Numerical methods for large eigenvalue problems

By Danny C. Sorensen. Acta Numerica, 2002

1,528 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/0bca04b13185bc4b3f6d08e3ac9da18ca7206fe9

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In Computational Theory and Mathematics

Citations in the corpus

Efficient Sampling Set Selection for Bandlimited Graph Signals Using Graph Spectral Proxies (2015, methodology): “This can be done using block-based Rayleigh quotient minimization methods [37], block-based Kryolov subspace methods such as Arnoldi/Lanczos iterations or deflation methods in conjunction with single eigen-pair solvers [38].”
Active Subspace Methods in Theory and Practice: Applications to Kriging Surfaces (2013, positive): “Common to both of these fields are methods for matrix factorizations and eigenvalue computations [35], which are replete with subspace oriented approaches.”
Modal Analysis of Fluid Flows: An Overview (2017, methodology): “The theories and numerical algorithms for eigenvalue and singular value decompositions are not provided here but are discussed extensively in textbooks by Horn and Johnson [30], Golub and Loan [31], Trefethen and Bau [29], and Saad [32].”
A New Truncation Strategy for the Higher-Order Singular Value Decomposition (2012, background): “An orthogonal projector [46, 47] is a linear transformation P that projects a vector u ∈ R onto a vector space U ⊆ R such that the residual u − Pu is orthogonal to U .”
On the generation of large passive macromodels for complex interconnect structures (2006, positive): “the basic shifted Arnoldi process (see [26], [36] for an excellent review and a complete bibliographic database).”
A two-directional Arnoldi process and its application to parametric model order reduction (2009, neutral): “…Arnoldi process play an important role in modern computational techniques for large-scale matrix computation problems, such as solving linear systems of equations [13] [15] [23] [24], computing a few selected eigenpairs [3] [25] [21] and reduced-order modeling of dynamical systems [18] [2] [16].”
Gramian-Based Reduction Method Applied to Large Sparse Power System Descriptor Models (2008, background): “A drawback of these techniques is the fact that one must compute spectral projections onto the left and right deflating subspaces corresponding to the finite eigenvalues of the related pencil [16], [24], which is computationally prohibitive for largescale systems.”
Randomized Block Krylov Methods for Stronger and Faster Approximate Singular Value Decomposition (2015, methodology): “We refer the reader to Saad’s standard ref e ence [27].”
Neon2: Finding Local Minima via First-Order Oracles (2017, background): “Let us recall how the shift-and-invert (SI) approach [26] solves the minimum eigenvector problem.”
Critical Eigenvalues Tracing for Power System Analysis via Continuation of Invariant Subspaces and Projected Arnoldi Method (2007, methodology): “The orthonormal basis in the precondition matrix may be calculated by the Gram–Schmidt method or by the modified Gram–Schmidt method [2] and other sophisticated variants [32] for improved robustness.”
Handbook of Semidefinite Programming (2000, methodology): “, a few matrix vector multiplications will su ce to produce reasonable approximations to max(A) and its eigenvector (see [313] for a quick introduction or [736] for a detailed description of Lanczos methods).”
Augmented Implicitly Restarted Lanczos Bidiagonalization Methods (2005, methodology): “These methods are commonly referred to as restarted partial Lanczos bidiagonalization methods.”
Ergodic Theory, Dynamic Mode Decomposition, and Computation of Spectral Properties of the Koopman Operator (2016, methodology): “The basic idea of the Hankel-DMD method is to extract the Koopman spectra from this sequence, which is analogous to the idea of Krylov subspace methods for computing the eigenvalues of large matrices [35].”
Iterative methods for sparse linear systems (2003, methodology): “It is known [180] that the characteristic polynomial of the tridiagonal matrix produced by the Lanczos algorithm minimizes the norm k:kv1 over the monic polynomials.”
Efficient model reduction in non-linear dynamics using the Karhunen-Loève expansion and dual-weighted-residual methods (2003, methodology): “is calculated using the Arnoldi or Lanczos algorithm, cf. [ 44 ] for details.”