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COMPLETE KAHLER METRICS WITH POSITIVE HOLOMORPHIC SECTIONAL CURVATURES ON CERTAIN LINE BUNDLES(RELATED TO A COHOMOGENEITY ONE POINT OF VIEW ON A YAU CONJECTURE)

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Acta Mathematica Scientia 2024年第1期44卷 78-102页

作者：段晓曼关庄丹 School of mathematics and Statistics Henan UniversityKaifeng475004China department of mathematics The University of California at RiversideRiversideCA92521USA

In this article,we study Kahler metrics on a certain line bundle over some compact Kahler manifolds to find complete Kahler metrics with positive holomorphic sectional(or bisectional)***,we apply a strategy to a famou... 详细信息

关键词： Kahler Metrics complete Riemannian metrics open complex manifolds holomorphic bisectional curvature C*bundle almost homogeneous manifolds

Lax-Oleinik-Type Formulas and Efficient Algorithms for Certain High-Dimensional Optimal Control Problems

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Communications on Applied mathematics and Computation 2024年第2期6卷 1428-1471页

作者： Paula Chen Jerome Darbon Tingwei Meng Division of Applied mathematics Brown UniversityProvidenceRIUSA department of mathematics UCLALos AngelesCAUSA

Two of the main challenges in optimal control are solving problems with state-dependent running costs and developing efficient numerical solvers that are computationally tractable in high *** this paper,we provide analytical solutions to certain optimal control problems whose running cost depends on the state variable and with constraints on the *** also provide Lax-Oleinik-type representation formulas for the corresponding Hamilton-Jacobi partial differential equations with state-dependent ***,we present an efficient,grid-free numerical solver based on our representation formulas,which is shown to scale linearly with the state dimension,and thus,to overcome the curse of *** existing optimization methods and the min-plus technique,we extend our numerical solvers to address more general classes of convex and nonconvex initial *** demonstrate the capabilities of our numerical solvers using implementations on a central processing unit(CPU)and a field-programmable gate array(FPGA).In several cases,our FPGA implementation obtains over a 10 times speedup compared to the CPU,which demonstrates the promising performance boosts FPGAs can *** numerical results show that our solvers have the potential to serve as a building block for solving broader classes of high-dimensional optimal control problems in real-time.

关键词： Optimal control Hamilton-Jacobi partial differential equations Grid-free numerical methods High dimensions Field-programmable gate arrays(FPGAs)

An Efficient Smoothing and Thresholding Image Segmentation Framework with Weighted Anisotropic-Isotropic Total Variation

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Communications on Applied mathematics and Computation 2024年第2期6卷 1369-1405页

作者： Kevin Bui Yifei Lou Fredrick Park Jack Xin department of mathematics University of CaliforniaIrvineIrvineCA 92697-3875USA department of mathematics University of North CarolinaChapel HillChapel HillNC 27599.USA department of mathematics and Computer Science Whittier CollegeWhittierCA 90602USA

In this paper,we design an efficient,multi-stage image segmentation framework that incorporates a weighted difference of anisotropic and isotropic total variation(AITV).The segmentation framework generally consists of two stages:smoothing and thresholding,thus referred to as smoothing-and-thresholding(SaT).In the first stage,a smoothed image is obtained by an AITV-regularized Mumford-Shah(MS)model,which can be solved efficiently by the alternating direction method of multipliers(ADMMs)with a closed-form solution of a proximal operator of the l_(1)-αl_(2) *** convergence of the ADMM algorithm is *** the second stage,we threshold the smoothed image by K-means clustering to obtain the final segmentation *** experiments demonstrate that the proposed segmentation framework is versatile for both grayscale and color images,effcient in producing high-quality segmentation results within a few seconds,and robust to input images that are corrupted with noise,blur,or *** compare the AITV method with its original convex TV and nonconvex TVP(Ocasing the qualitative and quantitative advantages of our proposed method.

关键词： Image segmentation Non-convex optimization Mumford-Shah(MS)model Alternating direction method of multipliers(ADMMs) Proximal operator

Convergence of Hyperbolic Neural Networks Under Riemannian Stochastic Gradient Descent

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Communications on Applied mathematics and Computation 2024年第2期6卷 1175-1188页

作者： Wes Whiting Bao Wang Jack Xin department of mathematics University of CaliforniaIrvineCAUSA department of mathematics Scientific Computing and Imaging InstituteUniversity of UtahSalt Lake CityUTUSA

We prove,under mild conditions,the convergence of a Riemannian gradient descent method for a hyperbolic neural network regression model,both in batch gradient descent and stochastic gradient *** also discuss a Riemann... 详细信息

关键词： Hyperbolic neural network Riemannian gradient descent Riemannian Adam(RAdam) Training convergence

Iterative Subregion Correction Preconditioners with Adaptive Tolerance for Problems with Geometrically Localized Stiffness

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Communications on Applied mathematics and Computation 2024年第2期6卷 811-836页

作者： Michael Franco Per-Olof Persson Will Pazner department of mathematics University of CaliforniaBerkeleyBerkeleyCA 94720-3840USA Fariborz Maseeh department of mathematics and Statistics Portland State UniversityPortlandOR 97201USA

We present a class of preconditioners for the linear systems resulting from a finite element or discontinuous Galerkin discretizations of advection-dominated *** preconditioners are designed to treat the case of geometrically localized stiffness,where the convergence rates of iterative methods are degraded in a localized subregion of the *** convergence may be caused by a number of factors,including the mesh size,anisotropy,highly variable coefficients,and more challenging *** approach taken in this work is to correct well-known preconditioners such as the block Jacobi and the block incomplete LU(ILU)with an adaptive inner subregion *** goal of these preconditioners is to reduce the number of costly global iterations by accelerating the convergence in the stiff region by iterating on the less expensive reduced *** tolerance for the inner iteration is adaptively chosen to minimize subregion-local work while guaranteeing global convergence *** present analysis showing that the convergence of these preconditioners,even when combined with an adaptively selected tolerance,is independent of discretization parameters(e.g.,the mesh size and diffusion coefficient)in the *** demonstrate significant performance improvements over black-box preconditioners when applied to several model convection-diffusion ***,we present performance results of several variations of iterative subregion correction preconditioners applied to the Reynolds number 2.25×10^(6)fluid flow over the NAca 0012 airfoil,as well as massively separated flow at 30°angle of attack.

