The block Kaczmarz method is an iterative algorithm for solving large-scale linear sys-tems.At each iteration,the current iteration points will be orthogonally projected to the solution space of the constraint subset.In the paper,according to the principle of the block Kaczmarz method,we propose the randomized block Kaczmarz method with the Gaussian mixture model(GMM-RBK(k)).The partition of blocks is carried out by the Gaussian mixture model.In order to avoid the computation of pseudo-inverse or the solution of the least-squares problem,we propose the randomized average block Kaczmarz method with the Gaussian mixture model(GMM-RABK(k)).We prove that the two algorithms converge when the linear system is consistent,and give the corresponding formula of convergence rate.The effectiveness of the GMM-RBK(k)method and GMM-RABK(k)method is also verified in the final numerical experiments,and avoiding the calculation of pseudo-inverse,the GMM-RABK(k)method is superior to the GMM-RBK(k)method.
In this paper,based on the active set identification,we propose a Proximal Newton method for l1 problems.One advantage of this method is that it uses the good support identification property of the ISTA algorithm to determine the free variables and active set variables,and another advantage is that it uses the information of the partial Hessian matrix to update the free variables.Under appropriate conditions,we demonstrate the global convergence of the algorithm with a nonmonotonic line search strategy.The results of numerical experiments show that our proposed algorithm is effective.
Gegenbauer's addition theorem is an important theory of Bessel functions.This paper studies the truncation error of Gegenbauer's addition theorem.Based on the limiting forms of Bessel and Neumann functions as the argument tend to zero,an explicit bound of the truncation error and its convergence order are derived.Numerical experiments show that the estimates are valid and sharp.
The two-subspace projection algorithm and the multi-step inertial randomized Kaczmarz algorithm are effective methods for solving the coherent linear system.By adding the soft shrinkage in the two-subspace projection algorithm and the multi-step inertial randomized Kaczmarz algorithm,this paper introduces the sparse two-subspace projection algorithm and the sparse multi-step inertial randomized Kaczmarz algorithm.The linear convergence rate in expectation of the proposed algorithms in noise and noiseless cases is presented.Finally,we give some numerical experiments to illustrate the effectiveness and advantage of our algorithms.
In the paper,for solving large sparse vertical linear complementarity problems,a two-step modulus-based matrix synchronous multisplitting parallel iteration method is construct-ed by using two-step multisplitting technique.The new method can be viewed as a gener-alization of the modulus-based matrix synchronous multisplitting iterative method and the two-step modulus-based matrix splitting iterative method in the existing literatures.Fur-thermore,under the assumption that the system matrices are H+-matrices,the convergence analysis of the proposed method is given,and the convergence domain of the parameter matrix is obtained,which extends the convergence results of the existing methods.Finally,numerical experiments are carried out under the OpenMP framework for two numerical ex-amples in the existing literatures.Numerical results show that the two-step multisplitting technique can improve the computational efficiency of the existing methods.
In this paper,we use the concatenating method to construct a local momentum-preserving algorithm for the Zakharov system based on the idea that the numerical algorithm should keep the intrinsic properties of the original problem as much as possible.The advantage of the proposed algorithm is that it preserves the local momentum conservation law and the local mass conservation law in any time-space regions,i.e.,independence on any boundary conditions.Certainly,if the boundary condition is suitable,such as homogeneous boundary or periodic boundary,the proposed algorithm also preserves the global momentum conser-vation law and the global mass conservation law.Numerical experiments do not only show the second-order convergence,but also verify the stability in long term calculations and the conservation of invariants for the proposed algorithm.
Compressed Sensing(CS)is an effective method for fast Magnetic Resonance Imaging(MRI),which aims to reconstruct MRI from a small amount of under-sampled data space and accelerate data acquisition in MRI.In order to improve the reconstruction accuracy and speed of current MRI,in this paper,we combine the traditional model-driven compressed sensing method and data-driven deep learning method to propose a new algorithm-ADMM-CNN.This proposed algorithm solves a subproblem of the alternating direction multiplier method(ADMM)by convolutional neural network,and then optimizes the penalty parameter and the update rate of Lagrange multiplizer by constructing the loss function and using the gradient descent method to obtain the trained ADMM model,which can effectively recover the MRI signal from the compressed sensing measurement.Compared with the traditional reconstruction algorithms,the proposed algorithm has fewer regularization parameters and lower computational complexity,at the same time,we introduce a trained convolutional neural network directly to greatly simplify the training process and computation,also we use GPU to accelerate the training.Finally,we give some experimental results to show that ADMM-CNN has faster calculation speed and better reconstruction accuracy compared with traditional reconstruction algorithms and other deep learning methods.
