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Eigenproblem Basics and Algorithms

SYMMETRY-BASEL (2023)

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Article Statistics & Probability

Eigen Selection in Spectral Clustering: A Theory-Guided Practice

Xiao Han et al.

Summary: Based on a Gaussian mixture type model, this paper proposes eigen selection procedures to improve spectral clustering algorithms in high-dimensional settings. The procedures drop eigenvectors with no clustering power and eigenvectors corresponding to smaller spiked eigenvalues due to estimation inaccuracy. The newly proposed algorithms, ESSC and GESSC, demonstrate better stability and superior performance compared to canonical alternatives in extensive simulation and real data studies.

JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION (2023)

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Article Environmental Sciences

Multiscale Feature Fusion for the Multistage Denoising of Airborne Single Photon LiDAR

Shuming Si et al.

Summary: Compared with existing LiDAR modes, single-photon LiDAR (SPL) is more efficient in acquiring terrain data. However, the collected point cloud data from SPL contain a large amount of noisy points due to the photon-sensitive detectors. Existing denoising techniques based on sparsity assumption do not work for SPL point clouds, making it ineffective to remove the noisy points. To address this issue, we propose a novel multistage denoising strategy with fused multiscale features. Experimental results show that our approach outperforms existing denoising methods and significantly improves the denoising precision of airborne SPL point clouds, especially in water areas and dense urban areas.

REMOTE SENSING (2023)

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Article Mathematics

A One-Parameter Memoryless DFP Algorithm for Solving System of Monotone Nonlinear Equations with Application in Image Processing

Najib Ullah et al.

Summary: This article proposes a one-parameter scaling memoryless DFP algorithm for solving a system of monotone nonlinear equations with convex constraints. The algorithm is matrix and derivative-free with low memory requirements and is globally convergent under some mild conditions. Numerical comparison and image restoration problems demonstrate the efficiency of the algorithm.

MATHEMATICS (2023)

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Article Engineering, Electrical & Electronic

Eigenvalue Decomposition of a Parahermitian Matrix: Extraction of Analytic Eigenvectors

Stephan Weiss et al.

Summary: By analyzing parahermitian matrices, we can obtain an eigenvalue decomposition with analytic eigenvalues and eigenvectors. However, determining analytic eigenvectors is not straightforward, and we propose an iterative algorithm that converges to the best stationary point for extracting analytic eigenvectors.

IEEE TRANSACTIONS ON SIGNAL PROCESSING (2023)

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Article Physics, Multidisciplinary

Trimmed sampling algorithm for the noisy generalized eigenvalue problem

Caleb Hicks et al.

Summary: Solving the generalized eigenvalue problem is a useful method for finding energy eigenstates of large quantum systems. However, the process is susceptible to small errors, especially when using stochastic methods with significant error bars. In this work, the trimmed sampling algorithm is introduced to overcome this problem by sampling prior probability distributions determined by uncertainty estimates and physics-informed constraints. The method provides a probability distribution for the eigenvectors and observables with reliable error estimates, outperforming standard regularization methods.

PHYSICAL REVIEW RESEARCH (2023)

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Article Quantum Science & Technology

Quantum algorithms for the generalized eigenvalue problem

Jin-Min Liang et al.

Summary: The generalized eigenvalue problems are important in various fields. This paper presents a variational quantum algorithm to solve this problem by using loss functions and a weighting matrix to combine the trial state and eigenvectors. In addition, values and derivatives of the loss functions can be calculated on shallow quantum circuits. Finally, a full quantum generalized eigensolver is proposed with a quantum gradient descent algorithm.

QUANTUM INFORMATION PROCESSING (2022)

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Article Mathematics, Applied

TWICE IS ENOUGH FOR DANGEROUS EIGENVALUES

Andrew Horning et al.

Summary: We analyze the stability of eigensolvers that use rational filters to target interior eigenvalues. The results show that subspace iteration with a rational filter remains robust even when an eigenvalue is close to a filter's pole, and these dangerous eigenvalues self-correct in subsequent iterations.

SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS (2022)

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Article Geochemistry & Geophysics

Pre-conditioned BFGS-based uncertainty quantification in elastic full-waveform inversion

Qiancheng Liu et al.

Summary: In this study, the BFGS algorithm was modified to perform uncertainty quantification for large-scale applications, with the combination of random singular value decomposition for a low-rank approximation of the inverse Hessian. Different initial, diagonal Hessian matrices were used as pre-conditioners in elastic FWI applications to compare performance.

GEOPHYSICAL JOURNAL INTERNATIONAL (2022)

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Article Multidisciplinary Sciences

Binomial Distributed Data Confidence Interval Calculation: Formulas, Algorithms and Examples

Lorentz Jantschi

Summary: This paper discusses the challenge of mathematically evaluating the exact confidence interval for sampled outcomes, which follow a binomial distribution. It proposes three alternative methods for calculating confidence intervals and provides descriptions and examples for each.

