流形上随机微分方程的中点投影算法
A midpoint projection algorithm for stochastic differential equations on manifolds
摘要：
投影到流形上的随机微分方程存在于物理、化学、生物、工程、纳米技术和优化中，具有跨学科的应用。流形上的内禀坐标随机方程在计算上往往是不切实际的，而数值投影在许多情况下是有用的。我们表明，Stratonovich解释的随机分析得到绝热消除与约束潜力。为了比较的目的，我们推导了球面和双曲面的内在随机方程，并回顾了一些早期的投影算法。本文提出了一种组合中点投影算法，该算法利用中点投影到切空间上，并结合随后的法向投影来满足约束条件。数值例子给出了一系列的流形，包括圆形，椭球形，双曲面，悬链线的情况下，以及高阶多项式的约束和一个十维超球。我们表明，在所有情况下，与使用组合欧拉投影方法或纯切向投影的方法相比，组合中点法则大大减少了误差。我们的技术可以处理多个约束。这允许流形包含几个守恒量。该算法准确、简单、高效。与其他方法相比，通常发现扩散距离的数量级误差减少，约束误差减少几个数量级。
英文摘要：
Stochastic differential equations projected onto manifolds occur in physics, chemistry, biology, engineering, nanotechnology and optimization, with interdisciplinary applications. Intrinsic coordinate stochastic equations on the manifold are often computationally impractical, and numerical projections are useful in many cases. We show that the Stratonovich interpretation of the stochastic calculus is obtained using adiabatic elimination with a constraint potential. We derive intrinsic stochastic equations for spheroidal and hyperboloidal surfaces for comparison purposes, and review some earlier projection algorithms. In this paper, a combined midpoint projection algorithm is proposed that uses a midpoint projection onto a tangent space, combined with a subsequent normal projection to satisfy the constraints. Numerical examples are given for a range of manifolds, including circular, spheroidal, hyperboloidal, and catenoidal cases, as well as higher-order polynomial constraints and a ten-dimensional hypersphere. We show that in all cases the combined midpoint method has greatly reduced errors compared to methods using a combined Euler projection approach or purely tangential projection. Our technique can handle multiple constraints. This allows manifolds that embody several conserved quantities. The algorithm is accurate, simple and efficient. An order of magnitude error reduction in diffusion distance is typically found compared to the other methods, with reductions of several orders of magnitude in constraint errors.
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