查看更多>>摘要：The time distributed-order diffusion-wave equation describes radial groundwater flow to or from a well. In the paper, an alternating direction implicit (ADI) Legendre-Laguerre spectral scheme is proposed for the two-dimensional time distributed-order diffusion wave equation on a semi-infinite domain. The Gauss quadrature formula has a higher computational accuracy than the Composite Trapezoid formula and Composite Simpson formula, which is presented to approximate the distributed order time derivative so that the considered equation is transformed into a multi-term fractional equation. Moreover, the transformed equation is solved by discretizing in space by the ADI Legendre-Laguerre spectral scheme to avoid introducing the artificial boundary and in time using the weighted and shifted Grunwald-Letnikov difference (WSGD) method. A stability and convergence analysis is performed for the numerical approximation. Some numerical results are illustrated to justify the theoretical analysis. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：In this paper we introduce a hybrid absorbing boundary condition (HABC) by combining perfectly matched layer (PML) and complete radiation boundary condition (CRBC) for solving a one-dimensional diffraction grating problem. The new boundary condition is devised in such a way that it can enjoy relative advantages from both methods. The well-posedness of the problem with HABC and the convergence of approximate solutions will be analyzed. Numerical examples to illustrate the efficiency of HABC are also presented. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：The Bernstein operators (BO) are not orthogonal, but they have duals, which are obtained by a linear combination of BO. In recent years dual BO have been adopted in computer graphics, computer aided geometric design, and numerical analysis. This paper presents a numerical method based on the Bernstein operational matrices to solve the time-space fractional convection-diffusion equation. A generalization of the derivative matrix operator of fractional order and the error analysis are discussed. Numerical examples compare the proposed approach with previous works, showing that the method is more accurate and efficient. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：This paper proposes a class of randomized Kaczmarz algorithms for obtaining isolated solutions of large-scale well-posed or overdetermined nonlinear systems of equations. This type of algorithm improves the classic Newton method. Each iteration only needs to calculate one row of the Jacobian instead of the entire matrix, which greatly reduces the amount of calculation and storage. Therefore, these algorithms are called matrix-free algorithms. According to the different probability selection patterns of choosing a row of the Jacobian matrix, the nonlinear Kaczmarz (NK) algorithm, the nonlinear randomized Kaczmarz (NRK) algorithm and the nonlinear uniformly randomized Kaczmarz (NURK) algorithm are proposed. In addition, the NURK algorithm is similar to the stochastic gradient descent (SGD) algorithm in nonlinear optimization problems. The only difference is the choice of step size. In the case of noise-free data, theoretical analysis and the results of numerical based on the classical tangential cone conditions show that the algorithms proposed in this paper are superior to the SGD algorithm in terms of iterations and calculation time. (C) 2021 Elsevier B.V. All rights reserved.
Zaman, SakhiSiraj-ul-IslamKhan, Muhammad MunibAhmad, Imtiaz...
14页
查看更多>>摘要：Accurate algorithms are proposed for approximation of integrals involving highly oscillatory Bessel function of the first kind over finite and infinite domains. Accordingly, Bessel oscillatory integrals having high oscillatory behavior are transformed into oscillatory integrals with Fourier kernel by using complex line integration technique. The transformed integrals contain an inner non-oscillatory improper integral and an outer highly oscillatory integral. A modified meshfree collocation method with Levin approach is considered to evaluate the transformed oscillatory type integrals numerically. The inner improper complex integrals are evaluated by either Gauss-Laguerre or multi-resolution quadrature. Inherited singularity of the meshfree collocation method at x = 0 is treated by a splitting technique. Error estimates of the proposed algorithms are derived theoretically in the inverse powers of omega and verified numerically. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：We introduce a tamed exponential time integrator which exploits linear terms in both the drift and diffusion for Stochastic Differential Equations (SDEs) with a one sided globally Lipschitz drift term. Strong convergence of the proposed scheme is proved, exploiting the boundedness of the geometric Brownian motion (GBM) and we establish order 1 convergence for linear diffusion terms. In our implementation we illustrate the efficiency of the proposed scheme compared to existing fixed step methods and utilize it in an adaptive time stepping scheme. Furthermore we extend the method to nonlinear diffusion terms and show it remains competitive. The efficiency of these GBM based approaches is illustrated by considering some well-known SDE models. (C) 2021 Elsevier B.V. All rights reserved.
Hill, P. A.Parker, J. T.Dickinson, D.Dudson, B. D....
