The dynamic formulation of optimal transport has attracted growing interests in scientific computing and machine learning, and its computation requires to solve a PDE-constrained optimization problem. The classical Eulerian discretization based approaches suffer from the curse of dimensionality, which arises from the approximation of high-dimensional velocity field. In this work, we propose a deep learning based method to solve the dynamic optimal transport in high dimensional space. Our method contains three main ingredients: a carefully designed representation of the velocity field, the discretization of the PDE constraint along the characteristics, and the computation of high dimensional integral by Monte Carlo method in each time step. Specifically, in the representation of the velocity field, we apply the classical nodal basis function in time and the deep neural networks in space domain with the H1-norm regularization. This technique promotes the regularity of the velocity field in both time and space such that the discretization along the characteristic remains to be stable during the training process. Extensive numerical examples have been conducted to test the proposed method. Compared to other solvers of optimal transport, our method could give more accurate results in high dimensional cases and has very good scalability with respect to dimension. Finally, we extend our method to more complicated cases such as crowd motion problem.

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We propose Gaussian-beam based Eulerian methods to compute semi-classical solutions of the Schrdinger equation. Traditional Gaussian beam type methods for the Schrdinger equation are based on the Lagrangian ray tracing. Based on the first Eulerian Gaussian beam framework proposed in Leung et al.[S. Leung, J. Qian, R. Burridge, Eulerian Gaussian beams for high frequency wave propagation, Geophysics 72 (2007) SM61SM76], we develop a new Eulerian Gaussian beam method which uses global Cartesian coordinates, level-set based implicit representation and Liouville equations. The resulting method gives uniformly distributed phases and amplitudes in phase space simultaneously. To obtain semi-classical solutions to the Schrdinger equation with different initial wave functions, we only need to slightly modify the summation formula. This yields a very efficient method for computing semi

Herein, we use hybrid resampling to address (a) the long-standing problem of inference on change times and changed parameters in change-point ARX-GARCH models, and (b) the challenging problem of valid confidence intervals, after variable selection under sparsity assumptions, for the parameters in linear regression models with high-dimensional stochastic regressors and asymptotically stationary noise. For the latter problem, we introduce consistent estimators of the selected parameters and a resampling approach to overcome the inherent difficulties of post-selection confidence intervals. For the former problem, we use a sequential Monte Carlo for the latent states (representing the change times and changed parameters) of a hidden Markov model. Asymptotic efficiency theory and simulation and empirical studies demonstrate the advantages of the proposed methods.

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