This paper proposes an efficient general alternating-direction implicit (GADI) framework for solving large sparse linear systems. The convergence property of the GADI framework is discussed. Most of the existing ADI methods can be viewed as particular schemes of the developed framework. Meanwhile the GADI framework can derive new ADI methods. Moreover, as the algorithm efficiency is sensitive to the splitting parameters, we offer a data-driven approach, the Gaussian process regression (GPR) method based on the Bayesian inference, to predict the GADI framework's relatively optimal parameters. The GPR method requires a small training data set to learn the regression prediction mapping, which has sufficient accuracy and high generalization capability. It allows us to efficiently solve linear systems with a one-shot computation, and does not require any repeated computations to obtain relatively optimal splitting parameters. Finally, we use the three-dimensional convection-diffusion equation and continuous Sylvester matrix equation to examine the performance of our proposed methods. Numerical results demonstrate that the proposed framework is faster tens to thousands of times than the existing ADI methods, such as (inexact) Hermitian and skew-Hermitian splitting type methods in which the consumption of obtaining relatively optimal splitting parameters is ignored. Due to the efficiency of the developed methods, we can solve much larger linear systems which these existing ADI methods have been not reached.
An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations
School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）；School of Mathematical Sciences, Queensland University of Technology（昆士兰科技大学理学院）
Alternating directions implicit higher-order finite element method for simulations of time-dependent electromagnetic wave propagation in non-regular biological tissues
我们专注于非平稳麦克斯韦方程组定义在一个定期补丁的元素考虑在等几何分析（IGA）。我们采用时间积分方案的思想发展的有限差分社区[M。Hochbruck，T.扬克河，巴西-地Schnaubelt，Convergence of an ADI splitting for Maxwell's equations，Numerishe Mathematik，2015]，以导出导致Kronecker乘积矩阵离散化的弱公式。我们采取的张量积结构的计算补丁的元素从IGA框架作为一个优势，允许在每个时间步长的线性计算成本因式分解。我们设计我们的求解器，以模拟电磁波在不规则的生物组织中的传播。我们表明，交替方向求解器的线性成本被保存时，我们任意改变材料的数据系数在整个计算域。我们验证求解器使用制造的解决方案和电磁波在人的头部的传播问题。
