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Optimization of Random Feature method in the High-Precision Regime

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Communications on Applied Mathematics and Computation 2024年第2期6卷 1490-1517页

作者： Jingrun Chen Weinan E Yifei Sun School of Mathematical Sciences and Suzhou Institute for Advanced Research Suzhou 215006JiangsuChina University of Science and Technology of China Hefei 230026AnhuiChina Center for Machine Learning Research and School of Mathematical Sciences Peking UniversityBeijing 100871China AI for Science Institute Beijing 100084China School of Mathematical Sciences Soochow UniversitySuzhou 215006JiangsuChina

Machine learning has been widely used for solving partial differential equations(PDEs)in recent years,among which the random feature method(RFM)exhibits spectral accuracy and can compete with traditional solvers in terms of both accuracy and ***,the optimization problem in the RFM is more difficult to solve than those that arise in traditional *** the broader machine-learning research,which frequently targets tasks within the low-precision regime,our study focuses on the high-precision regime crucial for solving *** this work,we study this problem from the following aspects:(i)we analyze the coeffcient matrix that arises in the RFM by studying the distribution of singular values;(ii)we investigate whether the continuous training causes the overfitting issue;(ii)we test direct and iterative methods as well as randomized methods for solving the optimization *** on these results,we find that direct methods are superior to other methods if memory is not an issue,while iterative methods typically have low accuracy and can be improved by preconditioning to some extent.

关键词： Random feature method(RFM) Partial differential equation(PDE) Least-squares problem Direct method iterative method

iterative methodS FOR THE FORWARD-BACKWARD HEAT EQUATION

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Journal of Computational Mathematics 2005年第4期23卷 419-424页

作者： Xiao-liang Cheng Jie Sun Department of Mathematics Zhejiang University Hangzhou 310028 China Department of Mathematics Zhejiang University of Finance and Economics Hangzhou 310012 China

In this paper we propose the finite difference method for the forward-backward heat equation. We use a coarse-mesh second-order central difference scheme at the middle line mesh points and derive the error estimate. Then we discuss the iterative method based on the domain decomposition for our scheme and derive the bounds for the rates of convergence. Finally we present some numerical experiments to support our analysis.

关键词： Forward-backward heat equation Finite difference method iterative method Coarse mesh

iterative methods for Solving the Nonlinear Balance Equation with Optimal Truncation

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Advances in Atmospheric Sciences 2021年第5期38卷 755-770页

作者： Qin XU Jie CAO NOAA/National Severe Storms Laboratory NormanOklahoma 73069USA Cooperative Institute for Mesoscale Meteorological Studies University of OklahomaNormanOklahoma 73072USA Institute of Atmospheric Physics Chinese Academy of SciencesBeijing 100029China

Two types of existing iterative methods for solving the nonlinear balance equation(NBE)are *** the first type,the NBE is rearranged into a linearized equation for a presumably small correction to the initial guess or the subsequent updated *** the second type,the NBE is rearranged into a quadratic form of the absolute vorticity with the positive root of this quadratic form used in the form of a Poisson equation to solve NBE *** two methods are rederived by expanding the solution asymptotically upon a small Rossby number,and a criterion for optimally truncating the asymptotic expansion is proposed to obtain the super-asymptotic approximation of the *** each rederived method,two iterative procedures are designed using the integral-form Poisson solver versus the over-relaxation scheme to solve the boundary value problem in each *** testing with analytically formulated wavering jet flows on the synoptic,sub-synoptic and meso-αscales,the iterative procedure designed for the first method with the Poisson solver,named M1a,is found to be the most accurate and *** the synoptic wavering jet flow in which the NBE is entirely elliptic,M1a is extremely *** the sub-synoptic wavering jet flow in which the NBE is mostly elliptic,M1a is sufficiently *** the meso-αwavering jet flow in which the NBE is partially hyperbolic so its boundary value problem becomes seriously ill-posed,M1a can effectively reduce the solution error for the cyclonically curved part of the wavering jet flow,but not for the anti-cyclonically curved part.

关键词： nonlinear balance iterative method optimal truncation

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Computer methodologies for the Comparison of Some Efficient Derivative FreeSimultaneous iterative methods for Finding Roots of Non-Linear Equations

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Computers, Materials & Continua 2021年第1期66卷 275-290页

作者： Yuming Chu Naila Rafiq Mudassir Shams Saima Akram Nazir Ahmad Mir Humaira Kalsoom Department of Mathematics Huzhou UniversityHuzhou313000China Department of Mathematics National University of Modern LanguagesIslamabadPakistan Department of Mathematics and Statistics Riphah International University I-14Islamabad44000Pakistan Center for Advanced Studies in Pure and Applied Mathematics Bahauddin Zakariya University MultanPakistan School of Mathematical Sciences Zhejiang UniversityHanghou310027China

In this article,we construct the most powerful family of simultaneous iterative method with global convergence behavior among all the existing methods in literature for finding all roots of non-linear *** analysis proved that the order of convergence of the family of derivative free simultaneous iterative method is *** main aim is to check out the most regularly used simultaneous iterative methods for finding all roots of non-linear equations by studying their dynamical planes,numerical experiments and CPU *** planes of iterative methods are drawn by using MATLAB for the comparison of global convergence properties of simultaneous iterative *** behavior of the higher order simultaneous iterative methods are also illustrated by residual graph obtained from some numerical test *** test examples,dynamical behavior and computational efficiency are provided to present the performance and dominant efficiency of the newly constructed derivative free family of simultaneous iterative method over existing higher order simultaneous methods in literature.

