Journal of Scientific Computing最新文献

This paper is concerned with guided modes of an acoustic wave propagation problem on a periodic array of axially symmetric obstacles. A guided mode refers to a quasi-periodic eigenfield that propagates along the obstacles but decays exponentially away from them in the absence of incidences. Thus, the problem can be studied in an unbound unit cell due to the quasi-periodicity. We truncate the unit cell onto a cylinder enclosing the interior obstacle in terms of utilizing Rayleigh’s expansion to design an exact condition on the lateral boundary. We derive a new boundary integral equation (BIE) only involving the free-space Green function on the boundary of each homogeneous region within the cylinder. Due to the axial symmetry of the boundaries, each BIE is decoupled via the Fourier transform to curve BIEs and they are discretized with high-accuracy quadratures. With the lateral boundary condition and the side quasi-periodic condition, the discretized BIEs lead to a homogeneous linear system governing the propagation constant of a guided mode at a given frequency. The propagation constant is determined by enforcing that the coefficient matrix is singular. The accuracy of the proposed method is demonstrated by a number of examples even when the obstacles have sharp edges or corners.

In this paper, we consider two different types of numerical schemes for the nonstationary stochastic Stokes–Darcy equations with multiplicative noise. Firstly, we consider the Chorin-type time-splitting scheme for the Stokes equation in the free fluid region. The Darcy equation and an elliptic equation for the intermediate velocity of free fluid coupled with the interface conditions are solved, and then the velocity and pressure in free fluid region are updated by an elliptic system. Secondly, we further consider the artificial compressibility method (ACM) which separates the fully coupled Stokes–Darcy model into two smaller subphysics problems. The ACM reduces the storage and the computational time at each time step, and allows parallel computing for the decoupled problems. The pressure in free fluid region only needs to be updated at each time step without solving an elliptic system. We utilize the RT(_1)-P(_1) pair finite element space and the interior penalty discontinuous Galerkin (IPDG) scheme based on the BDM(_1)-P(_0) finite element space in the spatial discretizations. Under usual assumptions for the multiplicative noise, we prove that both of the Chorin-type scheme and the ACM are unconditionally stable. We present the error estimates for the time semi-discretization of the Chorin-type scheme. Numerical examples are provided to verify the stability estimates for both of schemes. Moreover, we test the convergence rate for the velocity in time for both of schemes, and the convergence rate for the pressure approximation in time average is also tested.

In this paper, a robustness-enhanced reconstruction for the high-order finite volume scheme is constructed on the 2-D structured mesh, and both the high-order gas-kinetic scheme and the Lax-Friedrichs flux solver are considered to verify the effectiveness of this algorithm. The strategy of the successful weighted essentially non-oscillatory (WENO) reconstruction is adopted to select the smooth sub-stencils. However, there are cases where strong discontinuities exist in all sub-stencils of the WENO reconstruction, weakening its robustness. To improve the robustness of the algorithm in discontinuous regions in two-dimensional space, the hybrid reconstruction based on a combination of discontinuity feedback factor (Ji et al. in Int. J. Comput. Fluid Dyn. 35:485–509, 2021) and WENO reconstruction is developed to deal with the possible discontinuities. Numerical results from smooth to extreme cases have been presented, which validates that the new finite volume scheme is effective for robustness enhancement while maintaining high resolution compared with the WENO scheme.

This paper is dedicated to the numerical solution of a fourth-order singular perturbation problem using the interior penalty virtual element method (IPVEM). Compared with the original IPVEM proposed in Zhao et al. (Math Comp 92(342):1543–1574, 2023), the study introduces modifications to the jumps and averages in the penalty term, as well as presents a mesh-dependent selection of the penalty parameter. Drawing inspiration from the modified Morley finite element methods, we leverage the conforming interpolation technique to handle the lower part of the bilinear form in the error analysis. We establish the optimal convergence in the energy norm and provide a rigorous proof of uniform convergence concerning the perturbation parameter in the lowest-order case.

Multi-dimensional scaling (MDS) with incomplete distance information represents a significant challenging inverse problem in computational geometry. This technique finds expensive applications in the fields of surface, manifold, and cubicle reconstructions, and is also relevant in the context of social networks. While a majority of existing methodologies tend to provide accurate results primarily when the missing distance indices are chosen randomly or when the omission rate is below 50%, our research proposes an innovative approach. We present a robust MDS framework when distances to the k-nearest neighbors (kNN) are known, even in situations characterized by a high coherence of missing indices. Our proposed strategy starts with a local reconstruction phase based on local correlation. Subsequently, the global reconstruction phase is realized through two distinct models: one based on low-rank semi-definite programming (SDP) and the other rooted in a model utilizing the Frobenius norm. Throughout the global reconstruction, we incorporate the alternating direction method of multipliers (ADMM) and the Riemann gradient descent algorithm (RGrad). Numerical Simulations have demonstrated that for MDS from kNN distances, our proposed model and algorithm outperforms the existed SDP models in terms of the visual effect and error of Gram matrix. We further validate that our approach can reconstruct surfaces from as mere as 1% of kNN distances, which shows that the proposed model is robust to the high coherence of missing indices. Additionally, we propose another MDS model which is applicable from kNN distances with additive noise.

