- Answer of the Probabilistic Lambert Drawback: Connections with Optimum Mass Transport, Schrödinger Bridge and Response-Diffusion PDEs(arXiv)
Writer : Alexis M. H. Teter, Iman Nodozi, Abhishek Halder
Summary : Lambert’s drawback considerations with transferring a spacecraft from a given preliminary to a given terminal place inside prescribed flight time by way of velocity management topic to a gravitational power discipline. We contemplate a probabilistic variant of the Lambert drawback the place the data of the endpoint constraints in place vectors are changed by the data of their respective joint chance density capabilities. We present that the Lambert drawback with endpoint joint chance density constraints is a generalized optimum mass transport (OMT) drawback, thereby connecting this classical astrodynamics drawback with a burgeoning space of analysis in trendy stochastic management and stochastic machine studying. This newfound connection permits us to scrupulously set up the existence and uniqueness of resolution for the probabilistic Lambert drawback. The identical connection additionally helps to numerically remedy the probabilistic Lambert drawback by way of diffusion regularization, i.e., by leveraging additional connection of the OMT with the Schrödinger bridge drawback (SBP). This additionally exhibits that the probabilistic Lambert drawback with additive dynamic course of noise is actually a generalized SBP, and might be solved numerically utilizing the so-called Schrödinger components, as we do on this work. We clarify how the ensuing evaluation results in fixing a boundary-coupled system of reaction-diffusion PDEs the place the nonlinear gravitational potential seems because the response price. We suggest novel algorithms for a similar, and current illustrative numerical outcomes. Our evaluation and the algorithmic framework are nonparametric, i.e., we make neither statistical (e.g., Gaussian, first few moments, combination or exponential household, finite dimensionality of the adequate statistic) nor dynamical (e.g., Taylor collection) approximations.
2.On the Martingale Schrödinger Bridge between Two Distributions (arXiv)
Writer : Marcel Nutz, Johannes Wiesel
Summary : We examine a martingale Schrödinger bridge drawback: given two chance distributions, discover their martingale coupling with minimal relative entropy. Our most important consequence supplies Schrödinger potentials for this coupling. Particularly, beneath sure situations, the log-density of the optimum coupling is given by a triplet of actual capabilities representing the marginal and martingale constraints. The potentials are additionally described as the answer of a twin drawback