15页
查看更多>>摘要：Tridiagonal matrix inversion is an important operation with many applications. It arises frequently in solving discretized one-dimensional elliptic partial differential equations, and forms the basis for many algorithms for block tridiagonal matrix inversion for discretized PDEs in higher-dimensions. In such systems, this operation is often the scaling bottleneck in parallel computation. In this paper, we derive a hybrid multigridThomas algorithm designed to efficiently invert tridiagonal matrix equations in a highly-scalable fashion in the context of time evolving partial differential equation systems. We decompose the domain between processors, using multigrid to solve on a grid consisting of the boundary points of each processor's local domain. We then reconstruct the solution on each processor using a direct solve with the Thomas algorithm. This algorithm has the same theoretical optimal scaling as cyclic reduction and recursive doubling. We use our algorithm to solve Poisson's equation as part of the spatial discretization of a time-evolving PDE system. Our algorithm is faster than cyclic reduction per inversion and retains good scaling efficiency to twice as many cores. Crown Copyright (c) 2021 Published by Elsevier B.V. All rights reserved.
Yoon, Hyun ChulLee, SanghyunMallikarjunaiah, S. M.
21页
查看更多>>摘要：We investigate a quasi-static tensile fracture in nonlinear strain-limiting solids by coupling with the phase-field approach. A classical model for the growth of fractures in an elastic material is formulated in the framework of linear elasticity for deformation systems. This linear elastic fracture mechanics (LEFM) model is derived based on the assumption of small strain. However, the boundary value problem formulated within the LEFM and under traction-free boundary conditions predicts large singular crack-tip strains. Fundamentally, this result is directly in contradiction with the underlying assumption of small strain. In this work, we study a theoretical framework of nonlinear strain-limiting models, which are algebraic nonlinear relations between stress and strain. These models are consistent with the basic assumption of small strain. The advantage of such framework over the LEFM is that the strain remains bounded even if the crack-tip stress tends to the infinity. Then, employing the phase-field approach, the distinct predictions for tensile crack growth can be governed by the model. Several numerical examples to evaluate the efficacy and the performance of the model and numerical algorithms structured on finite element method are presented. Detailed comparisons of the strain, fracture energy with corresponding discrete propagation speed between the nonlinear strain-limiting model and the LEFM for the quasi-static tensile fracture are discussed. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：Jacobian-free Newton-Krylov (JFNK) method is a popular approach to solve nonlinear algebraic equations arising from computational physics. The key issue is the calculation of Jacobian-vector product, commonly done through finite difference methods. However, these approaches suffer from both truncation error and round-off error, and the accuracy heavily depends on a sophisticated choice of the difference step size. In some extreme cases, even with the best choice of the difference step size, the accuracy may still not meet the requirement for the inner Krylov iteration. In this paper, we extend the complex step finite difference (CSFD) method to the JFNK method. Some tips are presented for accelerating the method. Multiple examples are presented to reveal the performance of the JFNK with the CSFD, and different methods for approximating the Jacobian-vector product are compared. It is demonstrated with a relatively easy way of implementation that the CSFD method is well-suited for the JFNK method, leading to extremely accurate and stable numerical performance. In strong contrast to traditional finite difference approaches, it frees us from the disturbing choice for the difference step size, and one can fully rely on the method without any accuracy concerns. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：In this article, we develop a residual-driven adaptive Gaussian mixture approximation (RD-AGMA) for Bayesian inverse problems. The posterior distribution is often non-Gaussian in practical Bayesian inference. To obtain a good approximation of the posterior, we provide the adaptive Gaussian mixture approximation (GMA) based on a residual. For GMA, the clustering of ensemble samples provides the predictor of means, covariances and weights by smoothed expectation-maximization (SmEM). SmEM can overcome the singularity of covariance matrix for small ensemble size. Then the parameters of GMA are updated by an iterative ensemble smoother (IES). To enhance clustering efficiency, the ensemble samples for the clustering are updated by IES as well. Since the goal of inverse problems is to minimize the residual between the observation data and the model response, the adaptive GMA of the posterior is constructed through a residual threshold. The mixture components with large residuals will be discarded in the adaptive procedure. When the prior is incorporated into the likelihood model, small residuals can drive the AGMA close to the true posterior. In the proposed method, a large number of samples can be efficiently drawn from the posterior distribution of GMA. A few numerical examples are presented to demonstrate the efficacy of RD-AGMA with applications in multimodal inversion and channel identification for subsurface flow problems in porous media. (C) 2021 Elsevier B.V. All rights reserved.
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