Gradient Flow Algorithms for Density Propagation in Stochastic Systems
Gradient Flow Algorithms for Density Propagation in Stochastic Systems
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DOI：
10.1109/tac.2019.2951348
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发表时间：
2020-10-01
影响因子：
6.8
通讯作者：
Halder, Abhishek
中科院分区：
文献类型：
--
作者：
Caluya, Kenneth F.;Halder, Abhishek
我们发展了一种新的计算框架来求解控制连续时间随机非线性系统中联合概率密度函数(PDF)流动的偏微分方程组(PDE)。在不确定性传播、非线性滤波和随机控制中，需要计算服从先验动态的暂态联合PDF。我们的方法脱离了传统的空间离散化或函数近似的方法--这两种方法通常都受到“维度诅咒”的影响。在提出的框架中，我们对时间进行离散化，而不是状态空间离散化。我们在联合PDF流形上求解无限维近端递推，在小的时间步长限制下，理论上等价于求解底层的传输偏微分方程组。由此得到的计算结果对某些自由能泛函相对于瓦瑟斯坦度规的梯度流有几何解释，这些梯度流是由最佳质量传输理论产生的。我们证明了对偶化和熵正则化导致了保锥不动点递推，证明了它在Thompson度量下是压缩的。提出了一种块坐标迭代格式来求解所得到的非线性递推，并保证了收敛。这种方法使得非参数暂态联合PDF传播的计算速度非常快。文中给出了数值算例和各种扩展，以说明该方法的范围和有效性。
We develop a new computational framework to solve the partial differential equations (PDEs) governing the flow of the joint probability density functions (PDFs) in continuous-time stochastic nonlinear systems. The need for computing the transient joint PDFs subject to prior dynamics arises in uncertainty propagation, nonlinear filtering, and stochastic control. Our methodology breaks away from the traditional approach of spatial discretization or function approximation-both of which, in general, suffer from the "curse-of-dimensionality." In the proposed framework, we discretize time but not the state space. We solve infinite dimensional proximal recursions in the manifold of joint PDFs, which in the small time-step limit, is theoretically equivalent to solving the underlying transport PDEs. The resulting computation has the geometric interpretation of gradient flow of certain free energy functional with respect to the Wasserstein metric arising from the theory of optimal mass transport. We show that dualization along with an entropic regularization, leads to a cone-preserving fixed point recursion that is proved to be contractive in Thompson metric. A block co-ordinate iteration scheme is proposed to solve the resulting nonlinear recursions with guaranteed convergence. This approach enables remarkably fast computation for nonparametric transient joint PDF propagation. Numerical examples and various extensions are provided to illustrate the scope and efficacy of the proposed approach.