Article
Engineering, Multidisciplinary
Dionysios Panagiotopoulos, Wim Desmet, Elke Deckers
Summary: The Automatic Krylov subspaces Recycling algorithm (AKR) automates the selection of Krylov subspaces to be recycled and generates a basis for accurate approximations of the solution for a parametric system. The algorithm offers a balance between solution accuracy and memory requirements, providing a predefined residual level and permitting predetermination of a threshold for maximum memory employed. AKR proves to be efficient for systems with clustered eigenvalues and requires fewer system assemblies compared to alternative reduced basis methods.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Mathematics, Applied
Qiang Niu, Hui Zhang, Youzhou Zhou
Summary: In this note, the Vandermonde with Arnoldi method is extended to handle the confluent Vandermonde matrix. A theorem is established for finding Krylov subspaces for any order derivatives, enabling precise computations and various applications.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Multidisciplinary Sciences
Zhao-Li Shen, Hao Yang, Bruno Carpentieri, Xian-Ming Gu, Chun Wen
Summary: The PageRank model calculates the importance or centrality of nodes in a network and has broad applications in various fields. By finding the unit positive eigenvector corresponding to the largest eigenvalue of a transition matrix, the model represents the importance of nodes mathematically. A novel preconditioning approach is proposed to improve efficiency and parallelism in solving the PageRank model, demonstrating faster convergence speed and maintaining superiority over other methods in numerical experiments.
Article
Engineering, Mechanical
Marcos Souza Lenzi, Leandro Fleck Fadel Miguel, Rafael Holdorf Lopez, Humberto Brambila de Salles
Summary: Determining the dynamic response to earthquakes is crucial for seismic design of buildings, but it often involves a significant computational burden. Non-modal Model Order Reduction (MOR) strategies, originally developed in control engineering, have gained attention in earthquake engineering due to their ability to handle non-classically damped systems. This paper presents an original non-modal MOR framework for controlled and uncontrolled buildings subjected to seismic excitations, which combines second-order Krylov subspaces and the Adaptative Windowing Algorithm (AWA) to reduce computational burden. Application cases demonstrate that the proposed scheme significantly accelerates computational time, achieving a speed-up factor (SUF) of more than 50.
JOURNAL OF THE BRAZILIAN SOCIETY OF MECHANICAL SCIENCES AND ENGINEERING
(2023)
Article
Engineering, Multidisciplinary
Dionysios Panagiotopoulos, Wim Desmet, Elke Deckers
Summary: This article introduces a model order reduction technique for acoustic boundary element method (BEM) systems using a multiparameter Krylov subspaces recycling scheme. The proposed scheme enables fast solution of parametric acoustic systems and reduces the required sampling.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
(2022)
Article
Mathematics, Applied
Alessandro Buccini
Summary: The Alternating Direction Multipliers Method (ADMM) is a popular algorithm for solving optimization problems and has been widely used for ill-posed inverse problems. This work proposes a computationally attractive implementation of ADMM by utilizing Generalized Krylov Subspaces (GKS) to decrease the computational cost. The method shows good performances in various computed examples.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
M. A. Botchev
Summary: The AccuRT restarting method aims to address accuracy loss in computing matrix exponential actions of nonsymmetric matrices using the SAI Krylov subspace method. By adjusting the shift value, the algorithm improves accuracy and efficiency, while requiring only a single LU factorization or preconditioner setup of the shifted matrix. Numerical experiments demonstrate the enhanced performance of this approach.
COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
(2021)
Article
Mathematics, Applied
Akira Imakura, Tetsuya Sakurai
Summary: This paper proposes an improved method based on block Arnoldi iteration to enhance the convergence behavior of the block SS-CAA method in complex moment-based eigensolvers. The proposed method utilizes two Krylov subspaces for high-order complex moments and the iteration technique, resulting in a higher convergence rate compared to the block SS-CAA method and the FEAST eigensolver.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Computer Science, Software Engineering
Hussam Al Daas, Laura Grigori, Pascal Henon, Philippe Ricoux
Summary: This article discusses deflation strategies for recycling Krylov subspace methods in solving linear systems, introducing various techniques such as Ritz- and harmonic Ritz-based deflation. Through numerical experiments in reservoir simulation, the impact of these strategies is demonstrated.
ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE
(2021)
Article
Automation & Control Systems
Mohammad Fahim Shakib, Giordano Scarciotti, Alexander Yu. Pogromsky, Alexey Pavlov, Nathan van de Wouw
Summary: Model reduction by moment matching is a reduction technique for linear time-invariant models that has a clear interpretation in the Laplace domain. Specifically, for the multiple-input multiple-output case, Krylov subspace methods aim to match the transfer-function matrix of the reduced-order model to that of the full-order model along tangential directions at desired interpolation points. Direct application of time-domain moment matching to MIMO models does not result in this kind of match in the transfer-function matrix. In this paper, we derive a relation between the transfer-function matrices of the full- and reduced-order models in the MIMO case, and formulate conditions on the parameters of time-domain moment matching to ensure consistency with classical Krylov subspace methods.
Article
Computer Science, Interdisciplinary Applications
Ryan M. Zbikowski, Calvin W. Johnson
Summary: The Lanczos algorithm is an efficient matrix eigensolver for high-dimensional problems. The computational cost depends on the number of matrix-vector multiplications. Block Lanczos algorithm improves efficiency by using matrix-matrix multiplication instead, but requires more multiplications to converge. Using an initial block constructed from approximate eigenvectors reduces the number of multiplications and improves computational speed.
COMPUTER PHYSICS COMMUNICATIONS
(2023)
Article
Mathematics, Applied
Pengwen Chen, Chung-Kuan Cheng, Xinyuan Wang
Summary: The study introduces a stability preserved Arnoldi algorithm for matrix exponential in the time domain simulation of large-scale power delivery networks, utilizing the range and null space of the system operator. By modifying the orthogonality in the Krylov subspace and adjusting the numerical ranges, theoretical convergence analysis for computing phi-functions is obtained. Simulations on RLC networks demonstrate the effectiveness of the Arnoldi algorithm.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Divya Aggarwal, Samrith Ram
Summary: This paper discusses the problem of T-splitting subspaces for a linear operator T, which involves finding subspaces of a vector space V that can be expressed as the direct sum of a subspace W and the powers of T. The paper also explores the connection between this problem and Krylov spaces, as well as polynomial matrices.
FINITE FIELDS AND THEIR APPLICATIONS
(2022)
Article
Mathematics, Applied
Fatemeh P. A. Beik, Mehdi Najafi-Kalyani
Summary: A framework for left/right preconditioning of multi-linear systems with Einstein product was proposed in this paper, and the inverse of preconditioned tensor was analytically derived. The feasibility of preconditioned Krylov subspace methods based on Hessenberg process was experimentally illustrated, and their performances were compared with those based on the Arnoldi process.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
A. Buccini, P. Diaz de Alba, F. Pes, L. Reichel
Summary: The solution of nonlinear inverse problems is a challenging task in numerical analysis. We propose an efficient implementation of the popular Gauss-Newton method for large-scale problems. Our implementation utilizes Generalized Krylov Subspaces and incorporates secant updates to reduce computational cost. We demonstrate convergence of the proposed methods and present numerical examples to illustrate their performance.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
