避免对大过渡率矩阵的矩阵指数
Pedro Pessoa1,2, Max Schweiger1,2, Steve Pressé1,2,3
1Center for Biological Physics, Arizona State University, Tempe, Arizona 85287, USA.
这项研究引入了用于矩阵指数的新方法,这在科学计算中至关重要. 一种新的马尔科夫跳跃过程 (MJP) 方法和克里洛夫子空间方法为速率矩阵提供了改进的计算缩放.
科学领域:
- 计算数学 计算数学 计算数学
- 科学计算科学计算
- 生物物理学的生物物理.
背景情况:
- 矩阵指数化是计算密集的 (O(N^3) 时间,O(N^2) 内存).
- 速率矩阵在科学应用中经常被使用,特别是在模拟动态过程中.
- 现有的方法面临着计算成本和数值精度的挑战.
研究的目的:
- 探索和比较五种不同的矩阵指数化方法,重点关注速率矩阵.
- 识别用于矩阵指数化的新,计算高效的方法.
- 分析这些方法对下游计算任务 (如蒙特卡洛采样) 的影响.
主要方法:
- 在矩阵指数和马尔科夫跳跃过程 (MJPs) 之间利用数学类比.
- 通过限制"轨迹"跳跃,开发了一种基于MJP的新方法.
- 基准Runge-Kutta集成器和Krylov子空间方法用于一般矩阵指数化.
主要成果:
- 确定了一种基于MJP的新方法,具有改进的计算缩放.
- 克里洛夫子空间方法和新的MJP方法实现了O{\displaystyle O} N^2的时间复杂度.
- 在最具竞争力的方法中,实现了将内存需求降低到O(N).
结论:
- 新的基于MJP的方法和克里洛夫子空间方法为指数化速率矩阵提供了显著的计算优势.
- 这些优化的方法减少了计算时间和内存足迹,使得科学模拟更有效.
- 这些发现对改善蒙特卡洛后部样品的混合特性有意义.
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