关键词： Subregion correction Nested Krylov Geometrically localized stiffness

A REDUCED ORDER SCHWARZ METHOD FOR NONLINEAR MULTIScaLE ELLIPTIC EQUATIONS BASED ON TWO-LAYER NEURAL NETWORKS

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Journal of Computational mathematics 2024年第2期42卷 570-596页

作者： Shi Chen Zhiyan Ding Qin Li Stephen J.Wright mathematics department University of Wisconsin-MadisonMadisonWI 53706USA department of mathematics University of California-BerkeleyBerkeley CA 94720USA mathematics department and Discovery Institute University of Wisconsin-Madison MadisonWI 53706USA Computer Sciences department University of WisconsinMadisonWI 53706USA

Neural networks are powerful tools for approximating high dimensional data that have been used in many contexts,including solution of partial differential equations(PDEs).We describe a solver for multiscale fully nonlinear elliptic equations that makes use of domain decomposition,an accelerated Schwarz framework,and two-layer neural networks to approximate the boundary-to-boundarymap for the subdomains,which is the key step in the Schwarz ***,the boundary-to-boundary map requires solution of boundary-value elliptic problems on each *** leveraging the compressibility of multiscale problems,our approach trains the neural network offline to serve as a surrogate for the usual implementation of the boundary-to-boundary *** method is applied to a multiscale semilinear elliptic equation and a multiscale p-Laplace *** both cases we demonstrate significant improvement in efficiency as well as good accuracy and generalization performance.

关键词： Nonlinear homogenization Multiscale elliptic problem Neural networks Do-main decomposition

Model Change Active Learning in Graph-Based Semi-supervised Learning

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Communications on Applied mathematics and Computation 2024年第2期6卷 1270-1298页

作者： Kevin S.Miller Andrea L.Bertozzi department of mathematics University of CaliforniaLos Angeles520 Portola PlazaLos AngelesCA 90095USA

Active learning in semi-supervised classification involves introducing additional labels for unlabelled data to improve the accuracy of the underlying classifier.A challenge is to identify which points to label to best improve performance while limiting the number of new labels."Model Change"active learning quantifies the resulting change incurred in the classifier by introducing the additional label(s).We pair this idea with graph-based semi-supervised learning(SSL)methods,that use the spectrum of the graph Laplacian matrix,which can be truncated to avoid prohibitively large computational and storage *** consider a family of convex loss functions for which the acquisition function can be efficiently approximated using the Laplace approximation of the posterior *** show a variety of multiclass examples that illustrate improved performance over prior state-of-art.

关键词： Active learning Graph-based methods Semi-supervised learning(SSL) Graph Laplacian

Review of Computational Approaches to Optimization Problems in Inhomogeneous Rods and Plates

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Communications on Applied mathematics and Computation 2024年第1期6卷 236-256页

作者： Weitao Chen Chiu-Yen Kao department of mathematics University of California at Riverside900 University AveRiversideCA92521USA department of Mathematical Sciences Claremont McKenna College850 Columbia AveClaremontCA91711USA

In this paper,we review computational approaches to optimization problems of inhomogeneous rods and *** consider both the optimization of eigenvalues and the localization of *** problems are motivated by physical problems including the determination of the extremum of the fundamental vibration frequency and the localization of the vibration *** demonstrate how an iterative rearrangement approach and a gradient descent approach with projection can successfully solve these optimization problems under different boundary conditions with different densities given.

关键词： Inhomogeneous rods and plates Bi-Laplacian Optimization of eigenvalues Localization of eigenfunctions Rearrangement

Analytical study of atmospheric internal waves model with fractional approach

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Journal of Ocean Engineering and Science 2024年第4期9卷 326-337页

作者： Trushit Patel Hardik Patel Ramakanta Meher Computer Science University of the PeoplePasadenaCA 91101United States department of mathematics Uka Tarsadia UniversityBardoliGujaratIndia mathematics and Humanities department Sardar Vallabhbhai National Institute of TechnologySuratGujaratIndia

This study looks at the mathematical model of internal atmospheric waves,often known as gravity waves,occurring inside a fluid rather than on the *** the shallow-fluid assumption,internal atmospheric waves may be described by a nonlinear partial differential equation *** shallow flow model’s primary concept is that the waves are spread out across a large horizontal area before rising *** Fractional Reduced Differential Transform Method(FRDTM)is applied to provide approximate solutions for any given *** aids in the modelling of the global atmosphere,which has applications in weather and climate *** the integer-order value(α=1),the FRDTM solution is compared to the precise solution,EADM,and HAM to assess the correctness and efficacy of the proposed technique.

关键词： Shallow-fluid equations Atmospheric internal waves Time-fractional partial differential equations Fractional reduced differential transform method(FRDTM) Climate prediction