Iteration Method for the Solution of the Eigenvalue Problem of Linear Differential and Integral Operators

By David R. Lide. 2018

948 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Numerical Methods & Scientific Computing

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Citations in the corpus

An Investigation into Neural Net Optimization via Hessian Eigenvalue Density (2019, methodology): “We use Lanczos algorithm [17] (with full re-orthogonalization) to perform this computation in a numerically stable manner.”
Quantum chaos and the complexity of spread of states (2022, neutral): “[13] The Recursion Method: Application to Many-Body Dynamics, vol.”
Survey of multifidelity methods in uncertainty propagation, inference, and optimization (2018, methodology): “Solvers based on Krylov subspace methods [121, 9, 122, 184] and on Anderson relaxation [5, 211, 202] perform intermediate computations in low-dimensional subspaces that are updated as the computation proceeds.”
On the Relation Between the Sharpest Directions of DNN Loss and the SGD Step Length (2018, methodology): “Therefore, we approximate the top K (up to 50 ) eigenvalues using the Lanczos algorithm (Lanczos, 1950; Dauphin et al., 2014), an extension of the power method, on approximately 5% of the training data (using more data was not beneﬁcial).”
Operator complexity: a journey to the edge of Krylov space (2020, background): “These coefficients arise in the process of orthonormalizing3 the Krylov basis [18, 19].”
Analyzing ImageNet with Spectral Relevance Analysis: Towards ImageNet un-Hans'ed (2019, methodology): “Speciﬁ-cally, we use the computationally eﬃcient and iterative Lanczos algorithm 1 [32], which has in general a computational complexity of O ( n 3 ), which reduces to O ( qn 2 ) when only the q ≤ n most discriminative eigenvectors are to be computed.”
GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration (2018, background): “These techniques have been around for decades, and are covered in popular books and papers [11, 12, 17, 32, 36, 37, 42].”
The Break-Even Point on Optimization Trajectories of Deep Neural Networks (2020, methodology): “Following Dauphin et al. (2014); Alain et al. (2019), we estimate the top eigenvalues and eigenvectors of H on a small subset of the training set (e.g. 5% in the case of CIFAR-10) using the Lanczos algorithm (Lanczos, 1950).”
Approximating Spectral Densities of Large Matrices (2013, positive): “We then describe a second class of methods based on the use of the familiar Lanczos partial tridiagonalization procedure [24].”
LanczosNet: Multi-Scale Deep Graph Convolutional Networks (2019, methodology): “he corresponding g t(X;S;:::;St) lies in the Krylov subspace K t+1(S;X :;i) spanfX :;i;SX :;i;:::;StX :;ig. This naturally motivates the usage of Krylov subspace methods, like the Lanczos algorithm [13], since it provides an orthonormal basis for the above Krylov subspace, thus making the ﬁlter coe”
Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution (2019, methodology): “Another family of approaches is variants of the iterative Lanczos method 18 .”