Constrained Ulam Dynamic Mode Decomposition: Approximation of the Perron-Frobenius Operator for Deterministic and Stochastic Systems
Constrained Ulam Dynamic Mode Decomposition: Approximation of the Perron-Frobenius Operator for Deterministic and Stochastic Systems
复制标题
约束 Ulam 动态模式分解：确定性和随机系统的 Perron-Frobenius 算子的近似
作者：
Debdipta Goswami;Emma Thackray;D. Paley
由常微分方程和随机微分方程描述的动力系统可以通过 Perron-Frobenius (PF) 和 Koopman 传递算子的特征分解进行分析。虽然 Koopman 算子可以通过数据驱动技术（例如扩展动态模式分解 (EDMD)）来近似，但 PF 算子的近似使用 Ulam 方法中的单通道蒙特卡罗方法，这需要足够长的时间步长。这封信提出了一种 PF 算子的有限维近似技术，该技术使用多通道蒙特卡罗数据提出并求解类似 EDMD 的约束最小二乘问题，以在有限维的基础上逼近 PF 算子。用于投影 PF 算子的基函数是状态空间分区的特征函数。使用确定性和时间均匀随机系统对结果进行理论分析和说明。
Dynamical systems described by ordinary and stochastic differential equations can be analyzed through the eigen-decomposition of the Perron–Frobenius (PF) and Koopman transfer operators. While the Koopman operator may be approximated by data-driven techniques, e.g., extended dynamic mode decomposition (EDMD), the approximation of the PF operator uses a single-pass Monte Carlo approach in Ulam’s method, which requires a sufficiently long time step. This letter proposes a finite-dimensional approximation technique for the PF operator that uses multi-pass Monte Carlo data to pose and solve a constrained EDMD-like least-squares problem to approximate the PF operator on a finite-dimensional basis. The basis functions used to project the PF operator are the characteristic functions of the state-space partitions. The results are analyzed theoretically and illustrated using deterministic and time-homogeneous stochastic systems.