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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

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

Fractional Dissipative PDEs

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arXiv 2023年

作者： Achleitner, Franz Akagi, Goro Kuehn, Christian Melenk, Jens Markus Rademacher, Jens D.M. Soresina, Cinzia Yang, Jichen TU Wien Institute for Analysis and Scientific Computing Vienna1040 Austria Tohoku University Mathematical Institute Graduate School of Science Sendai980-8578 Japan Technical University of Munich Department of Mathematics Research Unit "Multiscale and Stochastic Dynamics" Garching b. München85748 Germany Garching b. München85748 Germany External Faculty Complexity Science Hub Vienna Vienna1080 Austria Universität Hamburg MIN faculty Department of Mathematics Applied Dynamical Systems Hamburg20146 Germany Universität Graz Department of Mathematics and Scientific Computing Graz8010 Austria Harbin Engineering University College of Mathematical Sciences Harbin150001 China

In this chapter we provide an introduction to fractional dissipative partial differential equations (PDEs) with a focus on trying to understand their dynamics. The class of PDEs we focus on are reaction-diffusion equations but we also provide an outlook on closely related classes of PDEs. To simplify the exposition, we only discuss the cases of fractional time derivatives and fractional space derivatives in the PDE separately. As our main tools, we describe analytical as well as numerical methods, which are generically necessary to study nonlinear dynamics. We start with the analytical study of steady states and local linear stability for fractional time derivatives. Then we extend this view to a global perspective and consider time-fractional PDEs and gradient flows. Next, we continue to steady states, linear stability analysis and bifurcations for space-fractional PDEs. As a final analytical consideration we discuss existence and stability of traveling waves for space-fractional PDEs. In the last parts, we provide numerical discretization schemes for fractional (dissipative) PDEs and we utilize these techniques within numerical continuation in applied examples of fractional reaction-diffusion PDEs. We conclude with a brief summary and outlook on open questions in the field. © 2023, CC BY.

关键词： Linear stability analysis

Electron-phonon interaction contribution to the total energy of group IV semiconductor polymorphs: Evaluation and implications

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arXiv 2022年

作者： Varma, Arjun Varma Ramasimha Paul, Shilpa Itale, Anup Pable, Pranav Tibrewala, Radhika Dodal, Samruddhi Yerunkar, Harshal Bhaumik, Saurav Shah, Vaishali Gururajan, Mogadalai Pandurangan Prasanna, Tiramkudlu R.S. Department of Metallurgical Engineering and Materials Science Indian Institute of Technology Bombay Mumbai400076 India Department of Mathematics Indian Institute of Technology Bombay Mumbai400076 India Department of Scientific Computing Modeling and Simulation Savitribai Phule Pune University Pune411007 India

In density functional theory (DFT) based total energy studies, the van der Waals (vdW) and zero-point vibrational energy (ZPVE) correction terms are included to obtain energy differences between polymorphs. We propose and compute a new correction term to the total energy, due to electron-phonon interactions (EPI). We rely on Allen's general formalism, which goes beyond the Quasi-Harmonic Approximation (QHA), to include the free energy contributions due to quasiparticle interactions. We show that, for semiconductors and insulators, the EPI contributions to the free energies of electrons and phonons are the corresponding zero-point energy contributions. Using an approximate version of Allen's formalism in combination with the Allen-Heine theory for EPI corrections, we calculate the zero-point EPI corrections to the total energy for cubic and hexagonal polytypes of Carbon, Silicon and Silicon Carbide. The EPI corrections alter the energy differences between polytypes. In SiC polytypes, the EPI correction term is more sensitive to crystal structure than the vdW and ZPVE terms and is thus essential in determining their energy differences. It clearly establishes that the cubic SiC-3C is metastable and hexagonal SiC-4H is the stable polytype. Our results are consistent with the experimental results of Kleykamp. Our study enables the inclusion of EPI corrections as a separate term in the free energy expression. This opens the way to go beyond the QHA by including the contribution of EPI on all thermodynamic properties. © 2022, CC BY.

关键词： Silicon carbide

Mixed Matrix Completion in Complex Survey Sampling under Heterogeneous Missingness

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arXiv 2024年

作者： Mao, Xiaojun Wang, Hengfang Wang, Zhonglei Yang, Shu School of Mathematical Sciences Ministry of Education Key Laboratory of Scientific and Engineering Computing Shanghai Jiao Tong University China School of Mathematics and Statistics Fujian Provincial Key Laboratory of Statistics and Artificial Intelligence Fujian Normal University China Wang Yanan Institute for Studies in Economics School of Economics Xiamen University China Department of Statistics North Carolina State University United States

Modern surveys with large sample sizes and growing mixed-type questionnaires require robust and scalable analysis methods. In this work, we consider recovering a mixed dataframe matrix, obtained by complex survey sampling, with entries following different canonical exponential distributions and subject to heterogeneous missingness. To tackle this challenging task, we propose a two-stage procedure: in the first stage, we model the entry-wise missing mechanism by logistic regression, and in the second stage, we complete the target parameter matrix by maximizing a weighted log-likelihood with a low-rank constraint. We propose a fast and scalable estimation algorithm that achieves sublinear convergence, and the upper bound for the estimation error of the proposed method is rigorously derived. Experimental results support our theoretical claims, and the proposed estimator shows its merits compared to other existing methods. The proposed method is applied to analyze the National Health and Nutrition Examination Survey data. Copyright © 2024, The Authors. All rights reserved.

