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For the discrete linear systems resulted from the discretization of the one‐dimensional anisotropic spatial fractional diffusion equations of variable coefficients with the shifted finite‐difference formulas of the Grünwald–Letnikov type, we propose a class of respectively scaled Hermitian and skew‐Hermitian splitting iteration method and establish its asymptotic convergence theory. The corresponding induced matrix splitting preconditioner, through further replacements of the involved Toeplitz matrices with certain circulant matrices, leads to an economic variant that can be executed by fast Fourier transforms. Both theoretical analysis and numerical implementations show that this fast respectively scaled Hermitian and skew‐Hermitian splitting preconditioner can significantly improve the computational efficiency of the Krylov subspace iteration methods employed as effective linear solvers for the target discrete linear systems. 相似文献
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Equivariant matrices, commuting with a group of permutation matrices, are considered. Such matrices typically arise from PDEs
and other computational problems where the computational domain exhibits discrete geometrical symmetries. In these cases,
group representation theory provides a powerful tool for block diagonalizing the matrix via the Generalized Fourier Transform
(GFT). This technique yields substantial computational savings in problems such as solving linear systems, computing eigenvalues
and computing analytic matrix functions such as the matrix exponential.
The paper is presenting a comprehensive self contained introduction to this field. Building upon the familiar special (finite
commutative) case of circulant matrices being diagonalized with the Discrete Fourier Transform, we generalize the classical
convolution theorem and diagonalization results to the noncommutative case of block diagonalizing equivariant matrices.
Applications of the GFT in problems with domain symmetries have been developed by several authors in a series of papers. In
this paper we elaborate upon the results in these papers by emphasizing the connection between equivariant matrices, block
group algebras and noncommutative convolutions. Furthermore, we describe the algebraic structure of projections related to
non-free group actions. This approach highlights the role of the underlying mathematical structures, and provides insight
useful both for software construction and numerical analysis. The theory is illustrated with a selection of numerical examples.
AMS subject classification (2000) 43A30, 65T99, 20B25 相似文献
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Jan Gregorovič 《Differential Geometry and its Applications》2012,30(5):450-476
First we introduce a generalization of symmetric spaces to parabolic geometries. We provide construction of such parabolic geometries starting with classical symmetric spaces and we show that all regular parabolic geometries with smooth systems of involutive symmetries can be obtained in this way. Further, we investigate the case of parabolic contact geometries in great detail and we provide the full classification of those with semisimple groups of symmetries without complex factors. Finally, we explicitly construct all non-trivial contact geometries with non-complex simple groups of symmetries. We also indicate geometric interpretations of some of them. 相似文献
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Lenka Zalabov 《Differential Geometry and its Applications》2009,27(5):605-622
We generalize the concept of affine locally symmetric spaces for parabolic geometries. We discuss mainly |1|-graded geometries and we show some restrictions on their curvature coming from the existence of symmetries. We use the theory of Weyl structures to discuss more interesting |1|-graded geometries which can carry a symmetry in a point with nonzero curvature. More concretely, we discuss the number of different symmetries which can exist at the point with nonzero curvature. 相似文献
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Zhong-Zhi Bai 《Numerical Linear Algebra with Applications》2024,31(3):e2510
For solving large sparse systems of linear equations, we construct a paradigm of two-step matrix splitting iteration methods and analyze its convergence property for the nonsingular and the positive-definite matrix class. This two-step matrix splitting iteration paradigm adopts only one single splitting of the coefficient matrix, together with several arbitrary iteration parameters. Hence, it can be constructed easily in actual applications, and can also recover a number of representatives of the existing two-step matrix splitting iteration methods. This result provides systematic treatment for the two-step matrix splitting iteration methods, establishes rigorous theory for their asymptotic convergence, and enriches algorithmic family of the linear iteration solvers, for the iterative solutions of large sparse linear systems. 相似文献
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《Journal of Pure and Applied Algebra》2022,226(9):106960
Building upon Vaughan Pratt's work on applications of Chu space theory to Stone duality, we develop a general theory of categorical dualities on the basis of Chu space theory and closure conditions, which encompasses a variety of dualities for topological spaces, convex spaces, closure spaces, and measurable spaces (some of which are new duality results on their own). It works as a general method to generate analogues of categorical dualities between frames (locales) and topological spaces beyond topology, e.g., for measurable spaces, convex spaces, and closure spaces. After establishing the Chu duality theory, we apply the state-observable duality between quantum lattices and closure spaces to coalgebraic representations of quantum symmetries, showing that the quantum symmetry groupoid fully embeds into a purely coalgebraic category, i.e., the category of Born coalgebras, which refines, through the quantum duality that follows from Chu duality theory, Samson Abramsky's fibred coalgebraic representations of quantum symmetries (which, in turn, builds upon his Chu representations of symmetries). 相似文献
