Density tracking by quadrature (DTQ) is a numerical procedure for computing solutions to Fokker-Planck equations that describe probability densities for stochastic differential equations (SDEs). In this paper, we extend upon existing tensorized DTQ procedures by utilizing a flexible quadrature rule that allows for unstructured, adaptive meshes. We propose and describe the procedure for $N$-dimensions, and demonstrate that the resulting adaptive procedure is significantly more efficient than a tensorized approach. Although we consider two-dimensional examples, all our computational procedures are extendable to higher dimensional problems.
翻译:以等离子体(DTQ)跟踪密度是一个计算Fokker-Planck方程式解决方案的数字程序,它描述了随机差分方程式(SDEs)的概率密度。 在本文中,我们通过使用允许无结构、适应性地 meshes 的灵活等离子体规则,扩展了现有的有弹性的DTQ 程序。我们提出并描述以美元计值的平方程式,并表明由此产生的适应程序比分解法效率要高得多。 尽管我们考虑了二维实例,但我们所有的计算程序都可以扩大到更高维度的问题。