In this paper, we combine deep learning concepts and some proper orthogonal decomposition (POD) model reduction methods for predicting flow in heterogeneous porous media. Nonlinear flow dynamics is studied, where the dynamics is regarded as a multi-layer network. The solution at the current time step is regarded as a multi-layer network of the solution at the initial time and input parameters. As for input, we consider various sources, which include source terms (well rates), permeability fields, and initial conditions. We consider the flow dynamics, where the solution is known at some locations and the data is integrated to the flow dynamics by modifying the reduced-order model. This approach allows modifying the reduced-order formulation of the problem. Because of the small problem size, limited observed data can be handled. We consider enriching the observed data using the computational data in deep learning networks. The basis functions of the global reduced order model are selected such that the degrees of freedom represent the solution at observation points. This way, we can avoid learning basis functions, which can also be done using neural networks. We present numerical results, where we consider channelized permeability fields, where the network is constructed for various channel configurations. Our numerical results show that one can achieve a good approximation using forward feed maps based on multi-layer networks.

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We study the generalized eigenvalue problems (GEPs) derived from modeling the surface acoustic wave in piezoelectric materials with periodic inhomogeneity. The eigenvalues appear in the reciprocal pairs due to periodic boundary conditions in the modeling. By transforming the GEP into a T-palindromic quadratic eigenvalue problem (TPQEP), the reciprocal relationship of the eigenvalues can be maintained. In this paper, we outline four recently developed structure-preserving algorithms, SA, SDA, TSHIRA and GTSHIRA, for solving the TPQEP. Numerical comparisons on the accuracy and the computational costs of these algorithm are presented. The eigenvalues close to unit circle on the complex plane are of interest in the area of filter and sensor designs. Our numerical results show that the Arnoldi-type structure-preserving algorithms TSHIRA and GTSHIRA with re-symplectic and re-bi-isotropic processes

Rerandomization is a strategy for improving balance on observed covariates in randomized controlled trials. It has been both advocated and advised against by renowned scholars of experimental design. However, the relationship and differences between stratification, rerandomization, and the combination of the two have not been previously investigated. In this paper, we show that stratified designs can be recreated by rerandomization and explain why, in most cases, stratification on binary covariates followed by rerandomization on continuous covariates is more efficient than rerandomization on all covariates at the same time.

Point cloud is the most fundamental representation of 3D geometric objects as it provides the most exact information of them. Analyzing and processing point cloud surfaces is important in computer graphics and computer vision. However, most of the existing algorithms for surface analysis require connectivity information. Therefore, it is desirable to develop a mesh structure on point clouds. This task can be simplified with the aid of a parameterization. In particular, conformal parameterizations are advantageous in preserving the geometric information of the point cloud data. In this paper, we extend a state-of-the-art spherical conformal parameterization algorithm for genus-0 closed meshes to the case of point clouds, using an improved approximation of the Laplace-Beltrami operator on data points. Then, we propose an iterative scheme called the North-South reiteration for achieving a spherical conformal parameterization. A balancing scheme is introduced to enhance the distribution of the spherical parameterization. High quality triangulations and quadrangulations can then be built on the point clouds with the aid of the parameterizations. Also, the meshes generated are guaranteed to be genus-0 closed meshes. Moreover, using our proposed spherical conformal parameterization, multilevel representations of point clouds can be easily constructed. Experimental results demonstrate the effectiveness of our proposed framework.