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This paper introduces a numerical method to localize inclusions having slightly different elastic coefficients than those of a fully saturated poroelastic matrix, whose detection is often difficult. This method can be used to find weakly stiffer or softer objects in saturated soils or diseased biological tissues at early stages. To this end, we propose a reduced model from the Biot’s equations by splitting the fluid pressure into two parts: one embedded into an elasticity model and the other one used as a corrector term. By applying the small amplitude homogenization method, we can successfully retrieve the position and extension of inclusions in poroelastic media employing this simplified model. Numerical results show a good agreement for the location of inclusions when the contrast is below 30% stiffer or softer than the matrix, and for a noise level up to 5% for frequencies below 50 Hz. 相似文献
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P. N. Vabishchevich 《Computational Mathematics and Mathematical Physics》2014,54(6):953-962
In this paper, the schemes of the alternating triangular method are set out in the class of splitting methods used for the approximate solution of Cauchy problems for evolutionary problems. These schemes are based on splitting the problem operator into two operators that are conjugate transposes of each other. Economical schemes for the numerical solution of boundary value problems for parabolic equations are designed on the basis of an explicit-implicit splitting of the problem operator. The alternating triangular method is also of interest for the construction of numerical algorithms that solve boundary value problems for systems of partial differential equations and vector systems. The conventional schemes of the alternating triangular method used for first-order evolutionary equations are two-level ones. The approximation properties of such splitting methods can be improved by transiting to three-level schemes. Their construction is based on a general principle for improving the properties of difference schemes, namely, on the regularization principle of A.A. Samarskii. The analysis conducted in this paper is based on the general stability (or correctness) theory of operator-difference schemes. 相似文献
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In this paper, we contribute an operator-splitting method improved by the Zassenhaus product. Zassenhaus products are of fundamental importance for the theory of Lie groups and Lie algebras. While their applications in physics and physical chemistry are important, novel applications in CFD (computational fluid dynamics) arose based on the fact that their sparse matrices can be seen as generators of an underlying Lie algebra. We apply this to classical splitting and the novel Zassenhaus product formula. The underlying analysis for obtaining higher order operator-splitting methods based on the Zassenhaus product is presented. The benefits of dealing with sparse matrices, given by spatial discretization of the underlying partial differential equations, are due to the fact that the higher order commutators are very quickly computable (their matrix structures thin out and become nilpotent). When applying these methods to convection-diffusion-reaction equations, the benefits of balancing time and spatial scales can be used to accelerate these methods and take into account these sparse matrix structures.The verification of the improved splitting methods is done with numerical examples. Finally, we conclude with higher order operator-splitting methods. 相似文献
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The modulus‐based matrix splitting algorithms for a class of weakly nonlinear complementarity problems 下载免费PDF全文
In this paper, we study a class of weakly nonlinear complementarity problems arising from the discretization of free boundary problems. By reformulating the complementarity problems as implicit fixed‐point equations based on splitting of the system matrices, we propose a class of modulus‐based matrix splitting algorithms. We show their convergence by assuming that the system matrix is positive definite. Moreover, we give several kinds of typical practical choices of the modulus‐based matrix splitting iteration methods based on the different splitting of the system matrix. Numerical experiments on two model problems are presented to illustrate the theoretical results and examine the numerical effectiveness of our modulus‐based matrix splitting algorithms. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
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Based on the fact that the Painlevé equations can be written as Hamiltonian systems with affine Weyl group symmetries, a canonical quantization of the Painlevé equations preserving such symmetries has been studied recently. On the other hand, since the Painlevé equations can also be described as isomonodromic deformations of certain second-order linear differential equations, a quantization of such Lax formalism is also a natural problem. In this paper, we introduce a canonical quantization of Lax equations for the Painlevé equations and study their symmetries. We also show that our quantum Lax equations are derived from Virasoro conformal field theory. 相似文献
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Summary. We present new theoretical results on two classes of multisplitting methods for solving linear systems iteratively. These
classes are based on overlapping blocks of the underlying coefficient matrix which is assumed to be a band matrix. We show that under suitable conditions the spectral radius of the iteration matrix does not depend on the weights of the method even if these weights are allowed to be negative. For a certain class of splittings
we prove an optimality result for with respect to the weights provided that is an M–matrix. This result is based on the fact that the multisplitting method can be represented by a single splitting
which in our situation surprisingly turns out to be a regular splitting. Furthermore we show by numerical examples that weighting
factors may considerably improve the convergence.