SYMMETRY-BASEL (2022)

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Article Computer Science, Interdisciplinary Applications

Comparative study of inner-outer Krylov solvers for linear systems in structured and high-order unstructured CFD problems

Mehdi Jadoui et al.

Summary: This paper investigates advanced Krylov subspace methods for solving large sparse linear systems in stiff adjoint-based aerodynamic shape optimization problems. The flexible inner-outer GMRES strategy combined with relevant preconditioning and deflation techniques is especially focused on. The results demonstrate the effectiveness and scalability of the proposed methods in terms of robustness and convergence compared to standard GMRES solvers.

COMPUTERS & FLUIDS (2022)

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Article Environmental Sciences

A Novel Method for SAR Ship Detection Based on Eigensubspace Projection

Gaofeng Shu et al.

Summary: Synthetic Aperture Radar (SAR) is widely used in various fields due to its high-resolution imaging capability and all-weather operation. Ship detection in SAR images is crucial for marine applications, but ships can be easily masked by sea clutter, leading to missed detection. This study proposes a ship detection method based on eigensubspace projection (ESSP) in SAR images, which accurately extracts ship targets and reduces false alarms, demonstrating significant advantages compared to other methods in terms of detection accuracy and timeliness.

REMOTE SENSING (2022)

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Article Mathematics, Applied

Low synchronization Gram-Schmidt and generalized minimal residual algorithms

Katarzyna Swirydowicz et al.

Summary: The Gram-Schmidt process uses orthogonal projection to construct the QR factorization of a matrix, and approximate projections with form P = I - QTQ(T) can improve orthogonality. New variants of modified Gram-Schmidt algorithms introduce a compact WY representation.

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS (2021)

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Article Environmental Sciences

Analysis of Factors Influencing the Lake Area on the Tibetan Plateau Using an Eigenvector Spatial Filtering Based Spatially Varying Coefficient Model

Zhexin Xiong et al.

Summary: This study utilized the ESF-SVC model to analyze the relationship between lake area and climate and terrain factors in the inner watershed of the Tibet Plateau. The results showed significant differences in the influence of various factors on lake expansion, with each factor having varying effects over time and across regions.

REMOTE SENSING (2021)

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Article Chemistry, Multidisciplinary

WAMs Based Eigenvalue Space Model for High Impedance Fault Detection

Gian Paramo et al.

Summary: High impedance faults pose challenges for detection and location in power system protection. This study introduces a hybrid data-driven and analytical model for detection, utilizing eigenvalue drift coefficient for generalization. Case studies on the IEEE 5-Bus system show promising results for real-life applications.

APPLIED SCIENCES-BASEL (2021)

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Article Plant Sciences

Characterization of Root System Architecture Traits in Diverse Soybean Genotypes Using a Semi-Hydroponic System

Shuo Liu et al.

Summary: Phenotypic variation and correlations among root traits in soybean varieties play a crucial role in plant growth and adaptation strategies. By using a semi-hydroponic system, this study explored the variability in root morphological traits of 171 soybean genotypes, identifying significant positive correlations between root length/mass and shoot mass. The reliability of the semi-hydroponic system in phenotyping root trait variability at the early growth stage in soybean germplasms was validated through consistent rankings of genotypes in important root traits.

PLANTS-BASEL (2021)

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Article Mathematics

On Spectral Decomposition of States and Gramians of Bilinear Dynamical Systems

Alexey Iskakov et al.

Summary: The article demonstrates the unique decomposition of the state of a bilinear control system into generalized modes corresponding to the eigenvalues of the dynamics matrix. It also introduces the relationship between sub-Gramians and modal controllability and observability. Furthermore, it proposes an algorithm for computing Gramians and sub-Gramians based on element-wise computation of the solution matrix and presents a new criterion for the existence of solutions to the generalized Lyapunov equation.

MATHEMATICS (2021)

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Article Mathematics

Comparison of Molecular Geometry Optimization Methods Based on Molecular Descriptors

Donatella Balint et al.

Summary: This study analyzes 39 methods for optimizing molecule structures and classifies the results into three main groups, allowing for the selection of the appropriate method based on the specific problem at hand.

MATHEMATICS (2021)

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Article Computer Science, Artificial Intelligence

A novel algorithm for computing interior eigenpairs of large non-symmetric matrices

Ali Jamalian et al.

Summary: This paper presents a novel and efficient method, named Jaya-HA, for computing interior eigenpairs of large non-symmetric matrices. Numerical experiments show that this algorithm converges faster and provides superior performance.

SOFT COMPUTING (2021)

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Article Automation & Control Systems

A generalized minimal residual based iterative back propagation algorithm for polynomial nonlinear models

Jing Chen et al.

