We improve Gilbert-Varshamov bound by graph spectral method. Gilbert graph $G_{q,n,d}$ is a graph with all vectors in $\mathbb{F}^n_q$ as vertices where two vertices are adjacent if their Hamming distance is less than $d.$ In this paper, we calculate the eigenvalues and eigenvectors of $G_{q,n,d}$ using the properties of Cayley graph. The improved bound is associated with the minimum eigenvalue of the graph. Finally we give an algorithm to calculate the bound and linear codes which satisfy the bound.

In this paper, we propose a network model, the multiclass classification-based reduced order model (MC-ROM), for solving time-dependent parametric partial differential equations (PPDEs). This work is inspired by the observation of applying the deep learning-based reduced order model (DL-ROM) [14] to solve diffusion-dominant PPDEs. We find that the DL-ROM has a good approximation for some special model parameters, but it cannot approximate the drastic changes of the solution as time evolves. Based on this fact, we classify the dataset according to the magnitude of the solutions and construct corresponding subnets dependent on different types of data. Then we train a classifier to integrate different subnets together to obtain the MC-ROM. When subsets have the same architecture, we can use transfer learning techniques to accelerate offline training. Numerical experiments show that the MC-ROM improves the generalization ability of the DL-ROM both for diffusion- and convection-dominant problems, and maintains the DL-ROM’s advantage of good approximation ability. We also compare the approximation accuracy and computational efficiency of the proper orthogonal decomposition (POD) which is not suitable for convection-dominant problems. For diffusion-dominant problems, the MC-ROM has better approximation accuracy than the POD in a small dimensionality reduction space, and its computational performance is more efficient than the POD’s.

We propose a splitting Hamiltonian Monte Carlo (SHMC) algorithm, which can be computationally efficient when combined with the random mini-batch strategy. By splitting the potential energy into numerically nonstiff and stiff parts, one makes a proposal using the nonstiff part of $U,$ followed by a Metropolis rejection step using the stiff part that is often easy to compute. The splitting allows efficient sampling from systems with singular potentials (or distributions with degenerate points) and/or with multiple potential barriers. In our SHMC algorithm, the proposal only based on the nonstiff part in the splitting is generated by the Hamiltonian dynamics, which can be potentially more efficient than the overdamped Langevin dynamics. We also use random batch strategies to reduce the computational cost to $\mathcal{O}(1)$ per time step in generating the proposals for problems arising from many-body systems and Bayesian inference, and prove that the errors of the Hamiltonian induced by the random batch approximation is $\mathcal{O}(\sqrt{∆t})$ in the strong and $\mathcal{O}(∆t)$ in the weak sense, where $∆t$ is the time step. Numerical experiments are conducted to verify the theoretical results and the computational efficiency of the proposed algorithms in practice.

Extremal problems on the 4-cycle $C_4$ played a heuristic important role in the development of extremal graph theory. A fundamental theorem of Füredi states that the Turán number ${\rm ex}(q^2+q+1,C_4)≤(1/2)q(q+1)^2$ holds for every $q≥14,$ which matches with the classic construction of Erdös-Rényi-Sόs and Brown from finite geometry for prime powers $q.$ Very recently, we obtained the first stability result on Füredi’s theorem, by showing that for large even $q,$ every $(q^2+q+1)$-vertex $C_4$-free graph with more than $(1/2)q(q+1)^2−0.2q$ edges must be a spanning subgraph of a unique polarity graph. Using new technical ideas in graph theory and finite geometry, we strengthen this by showing that the same conclusion remains true if the number of edges is lowered to $(1/2)q(q+1)^2−(1/2)q+o(q).$ Among other applications, this gives an immediate improvement on the upper bound of ${\rm ex}(n,C_4)$ for infinite many integers $n.$ A longstanding conjecture of Erdös and Simonovits states that every $n$-vertex graph with ${\rm ex}(n,C_4)+1$ edges contains at least $(1+o(1))\sqrt{n}$ 4-cycles. We proved an exact result and confirmed Erdös-Simonovits conjecture for infinitely many integers $n$. As the second main result of this paper, we further characterize all extremal graphs for which achieve the $ℓ$-th least number of copies of $C_4$ for any fixed positive integer $ℓ.$ This can be extended to more general settings and provides enhancements on the understanding of the supersaturation problem of $C_4.$

This paper concerns both the structural and dynamical stabilities of radially symmetric transonic shock solutions for two-dimensional Euler-Poisson system in an annulus. The density of fixed, positively charged background ions is allowed to be different constants in supersonic and subsonic regimes. First, the existence and structural stability of a steady transonic shock solution are obtained by the monotonicity between the shock location and the density on the outer circle. Second, any radially symmetric transonic shock solution with respect to small perturbations of the initial data is shown to be dynamically stable. The proof relies on the decay estimates and coupled effects from electric field and geometry of the annulus, together with the methods from [18]. These results generalize previous stability results on transonic shock solutions for constant background charge.

We present a mathematical and numerical investigation to the shrinking-dimer saddle dynamics for finding any-index saddle points in the solution landscape. Due to the dimer approximation of Hessian in saddle dynamics, the local Lipschitz assumptions and the strong nonlinearity for the saddle dynamics, it remains challenges for delicate analysis, such as the boundedness of the solutions and the dimer error. We address these issues to bound the solutions under proper relaxation parameters, based on which we prove the error estimates for numerical discretization to the shrinking-dimer saddle dynamics by matching the dimer length and the time step size. Furthermore, the Richardson extrapolation is employed to obtain a high-order approximation. The inherent reason of requiring the matching of the dimer length and the time step size lies in that the former serves a different mesh size from the later, and thus the proposed numerical method is close to a fully-discrete numerical scheme of some space-time PDE model with the Hessian in the saddle dynamics and its dimer approximation serving as a "spatial operator" and its discretization, respectively, which in turn indicates the PDE nature of the saddle dynamics.

We present a systematic two-step approach to derive temporal up to the eighth-order, unconditionally maximum-principle-preserving schemes for a semilinear parabolic sine-Gordon equation and its conservative modification. By introducing a stabilization term to an explicit integrating factor approach, and designing suitable approximations to the exponential functions, we propose a unified parametric two-step Runge-Kutta framework to conserve the linear invariant of the original system. To preserve the maximum principle unconditionally, we develop parametric integrating factor two-step Runge-Kutta schemes by enforcing the non-negativeness of the Butcher coefficients and non-decreasing constraint of the abscissas. The order conditions, linear stability, and convergence in the $L^∞$-norm are analyzed. Theoretical and numerical results demonstrate that the proposed framework, which is explicit and free of limiters, cut-off post-processing, or exponential effects, offers a concise, and effective approach to develop high-order inequality-preserving and linear-invariant-conserving algorithms.