Received July 18, 1994 / Revised version received November 20, 1995 相似文献
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Chuan-Long Wang 《Applied mathematics and computation》2010,216(6):1687-1693
In this paper, we generalize the saddle point problem to general symmetric indefinite systems, we also present a kind of convergent splitting iterative methods for the symmetric indefinite systems. A special divergent splitting is introduced. The sufficient condition is discussed that the eigenvalues of the iteration matrix are real. The spectral radius of the iteration matrix is discussed in detail, the convergence theories of the splitting iterative methods for the symmetric indefinite systems are obtained. Finally, we present a preconditioner and discuss the eigenvalues of preconditioned matrix. 相似文献
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In this paper, we propose a preconditioned general modulus-based matrix splitting iteration method for solving modulus equations arising from linear complementarity problems. Its convergence theory is proved when the system matrix is an H+-matrix, from which some new convergence conditions can be derived for the (general) modulus-based matrix splitting iteration methods. Numerical results further show that the proposed methods are superior to the existing methods. 相似文献
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Hsuan-Ku Liu 《Applied mathematics and computation》2011,217(12):5259-5264
In this paper, homotopy perturbation methods (HPMs) are applied to obtain the solution of linear systems, and conditions are deduced to check the convergence of the homotopy series. Moreover, we have adapted the Richardson method, the Jacobi method, and the Gauss-Seidel method to choose the splitting matrix. The numerical results indicate that the homotopy series converges much more rapidly than the direct methods for large sparse linear systems with a small spectrum radius. 相似文献
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Hao Chen 《Numerical Algorithms》2016,73(4):1037-1054
In this paper we study efficient iterative methods for solving the system of linear equations arising from the fully implicit Runge-Kutta discretizations of a class of partial differential-algebraic equations. In each step of the time integration, a block two-by-two linear system is obtained and needed to be solved numerically. A preconditioning strategy based on an alternating Kronecker product splitting of the coefficient matrix is proposed to solve such linear systems. Some spectral properties of the preconditioned matrix are established and numerical examples are presented to demonstrate the effectiveness of this approach. 相似文献
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J.S.W. Lamb 《Journal of Differential Equations》2003,191(2):377-407
We study local bifurcation in equivariant dynamical systems from periodic solutions with a mixture of spatial and spatiotemporal symmetries.In previous work, we focused primarily on codimension one bifurcations. In this paper, we show that the techniques used in the codimension one analysis can be extended to understand also higher codimension bifurcations, including resonant bifurcations and mode interactions. In particular, we present a general reduction scheme by which we relate bifurcations from periodic solutions to bifurcations from fixed points of twisted equivariant diffeomorphisms, which in turn are linked via normal form theory to bifurcations from equilibria of equivariant vector fields.We also obtain a general theory for bifurcation from relative periodic solutions and we show how to incorporate time-reversal symmetries into our framework. 相似文献