一种计算二维偏微分方程拟势的高效喷气推进器
An efficient jet marcher for computing the quasipotential for 2D SDEs
摘要：
本文提出了一种计算二维随机微分方程拟势及其梯度的新算法--高效喷流推进法（EJM）。准势是一个潜在的功能，为非梯度的SDES，给出渐近估计不变的概率测度，预期的逃逸时间从盆地的吸引子，和最大似然逃逸路径。准势是具有各向异性费用函数的最优控制问题的解决方案，其可以通过Dijkstra类标号设置方法数值求解。以前的Dijkstra类准势解算器一般在网格间距中显示一阶精度。然而，通过利用准势的高阶插值以及最小作用路径（MAP）的更精确近似，EJM实现了准势的二阶精度和其梯度的近二阶精度。此外，通过使用有针对性的搜索邻域的最快的特点以下的想法Mirebeau，EJM也享有减少计算时间。这个高度精确的求解器使我们能够计算前因子的WKB近似不变的概率测度和Bouchet-Reygner尖锐估计的预期逃逸时间的迈尔-斯坦时间。我们的代码可以在GitHub上找到。
英文摘要：
We present a new algorithm, the efficient jet marching method (EJM), for computing the quasipotential and its gradient for two-dimensional SDEs. The quasipotential is a potential-like function for nongradient SDEs that gives asymptotic estimates for the invariant probability measure, expected escape times from basins of attractors, and maximum likelihood escape paths. The quasipotential is a solution to an optimal control problem with an anisotropic cost function which can be solved for numerically via Dijkstra-like label-setting methods. Previous Dijkstra-like quasipotential solvers have displayed in general 1st order accuracy in the mesh spacing. However, by utilizing higher order interpolations of the quasipotential as well as more accurate approximations of the minimum action paths (MAPs), EJM achieves second-order accuracy for the quasipotential and nearly second-order for its gradient. Moreover, by using targeted search neighborhoods for the fastest characteristics following the ideas of Mirebeau, EJM also enjoys a reduction in computation time. This highly accurate solver enables us to compute the prefactor for the WKB approximation for the invariant probability measure and the Bouchet-Reygner sharp estimate for the expected escape time for the Maier-Stein SDE. Our codes are available on GitHub.
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