We consider the effect of small scale random fluctuations of the constitutive coefficients on boundary measurements of solutions to radiative transfer equations. As the correlation length of the random oscillations tends to zero, the transport solution is well approximated by a deterministic, averaged, solution. In this paper, we analyze the random fluctuations to the averaged solution, which may be interpreted as a central limit correction to homogenization. With the inverse transport problem in mind, we characterize the random structure of the singular components of the transport measurement operator. In regimes of moderate scattering, such components provide stable reconstructions of the constitutive parameters in the transport equation. We show that the random fluctuations strongly depend on the decorrelation properties of the random medium.

We study 2d and 3d gravity theories on spacetimes with causal (timelike or null) codimension one boundaries while allowing for variations in the position of the boundary. We construct the corresponding solution phase space and specify boundary degrees freedom by analysing boundary (surface) charges labelling them. We discuss Y and W freedoms and change of slicing in the solution space. For D dimensional case we find D + 1 surface charges, which are generic functions over the causal boundary. We show that there exist solution space slicings in which the charges are integrable. For the 3d case there exists an integrable slicing where charge algebra takes the form of Heisenberg ⊕ A3 where A3 is two copies of Virasoro at Brown-Henneaux central charge for AdS3 gravity and BMS3 for the 3d flat space gravity.

In this article, we propose a numerical method based on sparse Gaussian processes (SGPs) to solve nonlinear partial differential equations (PDEs). The SGP algorithm is based on a Gaussian process (GP) method, which approximates the solution of a PDE with the maximum a posteriori probability estimator of a GP conditioned on the PDE evaluated at a finite number of sample points. The main bottleneck of the GP method lies in the inversion of a covariance matrix, whose cost grows cubically with respect to the size of samples. To improve the scalability of the GP method while retaining desirable accuracy, we draw inspiration from SGP approximations, where inducing points are introduced to summarize the information of samples. More precisely, our SGP method uses a Gaussian prior associated with a low-rank kernel generated by inducing points randomly selected from samples. In the SGP method, the size of the matrix to be inverted is proportional to the number of inducing points, which is much less than the size of the samples. The numerical experiments show that the SGP method using less than half of the uniform samples as inducing points achieves comparable accuracy to the GP method using the same number of uniform samples, which significantly reduces the computational cost. We give the existence proof for the approximation to the solution of a PDE and provide rigorous error analysis.

In this article, we propose two numerical methods, the Gaussian Process (GP) method and the Fourier Features (FF) algorithm, to solve mean field games (MFGs). The GP algorithm approximates the solution of a MFG with maximum a posteriori probability estimators of GPs conditioned on the partial differential equation (PDE) system of the MFG at a finite number of sample points. The main bottleneck of the GP method is to compute the inverse of a square gram matrix, whose size is proportional to the number of sample points. To improve the performance, we introduce the FF method, whose insight comes from the recent trend of approximating positive definite kernels with random Fourier features. The FF algorithm seeks approximated solutions in the space generated by sampled Fourier features. In the FF method, the size of the matrix to be inverted depends only on the number of Fourier features selected, which is much less than the size of sample points. Hence, the FF method reduces the precomputation time, saves the memory, and achieves comparable accuracy to the GP method. We give the existence and the convergence proofs for both algorithms. The convergence argument of the GP method does not depend on any monotonicity condition, which suggests the potential applications of the GP method to solve MFGs with non-monotone couplings in future work. We show the efficacy of our algorithms through experiments on a stationary MFG with a non-local coupling and on a time-dependent planning problem. We believe that the FF method can also serve as an alternative algorithm to solve general PDEs.

This article considers experimental design based on the strategy of rerandomization to increase the efficiency in experiments. Two aspects of rerandomization are addressed. First, we propose a two-stage allocation sample scheme for randomization inference to the units in experiments that guarantees that the difference-in-mean estimator is an unbiased estimator of the sample average treatment effect for any experiment, conserves the exactness of randomization inference, and halves the time consumption of the rerandomization design. Second, we propose a rank-based covariate-balance measure which can take into account the estimated relative weight of each covariate. Several strategies for estimating these weights using pre-experimental data are proposed. Using Monte Carlo simulations, the proposed strategies are compared to complete randomization and Mahalanobis-based rerandomization. An empirical example is given where the power of a mean difference test of electricity consumption of 54 households is increased by 99%, in comparison to complete randomization, using one of the proposed designs based on high frequency longitudinal electricity consumption data. Supplementary materials for this article are available online.