Journal of Scientific Computing最新文献

We propose an Efficient Inexact Learned Descent-type Algorithm (ELDA) for a class of nonconvex and nonsmooth variational models, where the regularization consists of a sparsity enhancing term and non-local smoothing term for learned features. The ELDA improves the performance of the LDA in Chen et al. (SIAM J Imag Sci 14(4), 1532–1564, 2021) by reducing the number of the subproblems from two to one for most of the iterations and allowing inexact gradient computation. We generate a deep neural network, whose architecture follows the algorithm exactly for low-dose CT (LDCT) reconstruction. The network inherits the convergence behavior of the algorithm and is interpretable as a solution of the varational model and parameter efficient. The experimental results from the ablation study and comparisons with several state-of-the-art deep learning approaches indicate the promising performance of the proposed method in solution accuracy and parameter efficiency.

We present a higher-order space-time adaptive method for the numerical solution of the Richards equation that describes a flow motion through variably saturated media. The discretization is based on the space-time discontinuous Galerkin method, which provides high stability and accuracy and can naturally handle varying meshes. We derive reliable and efficient a posteriori error estimates in the residual-based norm. The estimates use well-balanced spatial and temporal flux reconstructions which are constructed locally over space-time elements or space-time patches. The accuracy of the estimates is verified by numerical experiments. Moreover, we develop the hp-adaptive method and demonstrate its efficiency and usefulness on a practically relevant example.

The Keller-Segel (KS) chemotaxis equation is a widely studied mathematical model for understanding the collective behavior of cells in response to chemical gradients. This paper investigates the direct discontinuous Galerkin method with interface correction (DDGIC) for one-dimensional and two-dimensional KS equations governing the cell density and chemoattractant concentration. We establish error estimates for the proposed scheme under suitable smoothness assumptions of the exact solutions. Numerical experiments are conducted to validate the theoretical results. We explore the impact of different coefficient settings in the numerical fluxes on the error of the DDGIC method on uniform and nonuniform meshes. Our findings reveal that the DDGIC method achieves optimal convergence rates with any admissible coefficients for polynomials of odd degrees, while the accuracy of the cell density is sensitive to the numerical flux coefficient in the chemoattractant concentration for polynomials of even degrees. These results hold regardless of whether the mesh is uniform or nonuniform.

Anomalous diffusion in the presence or absence of an external force field is often modelled in terms of the fractional evolution equations, which can involve the hyper-singular source term. For this case, conventional time stepping methods may exhibit a severe order reduction. Although a second-order numerical algorithm is provided for the subdiffusion model with a simple hyper-singular source term (t^{mu }), (-2<mu <-1) in [arXiv:2207.08447], the convergence analysis remain to be proved. To fill in these gaps, we present a simple and robust smoothing method for the hyper-singular source term, where the Hadamard finite-part integral is introduced. This method is based on the smoothing/IDm-BDFk method proposed by Shi and Chen (SIAM J Numer Anal 61:2559–2579, 2023) for the subdiffusion equation with a weakly singular source term. We prove that the kth-order convergence rate can be restored for the diffusion-wave case (gamma in (1,2)) and sketch the proof for the subdiffusion case (gamma in (0,1)), even if the source term is hyper-singular and the initial data is not compatible. Numerical experiments are provided to confirm the theoretical results.

We prove the optimal convergence in space and time for the linear acoustic wave equation in its second-order formulation in time, using the hybrid high-order method for space discretization and the leapfrog (central finite difference) scheme for time discretization. The proof hinges on energy arguments similar to those classically deployed in the context of continuous finite elements or discontinuous Galerkin methods, but some novel ideas need to be introduced to handle the static coupling between cell and face unknowns. Because of the close ties between the methods, the present proof can be readily extended to cover space semi-disretization using the hybridizable discontinuous Galerkin method and the weak Galerkin method.

In this paper, we investigate the local discontinuous Galerkin (LDG) methods coupled with multistep implicit–explicit (IMEX) time discretization to solve one-dimensional and two-dimensional nonlinear Schrödinger equations. In this approach, the nonlinear terms are treated explicitly, while the linear terms are handled implicitly. By the skew symmetry property of LDG operators and the properties of Gauss–Radau projections, we obtain error estimates for the prime and auxiliary variables, as well as the estimate for the time difference of the prime variables. These results, together with a carefully chosen numerical initial condition, allow us to obtain the optimal error estimate in both space and time for the fully discrete scheme. Numerical experiments are performed to verify the accuracy and efficiency of the proposed methods.

