我们评估了时间自适应高阶分裂方法在求解具有周期边界条件的非线性抛物型问题（系统）中的适用性与效率。我们特别讨论了几种生成复杂图案并显示非光滑解动力学的应用。首先，我们为抛物型问题在周期边界条件下的分裂方法提供了一般误差分析，并推导了精确解所需的平滑性要求，特别是针对Gray-Scott方程和Van der Pol方程。数值实验展示了方法的收敛性，并用于比较不同时间自适应分裂方案以及基于适当构造的后验局部误差估计器的分裂方法（分为两个或三个算子）的效率。
英文摘要：
We assess the applicability and efficiency of time-adaptive high-order splitting methods applied for the numerical solution of (systems of) nonlinear parabolic problems under periodic boundary conditions. We discuss in particular several applications generating intricate patterns and displaying nonsmooth solution dynamics. First we give a general error analysis for splitting methods for parabolic problems under periodic boundary conditions and derive the necessary smoothness requirements on the exact solution in particular for the Gray-Scott equation and the Van der Pol equation. Numerical examples demonstrate the convergence of the methods and serve to compare the efficiency of different time-adaptive splitting schemes and of splitting into either two or three operators, based on appropriately constructed a posteriori local error estimators.
Practical splitting methods for the adaptive integration of nonlinear evolution equations. Part I: Construction of optimized schemes and pairs of schemes
Technische Universität Wien（维也纳工业大学）；King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）；Universität Wien（维也纳大学）
University of California, Irvine（加利福尼亚大学尔湾分校）；Department of Mathematics（数学系）
We propose an adaptive projection-gradient descent-shrinkage-splitting method (APGDSSM) to integrate penalty based channel pruning into quantization-aware training (QAT). APGDSSM concurrently searches weights in both the quantized subspace and the sparse subspace. APGDSSM uses shrinkage operator and a splitting technique to create sparse weights, as well as the Group Lasso penalty to push the weight sparsity into channel sparsity. In addition, we propose a novel complementary transformed l1 penalty to stabilize the training for extreme compression.
We propose a new splitting method for strong numerical solution of the Cox-Ingersoll-Ross model. For this method, applied over both deterministic and adaptive random meshes, we prove a uniform moment bound and strong error results of order $1/4$ in $L_1$ and $L_2$ for the parameter regime $κθ>σ^2$. We then extend the new method to cover all parameter values by introducing a \emph{soft zero} region (where the deterministic flow determines the approximation) giving a hybrid type method to deal with the reflecting boundary. From numerical simulations we observe a rate of order $1$ when $κθ>σ^2$ r…
Arbitrary high order splitting methods for linear Schr{ö}dinger equations with non-trivial compatibility conditions
Nantes Université, CNRS, Laboratoire de Mathématiques Jean Leray, LMJL（南特大学、法国国家科学研究中心、让·勒雷数学实验室）；Université de Genève, Section de mathématiques（日内瓦大学数学系）
Temporal Error Growth of Strang Splitting Method for the Periodic Cubic NLS
School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）；School of Mathematical Sciences, Nanjing Normal University（南京师范大学数学科学学院）
An adaptive finite element PML method for the elastic wave scattering problem in periodic structures
Beijing University of Posts and Telecommunications（北京邮电大学）；Purdue University（普渡大学）；Jilin University（吉林大学）；Academy of Mathematics and System Sciences, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
Stochastic nonlinear dynamical systems can undergo rapid transitions relative to the change in their forcing, for example due to the occurrence of multiple equilibrium solutions for a specific interval of parameters. In this paper, we modify one of the methods developed to compute probabilities of such transitions, Trajectory-Adaptive Multilevel Sampling (TAMS), to be able to apply it to high-dimensional systems. The key innovation is a projected time-stepping approach, which leads to a strong reduction in computational costs, in particular memory usage. The performance of this new implementat…
An Adaptive Fast Algorithm for Periodic Coulomb Lattice Sums in Arbitrary Unit Cells
Center for Computational Mathematics, Flatiron Institute（计算数学中心，平顿研究所）；Simons Foundation（西蒙斯基金会）；Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）
A splitting approach for the Kadomtsev--Petviashvili equation
我们认为分裂方法的Kadomtsev-Petviashvili方程的周期性边界条件，并表明，必要的插值过程可以有效地实现。将该数值方案产生的误差与Klein和Roidot（SIAM J. Sci.计算：2011）对Kadomtsev-Petviashvili方程的刚性解表现最好。由于许多经典的高阶分裂方法不能很好地执行，我们提出了一个稳定的外推方法，以构建一个有效的四阶数值格式。此外，守恒性质和可能的降低的考虑中的数值格式的某些初始值的订单。
Linearized Alternating Direction Method with Parallel Splitting and Adaptive Penalty for Separable Convex Programs in Machine Learning
Key Lab. of Machine Perception (MOE), School of EECS, Peking University（机器感知重点实验室（教委）、电子工程学院、北京大学）；School of Software and Microelectronics, Peking University（软件与微电子学院、北京大学）
