发表机构
Department of Mathematical and Computing Science, School of Computing; Institute of Science Tokyo（计算学院数学与计算科学系; 东京科学大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种基于逆随机控制Galerkin估计器的方法，从离散不规则观测中恢复Langevin扩散的漂移势，通过有限基参数化值函数，避免转移密度和离散化偏差，并在实验中验证了其有效性。
AI 中文摘要
我们考虑从离散的、可能不规则的观测位置中恢复过阻尼Langevin扩散的漂移势这一不适定逆问题。通过将逆随机控制的次优性间隙专门化到漂移势恢复，我们在有限基中参数化相应的值函数。在基非退化条件下，所得的Galerkin估计器由一个正定线性系统获得，其条目是经验时间边际的平滑泛函；无需转移密度。该间隙等于到真实势能的平方能量距离的一半，从而在基变得完整时产生最佳逼近恒等式和能量半范数一致性。在固定基下，我们将样本波动与时间求积误差分离，从而避免了Euler–Maruyama $O(\Delta t)$漂移离散化偏差。我们还将有限基分析为正则化，给出了一个具体基族的依赖于维度的收敛速率，并界定了观测噪声偏差。一维和二维实验验证了预测的采样间隔行为。对不规则动物遥测的应用展示了该方法在单条长轨迹上的效果。
英文摘要
We consider the ill-posed inverse problem of recovering the drift potential of an overdamped Langevin diffusion from discrete, possibly irregular position observations. Specializing the suboptimality gap for inverse stochastic control to drift-potential recovery, we parametrize the associated value function in a finite basis. Under a basis nondegeneracy condition, the resulting Galerkin estimator is obtained from one positive-definite linear system whose entries are smooth functionals of the empirical time marginals; no transition density is required. The gap equals one half of the squared energy distance to the true potential, yielding a best-approximation identity and energy-seminorm consistency as the basis becomes complete. At a fixed basis, we separate the sample fluctuation from the time-quadrature error, thereby avoiding the Euler--Maruyama $O(Δt)$ drift-discretization bias. We also analyze the finite basis as regularization, give a dimension-dependent convergence rate for a concrete basis family, and bound observation-noise bias. One- and two-dimensional experiments verify the predicted sampling-interval behavior. An application to irregular animal telemetry illustrates the method on a single long trajectory.
