In this paper, we propose a novel high order unfitted finite element method on Cartesian meshes for solving the acoustic wave equation with discontinuous coefficients having complex interface geometry. The unfitted finite element method does not require any penalty to achieve optimal convergence. We also introduce a new explicit time discretization method for the ODE system resulting from the spatial discretization of the wave equation. The strong stability and optimal hp-version error estimates both in time and space are established. Numerical examples confirm our theoretical results.

In this paper, a novel structure-preserving scheme is proposed for solving the three-dimensional Maxwell’s equations. The proposed scheme can preserve all of the desired structures of the Maxwell’s equations numerically, including five energy conservation laws, two divergence-free fields, three momentum conservation laws and a symplectic conservation law. Firstly, the spatial derivatives of the Maxwell’s equations are approximated with Fourier pseudo-spectral methods. The resulting ordinary differential equations are cast into a canonical Hamiltonian system. Then, the fully discrete structure-preserving scheme is derived by integrating the Hamiltonian system using a sixth order average vector field method. Subsequently, an optimal error estimate is established based on the energy method, which demonstrates that the proposed scheme is of sixth order accuracy in time and spectral accuracy in space in the discrete $L^2$-norm. The constant in the error estimate is proved to be only $\mathcal{O}(T),$ where $T > 0$ is the time period. Furthermore, its numerical dispersion relation is analyzed in detail, and a customized fast solver is presented to efficiently solve the resulting discrete linear equations. Finally, numerical results are presented to validate our theoretical analysis.

In this paper, we consider the global well-posedness of solutions to a parabolic-parabolic Keller-Segel model with p-Laplace diffusion. We first establish a critical exponent p * = 3N/(N +1) and prove that when p > p * , the solution exists globally for arbitrary large initial value. When 1< p ≤ p * , there exists an uniformly bounded global strong solution for small initial value, and the solution decays to zero as t → ∞. This paper improves and expands the results of [Cong and Liu, Kinet. Relat. Models, 9(4), 2016], in which the parabolic-elliptic case is studied.

In this paper, we propose a class of stochastic Runge-Kutta (SRK) methods for solving semilinear parabolic equations. By using the nonlinear Feynman-Kac formula, we first write the solution of the parabolic equation in the form of the backward stochastic differential equation (BSDE) and then deduce an ordinary differential equation (ODE) containing the conditional expectations with respect to a diffusion process. The time semidiscrete SRK methods are then developed based on the corresponding ODE. Under some reasonable constraints on the time step, we theoretically prove the maximum bound principle (MBP) of the proposed methods and obtain their error estimates. By combining the Gaussian quadrature rule for approximating the conditional expectations, we further propose the first-and second-order fully discrete SRK schemes, which can be written in the matrix form. We also rigorously analyze the MBP-preserving and error estimates of the fully discrete schemes. Some numerical experiments are carried out to verify our theoretical results and to show the efficiency and stability of the proposed schemes.

In the partition of unity finite element method, the nodal basis of the standard linear Lagrange finite element is multiplied by the P k polynomial basis to form a local basis of an extended finite element space. Such a space contains the P 1 Lagrange element space, but is a proper subspace of the P k+1 Lagrange element space on triangular or tetrahedral grids. It is believed that the approximation order of this extended finite element is k, in H 1 -norm, as it was proved in the first paper on the partition of unity, by Babuska and Melenk. In this work we show surprisingly the approximation order is k+1 in H 1 -norm. In addition we extend the method to rectangular/cuboid grids and give a proof to this sharp convergence order. Numerical verification is done with various partition of unity finite elements, on triangular, tetrahedral, and quadrilateral grids.

This paper considers the case of a firm's dynamic pricing problem for a nonperishable product experiencing surging demand caused by rare events modelled by a marked point process. The firm aims to maximize its running revenue by selecting an optimal price process for the product until its inventory is depleted. Using the dynamic program and inspired by the viscosity solution technique, we solve the resulting integro-differential Hamilton-Jacobi-Bellman (HJB) equation and prove that the value function is its unique classical solution. We also establish structural properties for our problem and find that the optimal price always decreases with initial inventory level in the absence of surging demand. However, with surging demand, we find that the optimal price could increase rather than decrease at the initial inventory level.

This paper proposes a novel method to establish the well-posedness of uniaxial perfectly matched layer (UPML) method for a two-dimensional acoustic scattering from a compactly supported source in a two-layered medium. We solve a long standing problem by showing that the truncated layered medium scattering problem is always resonance free regardless of the thickness and absorbing strength of UPML. The main idea is based on analyzing an auxiliary waveguide problem obtained by truncating the layered medium scattering problem through PML in the vertical direction only. The Green function for this waveguide problem can be constructed explicitly based on the separation of variables and Fourier transform. We prove that such a construction is always well-defined regardless of the absorbing strength. The well-posedness of the fully UPML truncated scattering problem follows by assembling the waveguide Green function through periodic extension.

In the previous paper [CSIAM Trans. Appl. Math. 2 (2021), 1-55], the authors proposed a theoretical framework for the analysis of RNA velocity, which is a promising concept in scRNA-seq data analysis to reveal the cell state-transition dynamical processes underlying snapshot data. The current paper is devoted to the algorithmic study of some key components in RNA velocity workflow. Four important points are addressed in this paper: (1) We construct a rational time-scale fixation method which can determine the global gene-shared latent time for cells. (2) We present an uncertainty quantification strategy for the inferred parameters obtained through the EM algorithm. (3) We establish the optimal criterion for the choice of velocity kernel bandwidth with respect to the sample size in the downstream analysis and discuss its implications. (4) We propose a temporal distance estimation approach between two cell clusters along the cellular development path. Some illustrative numerical tests are also carried out to verify our analysis. These results are intended to provide tools and insights in further development of RNA velocity type methods in the future.

In this paper, we consider signal recovery in both noiseless and noisy cases via weighted ℓ p (0 < p ≤ 1) minimization when some partial support information on the signals is available. The uniform sufficient condition based on restricted isometry property (RIP) of order tk for any given constant t > d (d ≥ 1 is determined by the prior support information) guarantees the recovery of all k-sparse signals with partial support information. The new uniform RIP conditions extend the state-of-the-art results for weighted ℓ p -minimization in the literature to a complete regime, which fill the gap for any given constant t > 2d on the RIP parameter, and include the existing optimal conditions for the ℓ p -minimization and the weighted ℓ 1 -minimization as special cases.