One-dimensional unsteady seepage model is the basic model in well-test research,existing solution includes two categories such as analytic solution and numerical solution.Analytic solution is depend on Laplace transform and numerical Inversion,and Numerical solution is depend on mesh generation and equation discretization.Iterative solution of non-linear equation is present which is direct solution for One-dimensional unsteady seepage equation.Based on steady state sequential replacement method and mass conservation equation,the unsteady seepage equation is solved by transformed as steady seepage equation,the non-linear equation of investigation radius under different outer boundary were build,well-bore pressure and Investigation radius at each moment were obtained by Newton method.The error between iterative solution and analytic solution is small,and bigger error appeared in early stage.Compared with analytic solution and numerical solution,the iterative solution can be spread used in other One-dimensional unsteady seepage problems because of its simple algorithm,acceptable accuracy and fewer time-consuming.
基于贪婪准则和最大距离准则选择系数矩阵工作列的策略,提出两种求解大规模超定不相容线性系统的斜方向的Gauss-Seidel方法,即斜方向的贪婪随机Gauss-Seidel(GRGSO)方法和斜方向的快速最大距离Gauss-Seidel(FMDGSO)方法.当系数矩阵是列满秩时,理论表明这些方法收敛到线性系统的唯一的最小二乘解.特别是当矩阵A的列接近线性相关时,数值结果表明这些方法在求解性能方面比传统的Gauss-Seidel型方法更具优势.
贝叶斯推断正逐渐成为解决反问题的一个越来越流行的工具,这主要是因为它能够描述所获得的解的不确定性.在许多实际的贝叶斯反问题中,可能有多个潜在性能良好的模型可用来描述数据和感兴趣的参数.在这种情况下,基于贝叶斯的模型选择方法经常被用来选择"最佳"模型,而这种方法依赖于对后验分布中归一化常数的估计.因此,估计归一化常数成为贝叶斯推断中的一项重要任务,而这个问题对于标准抽样方法来说具有计算上的挑战性.在这项工作中,新提出的一种基于多正则蒙特卡洛技术(MMC)的归一化常数估计方法,可以很好的解决上述难题,这是一种自适应的重要性采样方法.该方法可以以黑箱方式估计归一化常数,使其特别适用于具有复杂基础模型的问题.最后,通过数值例子证明了所提出的方法可以有效且准确地计算出归一化常数的估计值.
本文通过引入一种新的增广变换,发展了改进的偏牛顿校正算法,建立并证明了一类四阶非线性偏微分方程边值问题的新解与该问题零核空间的密切关系,去掉了标准收敛假设,使证明更简洁明了.分情况验证了该方程满足在Nehari子流形上全局分离定理的条件,该分离定理为本文算法成功找到新解提供理论保障.提出了二维非线性四阶偏微分方程Dirichlet边值问题的插值投影Legendre-Galerkin谱方法,通过构造插值算子和投影算子,对线性算子以及非线性项的处理进行了优化,得到原问题的代数方程,通过验证,其与经典谱方法具有相同的条件数并都达到谱精度.实验结果表明,此方法与经典的谱方法或拟谱方法具有相同的收敛阶,但计算所需CPU时间更少,且能计算出更多的解.
本文考虑一个次扩散方程的反问题,即根据某个时刻解在空间区域上的积分反向优化方程中的分数阶参数.本文说明了当初值为0时,以上问题可能有多个解.基于时空有限元法,我们提出了一个参数优化算法,并利用离散Laplace变换技巧证明了该算法的全局收敛性.另外针对球形区域,利用-△算子的谱分解,我们还提出了一个快速算法.最后两个数值算例验证了算法的有效性.
描述芯片或电力系统运行规律的常用数学模型是高维微分代数方程组,其中的微分方程组太大,诸如线性多步法和Runge-Kutta(RK)法等经典数值方法均不能有效求解.为求解这些微分方程组,学者们提出了波形松弛(WR)方法.多数情况下,这些微分方程组是刚性的,求解他们需要稳定性好的隐式方法,尤其需要A-稳定的方法.此外,RK方法是使用最广泛的常微分方程的数值方法.然而,迄今为止尚未发现RK型WR方法A-稳定的研究.本文研究了 RK型WR方法的A-稳定性,获得了方法A-稳定的充分条件.常见A-稳定的RK方法有Gauss-Legendre方法、Radau ⅠA 方法、Radau ⅡA 方法、Lobatto ⅢA 方法、Lobatto ⅢB 方法、Lobatto ⅢC方法,而且并非RK方法A-稳定,相应的RK型WR方法也A-稳定.在一个假设下,本文所得结果说明当选择低阶Radau ⅠA方法,或Radau ⅡA方法,或Lobatto ⅢC方法为底层方法时,存在分裂方式使得RK型WR方法是A-稳定的.
