层次化分数步近似与并行动力学蒙特卡洛算法
Hierarchical fractional-step approximations and parallel kinetic Monte Carlo algorithms
AI总结：
提出一种基于马尔可夫算子空间分解和Trotter定理的数学框架，用于构建和分析晶格动力学蒙特卡洛（KMC）的并行算法，实现受控误差近似并保证数值与统计一致性。
摘要：
我们提出了一个数学框架，用于构建和分析晶格动力学蒙特卡洛（KMC）模拟的并行算法。所得算法能够模拟具有复杂化学和传输微观机制的空间分布非平衡物理化学过程中的广泛时空尺度。这些算法可以针对特定的层次化并行架构（如多核处理器或图形处理单元（GPU）集群）进行定制。所提出的并行算法是动力学蒙特卡洛算法的受控误差近似，偏离了创建与串行算法具有完全相同主方程的并行KMC算法的主流范式。我们的方法依赖于将KMC算法背后的马尔可夫算子空间分解为与并行架构中处理器结构相对应的算子层次结构。基于这种算子分解，我们利用Trotter定理及其随机变体制定了分数步近似方案；这些方案（a）确定处理器之间的通信调度，以及（b）在每个分数步时间窗口上通过串行KMC模拟（称为内核）在每个处理器上独立运行。此外，所提出的数学框架使我们能够严格证明所提出算法的数值和统计一致性，表明我们的近似方案收敛到原始串行KMC。该方法还提供了对不同处理器通信调度的系统评估。
英文摘要：
We present a mathematical framework for constructing and analyzing parallel algorithms for lattice Kinetic Monte Carlo (KMC) simulations. The resulting algorithms have the capacity to simulate a wide range of spatio-temporal scales in spatially distributed, non-equilibrium physiochemical processes with complex chemistry and transport micro-mechanisms. The algorithms can be tailored to specific hierarchical parallel architectures such as multi-core processors or clusters of Graphical Processing Units (GPUs). The proposed parallel algorithms are controlled-error approximations of kinetic Monte Carlo algorithms, departing from the predominant paradigm of creating parallel KMC algorithms with exactly the same master equation as the serial one. Our methodology relies on a spatial decomposition of the Markov operator underlying the KMC algorithm into a hierarchy of operators corresponding to the processors' structure in the parallel architecture. Based on this operator decomposition, we formulate Fractional Step Approximation schemes by employing the Trotter Theorem and its random variants; these schemes, (a) determine the communication schedule} between processors, and (b) are run independently on each processor through a serial KMC simulation, called a kernel, on each fractional step time-window. Furthermore, the proposed mathematical framework allows us to rigorously justify the numerical and statistical consistency of the proposed algorithms, showing the convergence of our approximating schemes to the original serial KMC. The approach also provides a systematic evaluation of different processor communicating schedules.
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