• Abstract

An efficient real-time computational method for a feedback control problem of the Boussinesq equations is studied. We consider a simple and effective feedback control law based on the relationship between the control and adjoint variables in the optimality system. We investigate a closure type modeling in reduced order model (ROM) of this problem for real-time computing. In order to improve the existing well-known POD-ROM method, the deep learning technique, which is currently being actively researched, is studied and applied. Computational results presented show that the suggested methods work well.

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• Abstract

In this paper, we propose a nonconforming virtual element method for the elliptic interface problem based on an unfitted polygonal mesh. On interface elements, the intersecting points of the interface and the edges of elements are considered as additional nodes of the mesh, and then we present a virtual element space satisfying the interface conditions. On non-interface elements, we use the usual nonconforming virtual element. By employing a computable operator, we introduce a discrete scheme and obtain optimal convergence results which are independent of the contrast of the coefficients. Numerical examples are presented to validate the theoretical results.

• General Summary

• Abstract

We study the (2+1)-dimensional nonlocal Fokas system by using the Hirota’s bilinear method. Firstly, a general tau-function of Kadomtsev-Petviashvili (KP) hierarchy satisfied with the bilinear equation under nonzero boundary condition is derived by considering the differential relations and the variable transformation. Secondly, two Gramtype solutions are utilized to the construction of multi-breather, high-order rogue wave and multi-bright-dark soliton solutions. Then the corresponding parameter restrictions of these solutions are given to satisfy with the complex conjugation symmetry. Furthermore, we find that if the parameter $p_{iI}$ takes different values, the rogue wave solution can be classified as three types of states, such as dark-dark, four-peak and bright-bright high-order rogue wave. If the parameter $c_i$ takes different values, the soliton solution can be classified as three types of states, including the multi-dark, multi-bright-dark and multi-bright solitons. By considering third-type of reduced tau-function to the Hirota’s bilinear equations, we give the collisions between the high-order rogue wave and the multi-bright-dark solitons on constant ($N$ is positive even) or periodic background ($N$ is positive odd). In order to understand the dynamics behaviors of the obtained solutions better, the various rich patterns are theoretically and graphically analyzed in detail.

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• Abstract

In this paper, we extend a diagonalization-based parallel-in-time (PinT) algorithm to the viscoelastic equation. The central difference method is used for spatial discretization, while for temporal discretization, we use the Crank-Nicolson scheme. Then an all-at-once system collecting all the solutions at each time level is formed and solved using a fixed point iteration preconditioned by an $α$-circulant matrix in parallel. Via a rigorous analysis, we find that the spectral radius of the iteration matrix is uniformly bounded by $α/(1 − α),$ independent of the model parameters (the damping coefficient $\varepsilon$ and the wave velocity $\sqrt{\gamma}$) and the discretization parameters (the time step $\tau$ and the spatial mesh size $h$). Unlike the classical wave equation with Dirichlet boundary condition where the upper bound $α/(1 − α)$ is very sharp, we find that the occurrence of the damping term $−\varepsilon∆y_t,$ as well as the large final time $T,$ leads to even faster convergence of the algorithm, especially when $α$ is not very small. We illustrate our theoretical findings with several numerical examples.

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• Abstract

A new delay estimation model using a round trip time as a weight of the asymmetry of each host pair is developed. It improves the estimation accuracy and is suitable for complex wide area network architecture. For large-scale scenarios in practice, we design a symmetric Gauss-Seidel alternating direction method of multipliers. It significantly reduces memory consumption and computational cost. Numerical experiments demonstrate the accuracy and efficiency of the model and algorithm.

• General Summary