, especially answerability, introduces incorrect bias to the model, resulting in poor question quality. Our code is publicly available at https://***/YuxiXie/RL-for-Question-Generation. © 2020 COLING 2020 - 28th International Conference on Computational Linguistics, Proceedings of the Conference. All rights reserved.

关键词： Reinforcement learning

A global optimization approach for multi-marginal optimal transport problems with Coulomb cost

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arXiv 2021年

作者： Hu, Yukuan Chen, Huajie Liu, Xin 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 School of Mathematical Sciences Beijing Normal University China

In this work, we construct a novel numerical method for solving the multi-marginal optimal transport problems with Coulomb cost. This type of optimal transport problems arises in quantum physics and plays an important role in understanding the strongly correlated quantum systems. With a Monge-like ansatz, the orginal high-dimensional problems are transferred into mathematical programmings with generalized complementarity constraints, and thus the curse of dimensionality is surmounted. However, the latter ones are themselves hard to deal with from both theoretical and practical perspective. Moreover in the presence of nonconvexity, brute-force searching for global solutions becomes prohibitive as the problem size grows large. To this end, we propose a global optimization approach for solving the nonconvex optimization problems, by exploiting an efficient proximal block coordinate descent local solver and an initialization subroutine based on hierarchical grid refinements. We provide numerical simulations on some typical physical systems to show the efficiency of our approach. The results match well with both theoretical predictions and physical intuitions, and give the first visualization of optimal transport maps for some two dimensional *** Codes 81V05 (Primary) 49M37, 65K05, 90C26, 90C30 (Secondary) Copyright © 2021, The Authors. All rights reserved.

关键词： Global optimization

HOMOGENIZATION OF THE DISTRIBUTION-DEPENDENT STOCHASTIC ABSTRACT FLUID MODELS

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arXiv 2023年

作者： Chen, Junlong Qiu, Zhaoyang Tang, Yanbin School of Sciences Great Bay University Great Bay Institute for Advanced Study Dongguan523000 China School of Mathematics University of Science and Technology of China Anhui Hefei230026 China School of Applied Mathematics Nanjing University of Finance and Economics Nanjing210046 China School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China

In this paper, we study the homogenization of the distribution-dependent stochastic abstract fluid models by combining the two−scale convergence and martingale representative approach. A general framework of the homogenization research is established for stochastic abstract fluid models which are the type of nonlinear partial differential equations including the (distribution-dependent) stochastic Navier-Stokes equations, stochastic magneto-hydrodynamic equations, stochastic Boussinesq equations, stochastic micropolar equations, stochastic Allen-Cahn equations. Copyright © 2023, The Authors. All rights reserved.

关键词： Navier Stokes equations

A Constraint Dissolving Approach for Nonsmooth Optimization over the Stiefel Manifold

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arXiv 2022年

作者： Hu, Xiaoyin Xiao, Nachuan Liu, Xin Toh, Kim-Chuan School of Computer and Computing Science Hangzhou City University Hangzhou310015 China The Institute of Operations 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

This paper focus on the minimization of a possibly nonsmooth objective function over the Stiefel manifold. The existing approaches either lack efficiency or can only tackle prox-friendly objective functions. We propose a constraint dissolving function named NCDF and show that it has the same first-order stationary points and local minimizers as the original problem in a neighborhood of the Stiefel manifold. Furthermore, we show that the Clarke subdifferential of NCDF is easy to achieve from the Clarke subdifferential of the objective function. Therefore, various existing approaches for unconstrained nonsmooth optimization can be directly applied to nonsmooth optimization problems over the Stiefel manifold. We propose a framework for developing subgradient-based methods and establish their convergence properties based on prior works. Furthermore, based on our proposed framework, we can develop efficient approaches for optimization over the Stiefel manifold. Preliminary numerical experiments further highlight that the proposed constraint dissolving approach yields efficient and direct implementations of various unconstrained approaches to nonsmooth optimization problems over the Stiefel manifold. Copyright © 2022, The Authors. All rights reserved.

