An inverse-stochastic-control Galerkin estimator for drift potentials
Abstract
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.
Related
- Time optimal control for the heat equation with inverse-square potentials
- Field-Modified Quantum Potentials from Tridiagonal Representations: Analytical Spectra and Galerkin Simulations
- A Novel Matrix-Free Approximate Elemental Inverse for Efficient Preconditioning of Discontinuous Galerkin Spectral Elements
- Trace Inverse Estimates and Kernel-based Nitsche Galerkin Methods for Elliptic Interface Problems
- Stochastic Galerkin Method for Fractional Boundary Value Problems: Convergence Analysis and Numerical Treatment
BibTeX
@misc{nakano2026inversestochasticcontrol,
title = {{An inverse-stochastic-control Galerkin estimator for drift potentials}},
author = {Yumiharu Nakano},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04268},
eprint = {2610.04268},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04268},
}
RIS
TY - UNPB
TI - An inverse-stochastic-control Galerkin estimator for drift potentials
AU - Yumiharu Nakano
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04268
UR - https://arxiv.org/abs/2610.04268
AB - 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.
ER -