開發以樹狀結構動態生成多重碎型場之演算法

Abstract

地球物理模型面臨著「尺度空缺」（scaling gap）問題，也就是計算能力的限制使其無法解析觀測極端事件時所必需的小尺度過程。統計降尺度（Statistical downscaling）為此提供了有效的解決方案，而離散乘法級聯（Discrete Multiplicative Cascades, DMCs）是此領域的常用工具。然而，DMCs 具有非平穩性（non-stationarity）的缺點，會產生人為的、與網格對齊的偽影。「鈍級聯」（Blunt Cascade）擴展雖能透過平滑化場來解決此問題，但其高昂的計算成本限制了它的實際應用。

本文介紹了一種名為「LazyTree」的演算法，這是一種用於生成動態多重碎形場的記憶體高效方法。該演算法採用惰性求值（lazy evaluation）策略，透過切片（slices）生成與處理場，從而大幅減少記憶體使用與計算時間。此方法能夠生成傳統方法難以處理的高解析度、多維度場。論文詳細說明了該演算法在標準 DMCs 及計算要求更高的鈍級聯擴展中的實作方式，並提出一種與此節省記憶體方法相容的即時多重碎形分析（Double Trace Moment）方案。

在 GPU 上進行的驗證實驗表明，對於一個三維時空級聯，LazyTree 與傳統方法相比，達成了近 1000 倍的加速，且記憶體使用量顯著減少。由 LazyTree 生成的場（包含經典級聯與鈍級聯）其統計特性也證實與普適多重碎形（UM）理論一致。透過克服過去的計算障礙，LazyTree 演算法使鈍級聯擴展得以實際應用於高要求的模擬中，例如模擬湍流風場中的雨滴軌跡，為多重碎形模型開闢了新的研究途徑。

Geophysical models face a "scaling gap" where computational limitations prevent the resolution of fine-scale processes, which are critical for observing extreme events . Statistical downscaling offers a solution, and Discrete Multiplicative Cascades (DMCs) are a common tool for this purpose . However, DMCs suffer from non-stationarity, which introduces artificial, grid-aligned patterns . The "Blunt Cascade" extension was proposed to resolve this by smoothing the field, but this comes at a significant computational cost that limits its practical use.

This paper introduces the "LazyTree" algorithm, a memory-efficient method for generating dynamic multifractal fields. The algorithm uses a lazy evaluation strategy, generating and processing the field in slices, which dramatically reduces memory usage and computational time. This approach allows for the generation of high-resolution, multi-dimensional fields that would be intractable with conventional methods. The paper details the algorithm's implementation for both standard DMCs and the more demanding Blunt Extension, proposing an on-the-fly adaptation for multifractal analysis (Double Trace Moment) that is compatible with the memory-saving approach.

Validation experiments, conducted on a GPU, demonstrate that LazyTree achieves nearly 1000-fold speed-up and uses significantly less memory compared to traditional methods for a 3D space-time cascade. The statistical properties of fields generated by LazyTree, for both classic and blunt cascades, are shown to be consistent with Universal Multifractal (UM) theory. By overcoming previous computational barriers, the LazyTree algorithm enables the practical application of the Blunt Extension to demanding simulations, such as modeling raindrop trajectories in turbulent wind fields, opening new avenues for multifractal modeling.