Advances in Computational Mathematics最新文献

Two types of asymptotically compatible energies are constructed for the variable-step L1 scheme for the time-fractional Allen-Cahn model with Caputo’s fractional derivative of order (alpha in (0,1)). An energy of the time-fractional Allen-Cahn model is called asymptotic compatibility if it approaches that of the classical Allen-Cahn model when the fractional order (alpha rightarrow 1^-). By splitting the L1 formula into a local part and a nonlocal part, we construct two discrete gradient structures by exploring the logarithmic convexity and algebraic convexity of associated kernels, respectively. The nonlinear implicit L1 and linearly implicit L1-SAV schemes are then investigated for the time-fractional Allen-Cahn model, and new discrete energy dissipation laws are established under some mild step-ratio constraints. Extensive numerical tests are provided to examine the accuracy, energy behaviors, and solution behaviors of our numerical solvers in the long-time simulations. They suggest that the asymptotically compatible energy constructed from the algebraic convexity approximates the original energy faster than that constructed from the logarithmic convexity. It seems that both methods monotonically converge to the correct steady-state solution for any initial data.

In this paper, we recast the variational formulation corresponding to the single layer boundary integral operator (varvec{{text {textsf{V}}}}) for the wave equation as a minimization problem in (varvec{L^2(Sigma )}), where (varvec{Sigma := partial Omega times (0,T)}) is the lateral boundary of the space-time domain (varvec{Q:= Omega times (0,T)}). For discretization, the minimization problem is restated as a mixed saddle point formulation. Unique solvability is established by combining conforming nested boundary element spaces for the mixed formulation such that the related bilinear form is discrete inf-sup stable. We analyze under which conditions the discrete inf-sup stability is satisfied, and moreover, we show that the mixed formulation provides a simple error indicator, which can be used for adaptivity. We present several numerical experiments showing the applicability of the method to different time-domain boundary integral formulations used in the literature.

Data embedding, as one of the dimension reduction methods in visualization and classification proposed in recent years, aims at maintaining the complete information of original data so that the difference between the original and the embedded data is imperceptible. Stochastic neighbor embedding(SNE) as a nonlinear manifold learning algorithm has received extensive attention. Considering the multimodality of actual data and the crowding problems in SNE, we propose an improved stochastic neighbor embedding based on spherical logistic distribution on three-dimensional Euclidean space, (mathbf {SL_3})-SNE. The technique is a variation of SNE that produces better clusterings by introducing spherical logistic distribution, which is more heavy-tailed than the normal distribution and is able to characterize the multimodality of data. Simulated and real experiment results show that the problems of crowding have been significantly alleviated, and the classification accuracy can be increased using the proposed algorithm in comparison to the existing t-SNE and vMF-SNE.

In this paper, we propose and analyze two novel fully discrete schemes for solving nonlinear stochastic parabolic equation with multiplicative noise. The conforming virtual element method is used for the spatial direction, and the semi-implicit Euler-Maruyama and two-step backward differentiation formula (BDF2)-Maruyama methods are used for the temporal direction, respectively. The proposed schemes offer flexibility in mesh processing and are capable of using general polygonal meshes. Additionally, both schemes are linear implicit methods that only require solving a linear system at each time step, significantly improving computational efficiency. We prove the mean-square stability of the two fully discrete schemes and derive strong approximation errors with optimal convergence rates in both time and space. As far as we know, this is the first attempt to solve time-dependent stochastic partial differential equations using the virtual element method. Finally, some numerical results are presented to validate the theoretical results and to demonstrate the efficiency of the numerical methods.

We present a high-order interior penalty discontinuous Galerkin method based on a reconstructed approximation to the biharmonic equation. The first contribution is that the approximation space is reconstructed from nodal values by solving a local least squares fitting problem per element. The numerical solution converges with optimal rates under error measurements. The second contribution is that an optimal preconditioned solver is proposed to solve the linear system efficiently that not only the condition number of the preconditioned system admits a uniform upper bound independent of the mesh size, but also the solver for the preconditioning matrix is of the optimal convergence rate. Such advantages for solvers to linear systems from penalty methods are seldom attained before.

We construct a new representation of entropy solutions to nonlinear scalar conservation laws with a smooth convex flux function in a single spatial dimension. The representation is a generalization of the method of characteristics and possesses a compositional form. While it is a nonlinear representation, the embedded dynamics of the solution in the time variable is linear. This representation is then discretized as a manifold of implicit neural representations where the feedforward neural network architecture has a low-rank structure. Finally, we show that the low-rank neural representation with a fixed number of layers and a small number of coefficients can approximate any entropy solution regardless of the complexity of the shock topology, while retaining the linearity of the embedded dynamics.

In this paper, by combining the two-grid decoupled strategy and an existing domain decomposition method, two novel two-grid domain decomposition methods are constructed and analyzed for the coupled Navier–Stokes–Darcy model with Beavers–Joseph–Saffman interface condition. The proposed algorithms can decouple the Navier–Stokes–Darcy model into two independent Navier–Stokes and Darcy subsystems on both the coarse and fine grids, respectively, which can be solved in parallel with existing code and efficient solvers; hence, they could significantly enhance the computational efficiency. Numerical analysis indicates that both algorithms could reach the same convergence order as that of the standard Galerkin method with a proper configuration between the coarse grid size and the fine mesh size. Some numerical results are reported to show the main features of the two proposed algorithms.