近年来,有关玻色-爱因斯坦凝聚态基态解的实验研究已经取得了一系列重要的成果.本文在相关研究成果的基础上,首先通过降维和无量纲化方法将玻色-爱因斯坦凝聚态基态解问题转换成能量泛函极值问题,在离散该方程时,我们使用有限差分法和傅里叶谱方法分别离散该能量泛函方程的一维和二维情形.其次,本文采用了一种高效的数值优化算法-SQP优化方法来求解玻色-爱因斯坦凝聚态基态解问题并运用该算法分别对一维和二维的能量泛函极小值问题进行数值模拟.最后通过分析实验数据结果和图像,得出该算法能提高能量函数值的精确性.
针对二维椭圆型界面问题的离散化方程,应用外推插值技巧和样条插值方法在细网格层上构造合适的迭代初始值,加快V型多重网格法求解离散化系统的速度,设计了外推完全多重网格(EXFMG)法.数值实验表明新算法有效降低了迭代次数,计算量更少.
本文在新的距离的框架下考虑p-Laplace方程的自适应有限元方法.在文献[13]中作者考虑了 2≤ p＜∞的情况,给出了具有上界的后验误差估计.本文在此基础上发展了 1＜p＜∞时具有上界和下界的后验误差估计.数值实验验证了理论分析的结果.
大气动力学问题的数值模拟在气象预报等领域具有广泛的应用.相关数值模拟依赖超级计算机平台实现高精度高分辨率的气象预报,隐式求解不受稳定性条件限制,相比显式求解更有优势.面向新的超级计算机架构特征研究隐式大气动力学问题中一系列算子操作的并行和优化方法是非常有必要的.本文在规则递推关系的理论框架下对大气动力学问题预条件阶段的稀疏三角回代求解以及ILU矩阵分解操作的特征进行了总结,并结合申威26010Pro处理器的架构特点,对现有结构化稀疏三角线性方程组问题的并行算法进行了推广,设计了一套面向单向规则递推关系的算法框架,解决了预条件阶段各类算子的并行加速问题.本文还面向申威26010Pro处理器对大气动力学问题的模板计算等算子进行了移植和优化.实验结果显示,本文的算法框架对预条件阶段的算子能够实现26-33倍不等的加速效果,对模板计算等算子的优化相比串行计算有10-152倍的加速比.在新的神威超级计算机上最大测试到1700多万核心,浮点性能达到20.5PFlop/s.在大规模测试条件下的强(弱)可扩展性维持在56.81％(41.87％)以上.
带有分数阶Laplacian算子的对流扩散方程常被用来刻画自然界与社会系统中的反常扩散现象.本文提出了一种新的格子Boltzmann模型,用于求解二维带分数阶Laplacian算子的对流扩散方程.首先,基于分数阶Laplacian算子的Fourier变换和Gauss型求积公式,得到控制方程的近似方程.然后,将速度空间、时间和空间进行离散,并构造合适的平衡态分布函数和离散作用力,建立有效的格子Boltzmann-BGK模型.通过Chapman-Enskog分析,可由建立的格子Boltzmann-BGK模型恢复出宏观方程,从而证明了模型的有效性.最后,将模型应用于求解带有解析解的数值算例和Allen-Cahn方程,数值结果进一步验证了模型的正确性和有效性.
We propose an affine-mapping based variational Ensemble Kalman filter for sequential Bayesian filtering problems with generic observation models. Specifically, the proposed method is formulated as to construct an affine mapping from the prior ensemble to the posterior one, and the affine mapping is computed via a variational Bayesian formulation, i.e., by minimizing the Kullback-Leibler divergence between the transformed distribution through the affine mapping and the actual posterior. Some theoretical properties of resulting optimization problem are studied and a gradient descent scheme is proposed to solve the resulting optimization problem. With numerical examples we demonstrate that the method has competitive performance against existing methods.
广义Dantzig选择器问题是解决参数估计的有效途径,其中任何范数都可以用于估计.本文采用对偶交替方向乘子法(dual Alternating Direction Method of Multipliers,简称 dADMM)求解?1范数,?2范数和?∞范数广义Dantzig选择器问题,并给出了 dADMM的全局收敛性和局部线性收敛速度.数值试验验证了 dADMM的有效性.
