Alexey VoroninYunhui HeScott MacLachlanLuke N. Olson...
17页
查看更多>>摘要：Abstract A well‐known strategy for building effective preconditioners for higher‐order discretizations of some PDEs, such as Poisson's equation, is to leverage effective preconditioners for their low‐order analogs. In this work, we show that high‐quality preconditioners can also be derived for the Taylor–Hood discretization of the Stokes equations in much the same manner. In particular, we investigate the use of geometric multigrid based on the ?1iso?2/?1 discretization of the Stokes operator as a preconditioner for the ?2/?1 discretization of the Stokes system. We utilize local Fourier analysis to optimize the damping parameters for Vanka and Braess–Sarazin relaxation schemes and to achieve robust convergence. These results are then verified and compared against the measured multigrid performance. While geometric multigrid can be applied directly to the ?2/?1 system, our ultimate motivation is to apply algebraic multigrid within solvers for ?2/?1 systems via the ?1iso?2/?1 discretization, which will be considered in a companion paper.
查看更多>>摘要：Abstract Recently, Zhao, Zheng, Liang and Xu (A locally and cubically convergent algorithm for computing ??‐eigenpairs of symmetric tensors. Numer Linear Algebra Appl, 2020, 27:e2284) studied on an efficient method for computing ??‐eigenpairs of symmetric tensors. Whereas, symmetric tensors are just special tensors. This article is concerned with the computation of ??‐eigenpairs of general real tensors. We propose a Rayleigh quotient‐gradient neural network model (RGNN) for computing ??‐eigenpairs of a general real tensor and the Euler‐type difference rule is used to discretize RGNN model. Theoretical analysis of the convergence for RGNN model is provided. Numerical experiments show that our method can capture all ??‐eigenpairs for some small‐scale general tensors and enjoys efficient computation for large‐scale tensors.
查看更多>>摘要：Abstract A two‐sided Rayleigh quotient iteration is proposed for the periodic eigenvalue problem of a formal matrix product. Local convergence analysis is presented and the cubic local convergence rate is proved. Numerical examples are provided for testing the proposed method.
查看更多>>摘要：Abstract In this article, we provide a bidiagonal decomposition of the Wronskian matrices of Bernstein bases of polynomials and other related bases such as the Bernstein basis of negative degree or the negative binomial basis. The mentioned bidiagonal decompositions are used to achieve algebraic computations with high relative accuracy for these Wronskian matrices. The numerical experiments illustrate the accuracy obtained using the proposed decomposition when computing inverse matrices, eigenvalues or singular values, and the solution of some related linear systems.
查看更多>>摘要：Abstract Compared to the classical Lanczos algorithm, the s‐step Lanczos variant has the potential to improve performance by asymptotically decreasing the synchronization cost per iteration. However, this comes at a price; despite being mathematically equivalent, the s‐step variant may behave quite differently in finite precision, potentially exhibiting greater loss of accuracy and slower convergence relative to the classical algorithm. It has previously been shown that the errors in the s‐step version follow the same structure as the errors in the classical algorithm, but are amplified by a factor depending on the square of the condition number of the O(s)‐dimensional Krylov bases computed in each outer loop. As the condition number of these s‐step bases grows (in some cases very quickly) with s, this limits the s values that can be chosen and thus can limit the attainable performance. In this work, we show that if a select few computations in s‐step Lanczos are performed in double the working precision, the error terms then depend only linearly on the conditioning of the s‐step bases. This has the potential for drastically improving the numerical behavior of the algorithm with little impact on per‐iteration performance. Our numerical experiments demonstrate the improved numerical behavior possible with the mixed precision approach, and also show that this improved behavior extends to mixed precision s‐step CG. We present preliminary performance results on NVIDIA V100 GPUs that show that the overhead of extra precision is minimal if one uses precisions implemented in hardware.
查看更多>>摘要：Abstract In optimization theory, to speed up the convergence of iterative procedures, many mathematicians often use the inertial extrapolation method. In this article, based on the three‐term derivative‐free method for solving monotone nonlinear equations with convex constraints [Calcolo, 2016;53(2):133‐145], we design an inertial algorithm for finding the solutions of nonlinear equation with monotone and Lipschitz continuous operator. The convergence analysis is established under some mild conditions. Furthermore, numerical experiments are implemented to illustrate the behavior of the new algorithm. The numerical results have shown the effectiveness and fast convergence of the proposed inertial algorithm over the existing algorithm. Moreover, as an application, we extend this method to solve the LASSO problem to decode a sparse signal in compressive sensing. Performance comparisons illustrate the effectiveness and competitiveness of the method.
查看更多>>摘要：Abstract In this article, we consider how to accurately solve the eigenvalue problem for a class of quasi‐rational Bernstein–Vandermonde (q‐RBV) matrices. This class of matrices belongs to generalized sign regular matrices with signature (1,…,1,?1). An algorithm is developed to accurately compute the parameter matrix for q‐RBV matrices. Based on the parameter matrix, all the eigenvalues of q‐RBV matrices have been computed to high relative accuracy. The perturbation theory for the eigenvalues of q‐RBV matrices and the error analysis of our proposed algorithm are provided. Numerical experiments are performed to confirm the claimed high relative accuracy.
NSTL
Wiley