Journal of Scientific Computing最新文献

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.

In this manuscript, we present a collective multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty, and develop a novel convergence analysis of collective smoothers and collective two-level methods. The multigrid algorithm is based on a collective smoother that at each iteration sweeps over the nodes of the computational mesh, and solves a reduced saddle-point system whose size is proportional to the number N of samples used to discretized the probability space. We show that this reduced system can be solved with optimal O(N) complexity. The multigrid method is tested both as a stationary method and as a preconditioner for GMRES on three problems: a linear-quadratic problem, possibly with a local or a boundary control, for which the multigrid method is used to solve directly the linear optimality system; a nonsmooth problem with box constraints and (L^1)-norm penalization on the control, in which the multigrid scheme is used as an inner solver within a semismooth Newton iteration; a risk-averse problem with the smoothed CVaR risk measure where the multigrid method is called within a preconditioned Newton iteration. In all cases, the multigrid algorithm exhibits excellent performances and robustness with respect to the parameters of interest.

The present article introduces, mathematically analyzes, and numerically validates a new weak Galerkin mixed finite element method based on Banach spaces for the stationary Navier–Stokes equation in pseudostress–velocity formulation. Specifically, a modified pseudostress tensor, which depends on the pressure as well as the diffusive and convective terms, is introduced as an auxiliary unknown, and the incompressibility condition is then used to eliminate the pressure, which is subsequently computed using a postprocessing formula. Consequently, to discretize the resulting mixed formulation, it is sufficient to provide a tensorial weak Galerkin space for the pseudostress and a space of piecewise polynomial vectors of total degree at most ’k’ for the velocity. Moreover, the weak gradient/divergence operator is utilized to propose the weak discrete bilinear forms, whose continuous version involves the classical gradient/divergence operators. The well-posedness of the numerical solution is proven using a fixed-point approach and the discrete versions of the Babuška–Brezzi theory and the Banach–Nečas–Babuška theorem. Additionally, an a priori error estimate is derived for the proposed method. Finally, several numerical results illustrating the method’s good performance and confirming the theoretical rates of convergence are presented.