查看更多>>摘要：Multi-degree splines are piecewise polynomial functions having sections of different degrees. They offer significant advantages over the classical uniform-degree framework, as they allow for modeling complex geometries with fewer degrees of freedom and, at the same time, for a more efficient engineering analysis. Moreover they possess a set of basis functions with similar properties to standard B-splines. In this paper we develop an algorithm for efficient evaluation of multi-degree B-splines, which, unlike previous approaches, is numerically stable. The proposed method consists in explicitly constructing a mapping between a known basis and the multi-degree B-spline basis of the space of interest, exploiting the fact that the two bases are related by a sequence of knot insertion and/or degree elevation steps and performing only numerically stable operations. In addition to theoretically justifying the stability of the algorithm, we will illustrate its performance through numerical experiments that will serve us to demonstrate its excellent behavior in comparison with existing methods, which, in some cases, suffer from apparent numerical problems. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：The Arnoldi method is a commonly used technique for finding a few eigenpairs of large, sparse and nonsymmetric matrices. Recently, a new variant of Arnoldi method (NVRA) was proposed. In NVRA, the modified Ritz vector is used to take the place of the Ritz vector by solving a minimization problem. Moreover, it was shown that if the refined Arnoldi method converges, then the NVRA method also converges. The contribution of this work is as follows. First, we point out that the convergence theory of the NVRA method is incomplete. More precisely, the cosine of the angle between the refined Ritz vector and the Ritz vector may not be uniformly lower-bounded, and it can be arbitrarily close to zero in theory. Consequently, the modified Ritz vector may fail to converge even when the search subspace is good enough. A remedy to the convergence of the NVRA method is given. Second, we show that the linear system for solving the modified Ritz vector in the NVRA method will become more and more ill-conditioned as the refined Ritz vector converges. If the Ritz vector also tends to converge as the refined Ritz vector does so, the ill-conditioning of the linear system will have little influence on the convergence of the modified Ritz vector, and the modified Ritz vector can improve the Ritz vector substantially. Otherwise, the ill-conditioning may have significant influence on the convergence of the modified Ritz vector. Third, to fix the NVRA method, we propose an improved refined Arnoldi method that uses improved refined Ritz vector to take the place of the modified Ritz vector. Theoretical results indicate that the improved refined Ritz method is often better than the refined Ritz method. Numerical experiments illustrate the numerical behavior of the improved refined Ritz method, and demonstrate the effectiveness of our theoretical analysis. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：We study efficient implicit methods to denoise low-field MR images using a nonlinear diffusion operator as a regularizer. This problem can be formulated as solving a nonlinear reaction-diffusion equation. After discretization, a lagged-diffusion approach is used which requires a linear system solve in every nonlinear iteration. The choice of diffusion model determines the denoising properties, but it also influences the conditioning of the linear systems. As a solution method, we use Conjugate Gradient (CG) in combination with a suitable preconditioner and deflation technique. We consider four different preconditioners in combination with subdomain deflation. We evaluate the methods for four commonly used denoising operators: standard Laplace operator, two Perona-Malik type operators, and the Total Variation (TV) operator. We show that a Discrete Cosine Transform (DCT) preconditioner works best for problems with a slowly varying diffusion coefficient, while Jacobi preconditioning with subdomain deflation works best for a strongly varying diffusion, as happens for the TV operator. This research is part of a larger effort that aims to provide low-cost MR imaging capabilities for low-resource settings. We have evaluated the algorithms on low-field MRI images using inexpensive commodity hardware. With a suitable preconditioner for the chosen diffusion model, we are able to limit the time to denoise three-dimensional images of more than 2 million pixels to less than 15 s, which is fast enough to be used in practice. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：Discontinuous Galerkin composite finite element methods (DGCFEM) are designed to tackle approximation problems on complicated domains. Partial differential equations posed on complicated domain are common when there are mesoscopic or local phenomena which need to be modelled at the same time as macroscopic phenomena. In this paper, an optical lattice will be used to illustrate the performance of the approximation algorithm for the ground state computation of a Gross-Pitaevskii equation, which is an eigenvalue problem with eigenvector nonlinearity. We will adapt the convergence results of Marcati and Maday 2018 to this particular class of discontinuous approximation spaces and benchmark the performance of the classic symmetric interior penalty hp-adaptive algorithm against the performance of the hp-DGCFEM. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：In this paper, we introduce fully implementable, adaptive Euler-Maruyama schemes for McKean-Vlasov stochastic differential equations (SDEs) assuming only a standard monotonicity condition on the drift and diffusion coefficients but no global Lipschitz continuity in the state variable for either, while global Lipschitz continuity is required for the measure component. We prove moment stability of the discretised processes and a strong convergence rate of 1/2. Several numerical examples, centred around a mean field model for FitzHugh-Nagumo neurons, illustrate that the standard uniform scheme fails and that the adaptive approach shows in most cases superior performance to tamed approximation schemes. In addition, we introduce and analyse an adaptive Milstein scheme for a certain sub-class of McKean-Vlasov SDEs with linear measure-dependence of the drift. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：The main purpose of this work is to study the numerical solution of a time fractional partial integro-differential equation of Volterra type, where the time derivative is defined in Caputo sense. Our method is a combination of the classical L1 scheme for temporal derivative, the general second order central difference approximation for spatial derivative and the repeated quadrature rule for integral part. The error analysis is carried out and it is shown that the approximate solution converges to the exact solution. Several examples are given in support of the theoretical findings. In addition, we have shown that the order of convergence is more high on any subdomain away from the origin compared to the entire domain. (C) 2021 Elsevier B.V. All rights reserved.
NSTL
Elsevier