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Traffic speed prediction is significant for intelligent navigation and congestion alleviation. However, making accurate predictions is challenging due to three factors: 1) traffic diffusion, i.e., the spatial and temporal causality existing between the traffic conditions of multiple neighboring roads, 2) the poor interpretability of traffic data with complicated spatio-temporal correlations, and 3) the latent pattern of traffic speed fluctuations over time, such as morning and evening rush. Jointly considering these factors, in this paper, we present a novel architecture for traffic speed prediction, called Interpretable Causal Spatio-Temporal Diffusion Network (ICST-DNET). Specifically, ICST-DENT consists of three parts, namely the Spatio-Temporal Causality Learning (STCL), Causal Graph Generation (CGG), and Speed Fluctuation Pattern Recognition (SFPR) modules. First, to model the traffic diffusion within road networks, an STCL module is proposed to capture both the temporal causality on each individual road and the spatial causality in each road pair. The CGG module is then developed based on STCL to enhance the interpretability of the traffic diffusion procedure from the temporal and spatial perspectives. Specifically, a time causality matrix is generated to explain the temporal causality between each road's historical and future traffic conditions. For spatial causality, we utilize causal graphs to visualize the diffusion process in road pairs. Finally, to adapt to traffic speed fluctuations in different scenarios, we design a personalized SFPR module to select the historical timesteps with strong influences for learning the pattern of traffic speed fluctuations. Extensive experimental results prove that ICST-DNET can outperform all existing baselines, as evidenced by the higher prediction accuracy, ability to explain causality, and adaptability to different scenarios.

We propose and analyze an inexact gradient method based on incremental proper orthogonal decomposition (iPOD) to address the data storage difficulty in time-dependent PDE-constrained optimization, particularly for a data assimilation problem as a detailed demonstration for the key ideas. The proposed method is proved robust by rigorous analysis. We first derive a sharp data compression error estimate of the iPOD with the help of Hilbert-Schmidt operators. Then we demonstrate a numerical PDE analysis to show how to properly choose the Hilbert space for the iPOD data compression so that the gradient error is under control. We further prove that for a convex problem with appropriately bounded gradient error, the inexact gradient method achieves the accuracy level of the optimal solution while not hurting the convergence rate compared with the usual gradient method. Finally, numerical experiments are provided to verify the theoretical results and validate the proposed method.

In this paper, we study the fractional critical Schr\"{o}dinger-Poisson system \[\begin{cases} (-\Delta)^su +\lambda\phi u= \alpha u+\mu|u|^{q-2}u+|u|^{2^*_s-2}u,&~~ \mbox{in}~{\mathbb R}^3,\\ (-\Delta)^t\phi=u^2,&~~ \mbox{in}~{\mathbb R}^3,\end{cases} \] having prescribed mass \[\int_{\mathbb R^3} |u|^2dx=a^2,\] where $ s, t \in (0, 1)$ satisfies $2s+2t > 3, q\in(2,2^*_s), a>0$ and $\lambda,\mu>0$ parameters and $\alpha\in{\mathbb R}$ is an undetermined parameter. Under the $L^2$-subcritical perturbation $q\in (2, 2+\frac{4s}{3})$, we derive the existence of multiple normalized solutions by means of the truncation technique, concentration-compactness principle and the genus theory. For the $L^2$-supercritical perturbation $q\in (2+\frac{4s}{3}, 2^*_s)$, by applying the constrain variational methods and the mountain pass theorem, we show the existence of positive normalized ground state solutions., Comment: 43 pages

In this article we consider the numerical modeling and simulation via the phase field approach of two-phase flows of different densities and viscosities in superposed fluid and porous layers. The model consists of the Cahn-Hilliard-Navier-Stokes equations in the free flow region and the Cahn-Hilliard-Darcy equations in porous media that are coupled by seven domain interface boundary conditions. We show that the coupled model satisfies an energy law. Based on the ideas of pressure stabilization and artificial compressibility, we propose an unconditionally stable time stepping method that decouples the computation of the phase field variable, the velocity and pressure of free flow, the velocity and pressure of porous media, hence significantly reduces the computational cost. The energy stability of the scheme effected with the finite element spatial discretization is rigorously established. We verify numerically that our schemes are convergent and energy-law preserving. Ample numerical experiments are performed to illustrate the features of two-phase flows in the coupled free flow and porous media setting.

This paper presents a recently developed particle simulation code package PIFE-PIC, which is a novel three-dimensional (3-D) Parallel Immersed-Finite-Element (IFE) Particle-in-Cell (PIC) simulation model for particle simulations of plasma-material interactions. This framework is based on the recently developed non-homogeneous electrostatic IFE-PIC algorithm, which is designed to handle complex plasma-material interface conditions associated with irregular geometries using a Cartesian-mesh-based PIC. Three-dimensional domain decomposition is utilized for both the electrostatic field solver with IFE and the particle operations in PIC to distribute the computation among multiple processors. A simulation of the orbital-motion-limited (OML) sheath of a dielectric sphere immersed in a stationary plasma is carried out to validate PIFE-PIC and profile the parallel performance of the code package. Furthermore, a large-scale simulation of plasma charging at a lunar crater containing 2 million PIC cells (10 million FE/IFE cells) and about 520 million particles, running for 20,000 PIC steps in about 109 wall-clock hours, is presented to demonstrate the high-performance computing capability of PIFE-PIC.

In this article we develop a multi-grid multi-level Monte Carlo (MGMLMC) method for the stochastic Stokes-Darcy interface model with random hydraulic conductivity both in the porous media domain and on the interface. Because the randomness through the interface affects the flow in the Stokes domain, we investigate the coupled stochastic Stokes-Darcy model to improve the fidelity as this model also considers the second and third porosity of the free flow. Then we prove the existence and uniqueness of the weak solution of the variational form. For the numerical solution, we adopt the Monte Carlo (MC) method and finite element method (FEM), for the discrete form in the probability space and physical space, respectively. In the traditional single-level Monte Carlo (SLMC) method, more accurate numerical approximate requires both larger number of samples in probability space and smaller mesh size in the physical space. Then the computational cost increase significantly as the mesh size becomes smaller for the more accurate numerical approximate. Therefore we adopt the multi-level Monte Carlo (MLMC) method to dramatically reduce the computational cost in the probability space, because the number of samples decays fast while the mesh size decreases. We also develop a strategy to calculate the number of samples needed in MLMC method for the stochastic Stokes-Darcy model. Furthermore MLMC naturally provides the hierarchial grids and sufficient information on these grids for multi-grid (MG) method, which can in turn improve the efficiency of MLMC. In order to fully make use of the dynamical interaction between this two methods, we propose the multi-grid multi-level Monte Carlo method for more efficiently solving the stochastic model. Numerical examples are provided to verify and illustrate the proposed method and the theoretical conclusions.