关键词： Non-linear equation iterative method simultaneous method basins of attractions computational efficiency

APPLICATIONS OF STAIR MATRICES AND THEIR GENERALIZATIONS TO iterative methodS

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Applied Mathematics and Mechanics(English Edition) 2006年第8期27卷 1115-1121页

作者：邵新慧沈海龙李长军 Department of Mathematics Northeastern UniversityShenyang 110004P.R.China

Stair matrices and their generalizations are introduced. The definitions and some properties of the matrices were first given by Lu Hao. This class of matrices provide bases of matrix splittings for iterative methods. The remarkable feature of iterative methods based on the new class of matrices is that the methods are easily implemented for parallel computation. In particular, a generalization of the accelerated overrelaxation method （GAOR） is introduced. Some theories of the AOR method are extended to the generalized method to include a wide class of matrices. The convergence of the new method is derived for Hermitian positive definite matrices. Finally, some examples are given in order to show the superiority of the new method.

关键词： stair matrices iterative method parallel computation generalization of the AOR method

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长方矩阵加权群逆的迭代法

高等学校计算数学学报 2012年第1期34卷 16-22页

作者：胡春梅刘晓冀丽江师范高等专科学校数理系广西民族大学数学与计算机科学学院

1引言与引理最近,文[1]定义了长方矩阵的一种加权群逆:设A∈C;,W∈C;.称满足下列矩阵方程组的矩阵X∈C;为A的加W权群逆:（W1）AWXWA=A,（W2）XWAWX=X,（W3）AWX=XWA通常记A的加W权群逆为A;.若A;存在,则它是唯一的.加权群逆A;不同于Cline... 详细信息

1引言与引理最近,文[1]定义了长方矩阵的一种加权群逆:设A∈C;,W∈C;.称满足下列矩阵方程组的矩阵X∈C;为A的加W权群逆:（W1）AWXWA=A,（W2）XWAWX=X,（W3）AWX=XWA通常记A的加W权群逆为A;.若A;存在,则它是唯一的.加权群逆A;不同于Cline与Greville在文[2]提出的长方矩阵的加权Drazin逆.文

关键词： rectangular matrix weighted group inverse iterative method

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超线性收敛的指数下降迭代法

高等学校计算数学学报 2000年第1期22卷 41-46页

作者：吴新元吴忠麟南京大学数学系!南京210093

This paper develops an iterative method of exponential descent with superlinear linear convegence x n+1 =x n exp (-f n(x n-x n-1 )/x n(f n-f n-1 )), x n≠0 for solving nonlinear equations. It becomes the well known secant method when we take linear approximation of the exponential function in the iterative method. Numerical tests show that exponential descent method has the advantage of secant method.

关键词： iterative method nonlinear equation numerical analysis.

线性矩阵方程组异类约束最小二乘解的迭代算法

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高等学校计算数学学报 2013年第3期35卷 250-262页

作者：刘晓敏张凯院西北工业大学应用数学系

1引言用R;表示m×n实矩阵集合,A（?）B表示矩阵A与B的Kronecker积（直积）,vec（A）表示将矩阵A按行拉直构成的列向量.定义m×n实矩阵A与B的内积为[A,B]=tr（A;B）,由此导出矩阵的Frobenius范数||A||=[A,A]/1/2.设P∈r（n×n）满足P;=P... 详细信息

1引言用R;表示m×n实矩阵集合,A（?）B表示矩阵A与B的Kronecker积（直积）,vec（A）表示将矩阵A按行拉直构成的列向量.定义m×n实矩阵A与B的内积为[A,B]=tr（A;B）,由此导出矩阵的Frobenius范数||A||=[A,A]/1/2.设P∈r（n×n）满足P;=P和p;=I;.若X∈R（n×n）满足PXPX=X,则称X为关于矩阵P的自反矩阵,记关于矩阵P的自反矩阵的集合为Ω;（P）;用Ω;表示R（n×n）,Ω;表示n阶实对称矩阵集合

关键词： linear matrix equations different constrained least square solution iterative method least-norm solution optimal approximation

A Family of Fifth-order iterative methods for Solving Nonlinear Equations

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Communications in Mathematical Research 2013年第3期29卷 255-260页

作者： Liu Tian-Bao Cai Hua Li Yong School of Mathematics Jilin University Fundamental Department Aviation University of Air Force

In this paper, we present and analyze a family of fifth-order iterative methods free from second derivative for solving nonlinear equations. It is established that the family of iterative methods has convergence order five. Numerical examples show that the new methods are comparable with the well known existing methods and give better results in many aspects.

关键词： Newton's method iterative method nonlinear equation order of convergence

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