This paper investigates the inverse source problem for fractional diffusion equations using terminal data from scattered point measurements. From a statistical perspective, we treat the measurements as stochastic variables and employ Tikhonov regularization to reconstruct the unknown source. Optimal error estimates are established, which explicitly depend on the noise level, the regularization parameter, and the number of observation points. The theoretical analysis is then extended to a numerical framework based on the finite element method, yielding convergence rates that balance discretization error, noise, and observation points. A fast POD-based algorithm is developed to reduce computational time and storage requirements for high-dimensional problems while avoiding the inverse crime. Numerical examples in one and two dimensions validate the theoretical results and demonstrate the effectiveness of the proposed method.
In this paper, we introduce a new class of energy-stable, fully decoupled numerical schemes for the binary phase-field crystal (BPFC) model, which is governed by a pair of coupled sixth-order Cahn–Hilliard-type PDEs. The key idea is to split the mobility operator using an operator splitting method. Our method decouples the two phase variables at each time step, significantly reducing computational complexity while preserving unconditional energy stability. We rigorously establish energy dissipation for both first- and second-order accurate schemes in time. Numerical convergence tests confirm the expected temporal accuracy. A series of numerical examples on regular domains, irregular geometries, surfaces, and 3D domains demonstrates the robustness and flexibility of the proposed schemes in capturing complex phase transitions in binary colloidal crystals. The results confirm the efficiency and scalability of our schemes for solving the BPFC model to simulate multi-component crystalline structures.
Tensor auto-regression has become an important tool for analyzing tensor time series. However, the high dimensionality and multi-way correlations present significant estimation challenges. To address these issues, we introduce a novel auto-regressive framework for third-order tensor time series based on the tensor-tensor product (T-product). The estimation of coefficient tensors is formulated as a low tubal rank optimization model. By introducing auxiliary variables, we derive a tractable relaxation and design an easy-to-operate and effective algorithm with guaranteed convergence. We prove that the sequence generated by the proposed algorithm converges to a stationary point of the established relaxation model. A series of numerical experiments on the prediction of third-order tensor time series, including synthetic and real-world tensor time series datasets, demonstrate that our prediction approach has more favorable performance than some state-of-the-art approaches.
This paper presents an hp-version direct discontinuous Galerkin analysis for a second-order elliptic problem with curved interface. Utilizing interface-fitted meshes, we develop a discrete scheme and establish fundamental results for curved elements under suitable assumptions. The bilinear form is proven to be coercive and continuous, with constants independent of the mesh size h, polynomial degree p, and diffusion coefficient α . A priori error estimates demonstrate optimal convergence in h for both the energy and L^2 norms, yet are suboptimal by half an order and one order in p, respectively. While numerical results confirm the optimal h-convergence, they also reveal a superconvergence in p for singular solutions, robustness against large coefficient contrasts and exponential convergence with respect to the polynomial degree for smooth solutions.
In this paper, we propose a novel approach to establish the convergence of the time-domain perfectly matched layer (PML) method for acoustic scattering problems. The time discretization employs the second-order backward differentiation formula (BDF2) combined with convolution quadrature (CQ), while the spatial domain is truncated by a circular PML layer. We establish a rigorous error estimate that accounts for both the temporal discretization error and the PML truncation error. The temporal error converges at the optimal rate O(τ ^2) under suitable regularity assumptions on the data. The PML error is shown to decay exponentially in σ _0 d/τ , where σ _0 is the absorption parameter, d is the layer thickness, and τ is the time step. A distinctive feature of the analysis is that setting the absorption coefficient proportional to 1/τ makes the PML attenuation automatically strengthen as the time step is refined, so that the exponential decay dominates any algebraic growth factors.