Numerical solving the Schrödinger equation with incommensurate potentials presents a great challenge since its solutions could be space-filling quasiperiodic structures without translational symmetry nor decay. In this paper, we propose two high-accuracy numerical methods to solve the time-dependent quasiperiodic Schrödinger equation. Concretely, we discretize the spatial variables by the quasiperiodic spectral method and the projection method, and the time variable by the second-order operator splitting method. The corresponding convergence analysis is also presented and shows that the proposed methods both have spectral convergence rates in space and second order accuracy in time, respectively. Meanwhile, we analyse the computational complexity of these numerical algorithms. One- and two-dimensional numerical results verify these convergence conclusions, and demonstrate that the projection method is more efficient.

Attribute to its powerful representation ability, block term decomposition (BTD) has recently attracted many views of multi-dimensional data processing, e.g., hyperspectral image unmixing and blind source separation. However, the popular alternating least squares algorithm for rank-(L, M, N) BTD (BTD-ALS) suffers expensive time and space costs from Kronecker products and solving low-rank approximation subproblems, hindering the deployment of BTD for real applications, especially for large-scale data. In this paper, we propose a fast sketching-based Kronecker product-free algorithm for rank-(L, M, N) BTD (termed as KPF-BTD), which is suitable for real-world multi-dimensional data. Specifically, we first decompose the original optimization problem into several rank-(L, M, N) approximation subproblems, and then we design the bilateral sketching to obtain the approximate solutions of these subproblems instead of the exact solutions, which allows us to avoid Kronecker products and rapidly solve rank-(L, M, N) approximation subproblems. As compared with BTD-ALS, the time and space complexities (mathcal {O}{(2(p+1)(I^3LR+I^2L^2R+IL^3R)+I^3LR)}) and (mathcal {O}{(I^3)}) of KPF-BTD are significantly cheaper than (mathcal {O}{(I^3L^6R^2+I^3L^3R+I^3LR+I^2L^3R^2+I^2L^2R)}) and (mathcal {O}{(I^3L^3R)}) of BTD-ALS, where (p ll I). Moreover, we establish the theoretical error bound for KPF-BTD. Extensive synthetic and real experiments show KPF-BTD achieves substantial speedup and memory saving while maintaining accuracy (e.g., for a (150times 150times 150) synthetic tensor, the running time 0.2 seconds per iteration of KPF-BTD is significantly faster than 96.2 seconds per iteration of BTD-ALS while their accuracies are comparable).

This study investigates the Poissonian image restoration problems. In particular, we propose a novel model that incorporates (L_1/L_2) minimization on the gradient as a regularization term combined with a box constraint and a nonlinear data fidelity term, specifically crafted to address the challenges caused by Poisson noise. We employ a splitting strategy, followed by the alternating direction method of multipliers (ADMM) to find a model solution. Furthermore, we show that under mild conditions, the sequence generated by ADMM has a sub-sequence that converges to a stationary point of the proposed model. Through numerical experiments on image deconvolution, super-resolution, and magnetic resonance imaging (MRI) reconstruction, we demonstrate superior performance made by the proposed approach over some existing gradient-based methods.

We introduce a novel class of ultra-convergent structures for the Finite Volume Element (FVE) method. These structures are characterized by asymmetric and optional superconvergent points. We establish a crucial relationship between ultra-convergence properties and the orthogonality condition. Remarkably, within this framework, certain FVE schemes achieve simultaneous superconvergence of both derivatives and function values at designated points, as demonstrated in Example 2. This is a phenomenon rarely observed in other numerical methods. Theoretical validation of these findings is provided through the proposed Generalized M-Decomposition (GMD). Numerical experiments effectively substantiate our results.

In this paper, we focus on the numerical solution of nonlinear inverse problems in applied geophysics. Our aim is to reconstruct the structure of the soil, i.e., either its electrical conductivity or the magnetic permeability distribution, by inverting frequency domain electromagnetic data. This is a very challenging task since the problem is nonlinear and severely ill-conditioned. To solve the nonlinear inverse problem, we propose an alternating direction multiplier method (ADMM), we prove its convergence, and propose an automated strategy to determine the parameters involved. Moreover, we present two heuristic variations of the ADMM that either improve the accuracy of the computed solutions or lower the computational cost. The effectiveness of the different proposed methods is illustrated through few numerical examples.