关键词： Machine learning

Post-Fault Power Grid Voltage Prediction via 1D-CNN with Spatial Coupling

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Post-Fault Power Grid Voltage Prediction via 1D-CNN with Spa...

Artificial Intelligence for Industries (AI4I)

作者： Carson Hu Guang Lin Bao Wang Meng Yue Jack Xin Department of Mathematics University of California Irvine CA USA Department of Mathematics School of Mechanical Engineering Purdue University West Lafayette IN USA Department of Mathematics Scientific Computing and Imaging Institute University of Utah Salt Lake City UT USA Sustainable Energy Technologies Department Brookhaven National Laboratory Upton NY USA

We propose a one-dimensional convolutional neural network (1D-CNN) with spatial coupling for post-fault power grid voltage prediction. Our proposed deep learning framework was inspired by the celebrated Prony’s method in classical signal processing. Our spatio-temporal model significantly outperforms existing benchmarks, including long short-term memory model, and is applicable to other strong transients in power industries.

关键词： Couplings Signal processing algorithms Voltage Benchmark testing Signal processing Prediction algorithms Power grids

Numerical Investigations of a Class of Biological Models on Unbounded Domain

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Numerical mathematics(Theory,Methods and Applications) 2019年第1期12卷 169-186页

作者： Qiumei Huang Dongfang Li Jiwei Zhang Beijing Institute for Scientific and Engineering Computing Beijing University of TechnologyBeijing 100124China School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074China Hubei Key Laboratoryof Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China Beijing Computational Science Research Center Beijing1000193China

This paper is concerned with numerical computations of a class of biologi-cal models on unbounded spatial *** overcome the unboundedness of spatial domain,we first construct efficient local absorbing boundary conditions(LABCs)to re-formulate the Cauchy problem into an initial-boundary value(IBV)*** that,we construct a linearized finite difference scheme for the reduced IVB problem,and provide the corresponding error estimates and stability *** delay-dependent dynamical properties on the Nicholson’s blowflies equation and the Mackey-Glass equa-tion are numerically ***,numerical examples are given to demonstrate the efficiency of our LABCs and theoretical results of the numerical scheme.

关键词： Biological models artificial boundary method absorbing boundary conditions finite difference method stability,convergence

Capturing the Influence of Conduction Velocity on Epicardial Activation Patterns Using Uncertainty Quantification

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Capturing the Influence of Conduction Velocity on Epicardial...

Computers in Cardiology (CinC)

作者： Anna Busatto Lindsay C Rupp Karli Gillette Akil Narayan Gernot Plank Rob S MacLeod Scientific Computing and Imaging Institute University of Utah SLC UT USA Nora Eccles Cardiovascular Research and Training Institute University of Utah SLC UT USA Department of Biomedical Engineering University of Utah SLC UT USA University of Utah Salt Lake City UT Institute of Biophysics Medical University of Graz Graz Austria Department of Mathematics University of Utah SLC UT USA

Individual variability in parameter settings, due to either user selection or disease states, can impact accuracy when simulating the electrical behavior of the heart. Here, we aim to test the impact of inevitable uncertainty in conduction velocities (CVs) on the output of simulations of cardiac propagation, given three stimulus locations on the left ventricular (LV) free wall. To understand the role of physiological variability in CV in simulations of cardiac activation, we generated detailed maps of the variability in propagation simulations by implementing bi-ventricular activation simulations and quantified the effects by deploying robust uncertainty quantification techniques based on polynomial chaos expansion (PCE). PCE allows efficient stochastic exploration with reduced computational demand by utilizing an emulator for the underlying forward model. Our results suggest that CV within healthy physiological ranges plays a small role in the activation times across all stimulation locations. However, we noticed differences in variation coefficients depending on the stimulation site, i.e., LV endocardium, midmyocardium, and epicardium. We observed low levels of variation in activation times near the earliest activation sites, whereas there was higher variation toward the termination sites. These results suggest that CV variability can play a role when simulating healthy and diseased states.

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Transductive Matrix Completion with Calibration for Multi-Task Learning

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arXiv 2023年

作者： Wang, Hengfang Zhang, Yasi Mao, Xiaojun Wang, Zhonglei School of Mathematics and Statistics Fujian Normal University China Department of Statistics University of California Los Angeles United States School of Mathematical Sciences Ministry of Education Key Laboratory of Scientific and Engineering Computing Shanghai Jiao Tong University China Wang Yanan Institute for Studies in Economics School of Economics Xiamen University China

Multi-task learning has attracted much attention due to growing multi-purpose research with multiple related data sources. Moreover, transduction with matrix completion is a useful method in multi-label learning. In this paper, we propose a transductive matrix completion algorithm that incorporates a calibration constraint for the features under the multi-task learning framework. The proposed algorithm recovers the incomplete feature matrix and target matrix simultaneously. Fortunately, the calibration information improves the completion results. In particular, we provide a statistical guarantee for the proposed algorithm, and the theoretical improvement induced by calibration information is also studied. Moreover, the proposed algorithm enjoys a sub-linear convergence rate. Several synthetic data experiments are conducted, which show the proposed algorithm out-performs other existing methods, especially when the target matrix is associated with the feature matrix in a nonlinear way. Copyright © 2023, The Authors. All rights reserved.

关键词： Calibration