The smoothed finite element method (S-FEM) has been found to be an effective solution method for solid mechanics problems. This paper proposes an effective approach to compute the lower bound solution of free vibratio...
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The smoothed finite element method (S-FEM) has been found to be an effective solution method for solid mechanics problems. This paper proposes an effective approach to compute the lower bound solution of free vibration and the upper bound solution of the forced vibration of solid structures, by making use of the important softening effects of the node-based smoothed finite element method (NS-FEM). Through the gradient smoothing technique, the strain-displacement matrix is obtained in the smoothing domain based on the element mesh nodes. Subsequently, the stiffness matrix is computed in a manner consistent with the standard finite element method (FEM). Here, the practical lanczos algorithm and the modal superposition technique are employed to obtain the frequencies, modes, and transient responses of a given homogeneous structure. For three-dimensional (3D) solid structures, the automatically generated four-node tetrahedron (T4) element meshes are utilized. The results obtained from the NS-FEM are compared with the standard FEM in terms of accuracy, convergence and computational efficiency.
Spectral clustering and matrix factorization are two widely utilized algorithms for community detection. On the one hand, most existing spectral clustering algorithms focus on learning node representations from a well...
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Spectral clustering and matrix factorization are two widely utilized algorithms for community detection. On the one hand, most existing spectral clustering algorithms focus on learning node representations from a well-designed similarity matrix that thoroughly incorporates data attribute information. However, they often overlook the intrinsic attribute information embedded within the algorithm itself. On the other hand, the node representations generated by most existing matrix factorization algorithms often exhibit a lack of linear independence, leading to the presence of redundant information. Motivated by them, we propose an algorithm, SOCD (Selective Orthogonalization for Community Detection), which leverages the orthogonality loss of the lanczos algorithm to monitor whether eigenvalues converge or not, and saves the already converged eigenpairs, then the corresponding converged eigenvectors are employed for community structure detection. Meanwhile, the already converged eigenvectors continue to be orthogonalized against the lanczos vector to preserve the orthogonality of the lanczos algorithm. A large number of experiments conducted on real datasets with community labels demonstrate the superiority of our algorithm over its competitors. The experimental code is available for download at https://***/AnonSimRank/SOCD/.
We report energy levels of H $ _2 $ 2O-CO computed with a large product contracted (PC) basis. Intra-molecular levels are obtained using a large basis of products of contracted intra-molecular functions and contracted...
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We report energy levels of H $ _2 $ 2O-CO computed with a large product contracted (PC) basis. Intra-molecular levels are obtained using a large basis of products of contracted intra-molecular functions and contracted inter-molecular functions. To determine the size of the contracted inter-molecular basis required to achieve convergence, we compute inter-molecular levels without contracting the inter-molecular basis. We call this a $ \vert v, L \rangle $ |v,L > basis. We find that a large contracted inter-molecular basis is necessary. Previous calculations with a smaller inter-molecular basis have convergence errors similar to those caused by using the rigid-monomer approximation. For example, the largest relative splitting error is 37%. Owing to the size of the inter-molecular basis, a quadrature-point intermediate matrix $ \boldsymbol {F} $ F, rather than a DVR-point intermediate matrix $ \boldsymbol {F} $ F, and MPI parallelisation are important for reducing the calculation time.
We demonstrate how different computational approaches affect the performance of solving the generalized eigenvalue problem (GEP). The layered piezo device is studied for resonance frequencies using different meshes, s...
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We demonstrate how different computational approaches affect the performance of solving the generalized eigenvalue problem (GEP). The layered piezo device is studied for resonance frequencies using different meshes, sparse matrix representations, and numerical methods in COMSOL Multiphysics and ACELAN-COMPOS packages. Specifically, the matrix-vector and matrix-matrix product implementation for large sparse matrices is discussed. The shift-and-invert lanczos method is used to solve the partial symmetric GEP numerically. Different solvers are compared in terms of the efficiency. The results of numerical experiments are presented.
The dynamics of quantum systems unfolds within a subspace of the state space or operator space, known as the Krylov space. This review presents the use of Krylov subspace methods to provide an efficient description of...
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The lanczos algorithm is one of the principal methods for the computation of a small part of the eigenspectrum of large, sparse, real symmetric matrices. A single-vector, explicitly restarted variant of the lanczos me...
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The lanczos algorithm is one of the principal methods for the computation of a small part of the eigenspectrum of large, sparse, real symmetric matrices. A single-vector, explicitly restarted variant of the lanczos method is proposed in this paper. The algorithm finds only one eigenpair at a time using a deflation technique in which the lanczos factorization for the current eigenpair is generated in the null space of all previously computed eigenvectors. This approach yields a fixed k-step restarting scheme which permits short lanczos factorizations and almost completely eliminates the reorthogonalization among the lanczos vectors. The orthogonalization strategy developed falls naturally into the class of selective orthogonalization strategies as classified by Simon. 'Reverse communication' software for the implementation of the proposed variant on a Connection Machine CM-200 with 8K processors and on a Gray T3D with 32 processors is discussed. Test results on the CM-200 using examples from the Harvell-Boeing collection of sparse matrices show the method to be very effective when compared with Sorensen's state-of-the-art routine taken from the ARPACK library. The method has fixed, small storage requirements, copes easily with genuinely multiple eigenvalues and is guaranteed to converge to the desired eigenvalues. (C) 1999 Elsevier Science B.V. All rights reserved.
A convergence analysis for the nonsymmetric lanczos algorithm is presented. By using a tridiagonal structure of the algorithm, some identities concerning Ritz values and Ritz vectors are established and used to derive...
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A convergence analysis for the nonsymmetric lanczos algorithm is presented. By using a tridiagonal structure of the algorithm, some identities concerning Ritz values and Ritz vectors are established and used to derive approximation bounds. In particular, the analysis implies the classical results for the symmetric lanczos algorithm.
The finite or the semi-infinite discrete-time Toda lattice has many applications to various areas in applied mathematics. The purpose of this paper is to review how the Toda lattice appears in the lanczos algorithm th...
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The finite or the semi-infinite discrete-time Toda lattice has many applications to various areas in applied mathematics. The purpose of this paper is to review how the Toda lattice appears in the lanczos algorithm through the quotient-difference algorithm and its progressive form (pqd). Then a multistep progressive algorithm (MPA) for solving linear systems is presented. The extended lanczos parameters can be given not by computing inner products of the extended lanczos vectors but by using the pqd algorithm with highly relative accuracy in a lower cost. The asymptotic behavior of the pqd algorithm brings us some applications of MPA related to eigenvectors.
Nearest neighbor graphs are widely used in data mining and machine learning. A brute-force method to compute the exact kNN graph takes Theta(dn(2)) time for n data points in the d dimensional Euclidean space. We propo...
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Nearest neighbor graphs are widely used in data mining and machine learning. A brute-force method to compute the exact kNN graph takes Theta(dn(2)) time for n data points in the d dimensional Euclidean space. We propose two divide and conquer methods for computing an approximate kNN graph in Theta(dn(t)) time for high dimensional data (large d). The exponent t is an element of (1, 2) is an increasing function of an internal parameter a which governs the size of the common region in the divide step. Experiments show that a high quality graph can usually be obtained with small overlaps, that is, for small values of t. A few of the practical details of the algorithms are as follows. First, the divide step uses an inexpensive lanczos procedure to perform recursive spectral bisection. After each conquer step, an additional refinement step is performed to improve the accuracy of the graph. Finally, a hash table is used to avoid repeating distance calculations during the divide and conquer process. The combination of these techniques is shown to yield quite effective algorithms for building kNN graphs.
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