A Novel Symmetric Skew-Hamiltonian Isotropic Lanczos Algorithm for Spectral Conformal Parameterizations

Huang, Wei-Qiang; Gu, Xianfeng; Lin, Wen‐Wei; Yau, Shing–Tung

doi:10.1007/s10915-014-9840-2

Cited by 9 publications

(9 citation statements)

References 32 publications

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“…Furthermore, we apply the deflating approach to exclude the influence of convergence eigenvalues so as to focus on capturing the smallest positive one and detect a few desired eigenvalues in order. Numerical experiments show that, compared with the numerical implementations of traditional diagonal perturbation eigs and the null-space deflation eigensolver ndeigs [11] using direct methods for inner linear systems, the new derived algorithm iSIRA, integrated SIRA (SIRA with trimming and deflating techniques), reveals more efficient and feasible when a LU factorization of the graph Laplacian matrix is not available.…”

Section: Discussionmentioning

confidence: 99%

“…and ndeigs methods. It is worth to mention that the rank-two correction strategy in ndeigs [11], our previous method, makes the performance be superior to eigs.…”

Section: Codementioning

confidence: 99%

“…In this paper we first remedy the problem of singularity of the graph Laplacian matrix. Then, we introduce a technique of eigenvalue deflations as shown in [9,11,28,34] so that we can find the desired eigenvalues in order and exclude the influence of the convergent ones. Finally, we integrate these approaches into the eigensolver called shift-invert residual Arnoldi (SIRA) method [18,19].…”

Section: The Graph Laplacian Matrixmentioning

confidence: 99%

“…Beside, we also apply the idea of null-space deflation in [11], proposed by the part authors (Huang, Lin and Yau) of this paper, to the graph Laplacian matrix. In short, using a rank-two correction, the matrix L is transformed to L d + e 1 1 n −e 1 1 n , where L d := L + e 1 e 1 .…”

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confidence: 99%

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iSIRA: Integrated shift–invert residual Arnoldi method for graph Laplacian matrices from big data

Wei

Lin

et al. 2019

Journal of Computational and Applied Mathematics

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The eigenvalue problem of a graph Laplacian matrix L arising from a simple, connected and undirected graph has been given more attention due to its extensive applications, such as spectral clustering, community detection, complex network, image processing and so on. The associated graph Laplacian matrix is symmetric, positive semi-definite, and is usually large and sparse. Computing some smallest positive eigenvalues and corresponding eigenvectors is often of interest.However, the singularity of L makes the classical eigensolvers inefficient since we need to factorize L for the purpose of solving large and sparse linear systems exactly. The next difficulty is that it is usually time consuming or even unavailable to factorize a large and sparse matrix arising from real network problems from big data such as social media transactional databases, and sensor systems because there is in general not only local connections.In this paper, we propose an eignsolver based on the inexact residual Arnoldi [18,19] method together with an implicit remedy of the singularity and an effective deflation for convergent eigenvalues. Numerical experiments reveal that the integrated eigensolver outperforms the classical Arnoldi/Lanczos method for computing some smallest positive eigeninformation provided the LU factorization is not available.

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Section: Discussionmentioning

confidence: 99%

“…and ndeigs methods. It is worth to mention that the rank-two correction strategy in ndeigs [11], our previous method, makes the performance be superior to eigs.…”

Section: Codementioning

confidence: 99%

Section: The Graph Laplacian Matrixmentioning

confidence: 99%

mentioning

confidence: 99%

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iSIRA: Integrated shift–invert residual Arnoldi method for graph Laplacian matrices from big data

Wei

Lin

et al. 2019

Journal of Computational and Applied Mathematics

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“…In addition, for a simply connected open surface, some efficient numerical methods including the fast disk mapping [19], the linear disk mapping [16], and the conformal energy minimization (CEM) [75] have been recently developed to map it conformally to a unit disk. Furthermore, classical methods including discrete conformal parameterization [21], least-squares conformal mapping [50], angle-based flattening [62,63,77], discrete conformal equivalence [66], spectral conformal parameterization [53,44], and orbifold Tutte embedding [8,9,7] have been proposed for the conformal parameterization of topological disks with free or mild shape constraints of boundaries. For conformal parameterizations of higher genus surfaces, some well-known methods have been developed, including the holomorphic one-form [36,37,46], the discrete Ricci flow [45,78], and the discrete Calabi flow [79].…”

Section: Previous Workmentioning

confidence: 99%

A Novel Algorithm for Volume-Preserving Parameterizations of 3-Manifolds

Yueh¹,

Li²,

Lin³

et al. 2019

SIAM J. Imaging Sci.

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Manifold parameterizations have been applied to various fields of commercial industries. Several efficient algorithms for the computation of triangular surface mesh parameterizations have been proposed in recent years. However, the computation of tetrahedral volumetric mesh parameterizations is more challenging due to the fact that the number of mesh points would become enormously large when the higher-resolution mesh is considered and the bijectivity of parameterizations is more difficult to guarantee. In this paper, we develop a novel volumetric stretch energy minimization algorithm for volume-preserving parameterizations of simply connected 3-manifolds with a single boundary under the restriction that the boundary is a spherical area-preserving mapping. In addition, our algorithm can also be applied to compute spherical angle-and area-preserving parameterizations of genus-zero closed surfaces, respectively. Several numerical experiments indicate that the developed algorithms are more efficient and reliable compared to other existing algorithms. Numerical results on applications of the manifold partition and the mesh processing for three-dimensional printing are demonstrated thereafter to show the robustness of the proposed algorithm.

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An Efficient Energy Minimization for Conformal Parameterizations

Yueh

Lin

et al. 2017

J Sci Comput

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A Novel Symmetric Skew-Hamiltonian Isotropic Lanczos Algorithm for Spectral Conformal Parameterizations

Cited by 9 publications

References 32 publications

iSIRA: Integrated shift–invert residual Arnoldi method for graph Laplacian matrices from big data

iSIRA: Integrated shift–invert residual Arnoldi method for graph Laplacian matrices from big data

A Novel Algorithm for Volume-Preserving Parameterizations of 3-Manifolds

An Efficient Energy Minimization for Conformal Parameterizations

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