关键词： Optimization

The contribution of dilatational motion to energy flux in homogeneous compressible turbulence

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arXiv 2025年

作者： Luo, Chensheng Fang, Le Fang, Jian Xu, Haitao Pumir, Alain Yang, Ping-Fan School of Aeronautics Institute of Extreme Mechanics Northwestern Polytechnical University Xi’an710072 China Research Institute of Aero-Engine Beihang University Beijing100191 China Laboratory of Complex System Ecole Centrale de Pékin School of General Engineering Beihang University Beijing100191 China Scientific Computing Department STFC Daresbury Laboratory WarringtonWA4 4AD United Kingdom Center for Combustion Energy School of Aerospace Engineering Tsinghua University Beijing100084 China Laboratoire de Physique Ecole Normale Supérieure de Lyon CNRS Université de Lyon LyonF-69007 France National Key Laboratory of Aircraft Configuration Design Xi’an710072 China

We analyze the energy flux in compressible turbulence by generalizing the exact decomposition recently proposed by Johnson (Phys. Rev. Lett., vol. 124, 2020. 104501) to study incompressible turbulent flows. This allows us to characterize the effect of dilatational motion on the inter-scale energy transfer in three-dimensional compressible turbulence. Our analysis reveals that the contribution of dilatational motion to energy transfer is due to three different physical mechanisms: the interaction between dilatation and strain, between dilatation and vorticity, and the self-interaction of dilatational motion across scales. By analyzing numerical simulations of flows at moderate turbulent Mach numbers (Mat 0.3), we validate our theoretical derivations and provide a quantitative description of the role of dilatational motion in energy transfer. In particular, we determine the scaling dependence of the dilatational contributions on the turbulent Mach number. Moreover, our findings reveal that the eddy-viscosity assumption often used in large-eddy simulations, in the spirit of the approach used for incompressible flows, effectively neglects the interaction between solenoidal-dilatational energy transfer and overestimate dilatational effects. Copyright © 2025, The Authors. All rights reserved.

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On the periodic solutions of the three-body problem

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National Science Review 2019年第6期6卷 1070-1071页

作者： Shijun Liao Xiaoming Li Advanced Center of Computing School of Naval Architecture Ocean and Civil EngineeringShanghai Jiao Tong University School of Physics and Astronomy Shanghai Jiao Tong University MOE Key Laboratory of Disaster Forecast and Control in Engineering School of Mechanics and Construction Engineering Jinan University

The motion of three bodies that attract each other by universal gravitation was first investigated by Newton [1] and was reconsidered by many great mathematicians and scientists, including Euler, Lagrange, Poincaré [2] and Hilbert. The famous three-body problem has had a great influence on physics, mathematics and non-linear dynamics. It has led to a new

关键词： On the periodic solutions of the three-body problem

Cascaded adaptive aberration-eliminating multimode fiber imaging

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Advanced Photonics 2025年第2期 135-143页

作者： Zhong Wen Qilin Deng Quanzhi Li Yizhou Tan Jingshan Zhong Chiming Zhang Jiahe Zhang Clemens F.Kaminski Ying Gu Xu Liu Qing Yang Zhejiang University College of Optical Science and Engineering State Key Laboratory of Extreme Photonics and Instrumentation Zhejiang University International Research Center for Advanced Photonics Zhejiang University ZJU-Hangzhou Global Scientific and Technological Innovation Center Chinese PLA General Hospital The First Medical Center Department of Laser Medicine Chinese PLA General Hospital Hainan HospitalLaser Medicine Center Zhejiang Laboratory Research Center for Intelligent Manufacturing Computing University of Cambridge Department of Chemical Engineering and Biotechnology

In vivo microscopic imaging inside a biological lumen such as the gastrointestinal tract, respiratory airways, or within blood vessels has faced significant technological challenges for decades. A promising candidate technology is the multimode fiber(MMF) endoscope, which enables minimally invasive diagnostics at a resolution reaching the cellular level. However, for in vivo imaging applications deep inside a biological lumen, sample-induced aberrations and the dynamic dispersion in the MMF make the MMF endoscope a chaotic system with many unknowns, where multiple minor fluctuations can couple and compound into intractable problems. We introduce a dynamically encoding, cascaded, optical, and ultrathin polychromatic light-field endoscopy(DECOUPLE) to tackle this challenge. DECOUPLE includes an adaptive aberration correction that can accurately track and control MMF behavior in the spatial-frequency domain to compensate for chaos introduced during complex dynamic imaging processes. We demonstrate the flexibility and practicality of DECOUPLE for noninvasive volumetric imaging in two colors for light passing through various highly aberrating samples including 120-μm-thick onion epidermal slices and 80-μm-thick layers of fat emulsions. To summarize, we represent a significant step toward practical in vivo imaging deep within biological tissue.

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