In the asymptotic theory of quantum hypothesis testing, the minimal error probability of the first kind jumps sharply from zero to one when the error exponent of the second kind passes by the point of the relative entropy of the two states in an increasing way. This is well known as the direct part and strong converse of quantum Stein's lemma. Here we look into the behavior of this sudden change and have make it clear how the error of first kind grows smoothly according to a lower order of the error exponent of the second kind, and hence we obtain the second-order asymptotics for quantum hypothesis testing. This actually implies quantum Stein's lemma as a special case. Meanwhile, our analysis also yields tight bounds for the case of finite sample size. These results have potential applications in quantum information theory. Our method is elementary, based on basic linear algebra and probability theory. It deals with the achievability part and the optimality part in a unified fashion.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1185 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A state-of-the-art deep domain decomposition method (D3M) based on the variational principle is proposed for partial differential equations (PDEs). The solution of PDEs can be formulated as the solution of a constrained optimization problem, and we design a multi-fidelity neural network framework to solve this optimization problem. Our contribution is to develop a systematical computational procedure for the underlying problem in parallel with domain decomposition. Our analysis shows that the D3M approximation solution converges to the exact solution of underlying PDEs. Our proposed framework establishes a foundation to use variational deep learning in large-scale engineering problems and designs. We present a general mathematical framework of D3M, validate its accuracy and demonstrate its efficiency with numerical experiments.

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A new approach to passive microwave retrievals of precipitation is described that relies on an objective dimensional reduction procedure to filter, normalize, and decorrelate geophysical background noise while retaining the majority of radiometric information concerning precipitation. The dimensional reduction also sharply increases the effective density of any a priori database used in a Bayesian retrieval scheme.The method is applied to passive microwave data from the Tropical Rainfall Measuring Mission (TRMM), reducing the original nine channels to three ''pseudochannels'' that are relatively insensitive to most background variations occurring within each of seven surface classes (one ocean plus six land and coast) for which they are defined. These pseudochannels may be used in any retrieval algorithm, including the current standard Goddard profiling algorithm (GPROF), in place of the original channels. The same methods are also under development for the Global Precipitation Measurement (GPM) Core Observatory Microwave Imager (GMI).Starting with the pseudochannel definitions, a new Bayesian algorithm for retrieving the surface rain rate is described. The algorithm uses an a priori database populated with matchups between the TRMM precipitation radar (PR) and the TRMM Microwave Imager (TMI). The explicit goal of the algorithm is to retrieve the PR-derived best estimate of the surface rain rate in portions of the TMI swath not covered by the PR. A unique feature of the new algorithm is that it provides robust posterior Bayesian probabilities of pixelaveraged rain rate exceeding various thresholds.Validation and intercomparison of the new algorithm is the subject of a companion paper.