We propose, analyze, and test a penalty projection-based robust efficient and accurate algorithm for the Uncertainty Quantification (UQ) of the time-dependent Magnetohydrodynamic (MHD) flow problems in convection-dominated regimes. The algorithm uses the Elsässer variables formulation and discrete Hodge decomposition to decouple the stochastic MHD system into four sub-problems (at each time-step for each realization) which are much easier to solve than solving the coupled saddle point problems. Each of the sub-problems is designed in a sophisticated way so that at each time-step the system matrix remains the same for all the realizations but with different right-hand-side vectors which allows saving a huge amount of computer memory and computational time. Moreover, the scheme is equipped with Ensemble Eddy Viscosity (EEV) and grad-div stabilization terms. The unconditional stability with respect to the time-step size of the algorithm is proven rigorously. We prove the proposed scheme converges to an equivalent non-projection-based coupled MHD scheme for large grad-div stabilization parameter values. We examine how Stochastic Collocation Methods (SCMs) can be combined with the proposed penalty projection UQ algorithm. Finally, a series of numerical experiments are given which verify the predicted convergence rates, show the algorithm’s performance on benchmark channel flow over a rectangular step, a regularized lid-driven cavity problem with high random Reynolds number and high random magnetic Reynolds number, and the impact of the EEV stabilization in the MHD UQ algorithm.

This paper is devoted to the numerical solution of constrained energy minimization problems arising in computational physics and chemistry such as the Gross–Pitaevskii and Kohn–Sham models. In particular, we introduce Riemannian Newton methods on the infinite-dimensional Stiefel and Grassmann manifolds. We study the geometry of these two manifolds, its impact on the Newton algorithms, and present expressions of the Riemannian Hessians in the infinite-dimensional setting, which are suitable for variational spatial discretizations. A series of numerical experiments illustrates the performance of the methods and demonstrates their supremacy compared to other well-established schemes such as the self-consistent field iteration and gradient descent schemes.

In this study, we present a new alternative formulation of a conservative weighted essentially non-oscillatory (WENO) scheme that improves the performance of the known fifth-order alternative WENO (AWENO) schemes. In the formulation of the fifth-order AWENO scheme, the numerical flux can be written in two terms: a low-order flux and a high-order correction flux. The low-order numerical flux is constructed by a fifth-order WENO interpolator, and the high-order correction flux includes terms of the second and fourth derivatives, yielding the sixth-order truncation error. Noticing the difference in the convergence rates between these two approximations, this study first aims to fill the accuracy gap by enhancing the approximation order of the low-order numerical flux. To this end, the WENO interpolator for the low-order term is implemented using exponential polynomials with a shape parameter. Selecting a locally optimized shape parameter, the proposed WENO interpolator achieves an additional order of improvement, resulting in the overall sixth order of accuracy of the final reconstruction, under the same fifth-order AWENO framework. In addition, since a linear approximation to the high-order correction term may cause some oscillations in the vicinity of strong shocks, we present a new strategy for the limiting procedure to deal with the second derivative term in the high-order correction flux. Several numerical results for the well-known benchmark test problems confirm the reliability of our AWENO method.

In recent years, tensor networks have emerged as powerful tools for solving large-scale optimization problems. One of the most promising tensor networks is the tensor ring (TR) decomposition, which achieves circular dimensional permutation invariance in the model through the utilization of the trace operation and equitable treatment of the latent cores. On the other hand, more recently, quaternions have gained significant attention and have been widely utilized in color image processing tasks due to their effectiveness in encoding color pixels by considering the three color channels as a unified entity. Therefore, in this paper, based on the left quaternion matrix multiplication, we propose the quaternion tensor left ring (QTLR) decomposition, which inherits the powerful and generalized representation abilities of the TR decomposition while leveraging the advantages of quaternions for color pixel representation. In addition to providing the definition of QTLR decomposition and an algorithm for learning the QTLR format, the paper further proposes a low-rank quaternion tensor completion (LRQTC) model and its algorithm for color image inpainting based on the defined QTLR decomposition. Finally, extensive experiments on color image inpainting demonstrate that the proposed LRQTC method is highly competitive.