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Daniel Sanz-Alonso 博士将于近期访问我院，并作以下学术报告：
报告题目：The Intrinsic Dimension of Importance Sampling
报告时间：2015年8月6日上午10：00
报告地点：理科楼407
报告摘要：
Importance sampling is a method that allows to compute expectations w.r.t. a complicated target measure by using weighted samples from a simpler proposal measure. We study importance sampling in a general abstract framework, and in the context of inverse problems and filtering. In the general setting we show a non-asymptotic theorem that holds in arbitrary measurable space, and with very mild assumptions on the change of measure. We prove that the mean squared error of the method is, non-asymptotically, of order $O(N^{-1})$ where $N$ is the number of samples from the proposal. However, in the context of inverse problems and filtering, importance sampling has been often claimed to fail in high dimensional or small noise regimes. We investigate how these claims reconcile with our general theorem. We introduce a notion of intrinsic dimension for importance sampling, which takes into account the dimension of the state space, the regularity of the operators involved, and the strength of the noise in the observations. This effective dimension governs the degeneracy of importance sampling for large state-space dimension and small noise limits. The intrinsic dimension is finite as long as the target is absolutely continuous with respect to the proposal. Absolute continuity is key when discretising inverse or filtering problems arising from PDEs in geophysical applications. This is joint work in progress with Sergios Agapiou, Omiros Papaspilioupoulos and Andrew Stuart.
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