Summary: A back propagation algorithm for polynomial nonlinear models using generalized minimal residual method is proposed in this paper, which offers several advantages over the traditional gradient descent iterative algorithm. The algorithm is based on Arnoldi's method and can adaptively compute the step-length, making it suitable for large-scale system identification.

SYSTEMS & CONTROL LETTERS (2021)

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Review Cardiac & Cardiovascular Systems

The Myosin Myocardial Mesh Interpreted as a Biological Analogous of Nematic Chiral Liquid Crystals

Pierre-Simon Jouk et al.

Summary: The article discusses the myoarchitecture of the ventricular wall using polarised light imaging, highlighting the main direction of myosin molecules and a regional isotropy index. This method, referred to as PLI-MMA, shows similarities between the myosin mesh and a chiral liquid crystal, providing a more accurate characterization of the myoarchitecture compared to previous descriptions.

JOURNAL OF CARDIOVASCULAR DEVELOPMENT AND DISEASE (2021)

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Article Geochemistry & Geophysics

Pre-conditioned BFGS-based uncertainty quantification in elastic full-waveform inversion

Qiancheng Liu et al.

Summary: Full-waveform inversion is crucial in mapping geophysical subsurface structures, but lacks proper uncertainty quantification in current applications. This study enhances the BFGS algorithm for uncertainty quantification in large-scale applications, combining it with randomized singular value decomposition to obtain a low-rank approximation of the inverse Hessian. This approach is efficient and memory-affordable, demonstrating applicability for large-scale FWI and uncertainty quantification.

GEOPHYSICAL JOURNAL INTERNATIONAL (2021)

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Article Mathematics, Applied

Arnoldi Method for Large Quaternion Right Eigenvalue Problem

Qing-Wen Wang et al.

JOURNAL OF SCIENTIFIC COMPUTING (2020)

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Article Engineering, Electrical & Electronic

Admittance-Based Stability Analysis: Bode Plots, Nyquist Diagrams or Eigenvalue Analysis?

Lingling Fan et al.

IEEE TRANSACTIONS ON POWER SYSTEMS (2020)

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Article Multidisciplinary Sciences

An Optimal Fourth Order Derivative-Free Numerical Algorithm for Multiple Roots

Sunil Kumar et al.

SYMMETRY-BASEL (2020)

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Article Mathematics, Applied

Data-driven spectral decomposition and forecasting of ergodic dynamical systems

Dimitrios Giannakis

APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS (2019)

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Article Multidisciplinary Sciences

The Eigenproblem Translated for Alignment of Molecules

Lorentz Jantschi

SYMMETRY-BASEL (2019)

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On a Class of Optimal Fourth Order Multiple Root Solvers without Using Derivatives

Janak Raj Sharma et al.

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Article Mathematics, Applied

A double parameter scaled BFGS method for unconstrained optimization

Neculai Andrei

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS (2018)

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A new real structure-preserving quaternion QR algorithm

Zhigang Jia et al.

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS (2018)

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Article Automation & Control Systems

Stability Analysis of Polynomially Dependent Systems by Eigenvalue Perturbation

Jie Chen et al.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL (2017)

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Article Chemistry, Physical

A new efficient method for the calculation of interior eigenpairs and its application to vibrational structure problems

Taras Petrenko et al.

JOURNAL OF CHEMICAL PHYSICS (2017)

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Extending the Characteristic Polynomial for Characterization of C20 Fullerene Congeners

Dan-Marian Joita et al.

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Randomized algorithms for generalized Hermitian eigenvalue problems with application to computing Karhunen-Loeve expansion

Arvind K. Saibaba et al.

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS (2016)

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Article Mathematics, Applied

Accurate eigenvalue decomposition of real symmetric arrowhead matrices and applications

Nevena Jakovcevic Stor et al.

LINEAR ALGEBRA AND ITS APPLICATIONS (2015)

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Article Chemistry, Medicinal

Molecular Design and QSARs/QSPRs with Molecular Descriptors Family

Sorana D. Bolboaca et al.

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Article Statistics & Probability

Maximum likelihood estimation of parameter structures for the Wishart distribution using constraints

H. F. Strydom et al.

JOURNAL OF STATISTICAL PLANNING AND INFERENCE (2013)

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Article Mathematics, Applied

Every invertible matrix is diagonally equivalent to a matrix with distinct eigenvalues

Man-Duen Choi et al.

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Article Mathematics

Francis's Algorithm

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AMERICAN MATHEMATICAL MONTHLY (2011)

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Predictivity Approach for Quantitative Structure-Property Models. Application for Blood-Brain Barrier Permeation of Diverse Drug-Like Compounds

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INTERNATIONAL JOURNAL OF MOLECULAR SCIENCES (2011)

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A numerical method for polynomial eigenvalue problems using contour integral

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JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS (2010)

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An algebraic method for Schrodinger equations in quaternionic quantum mechanics

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Chromatographic retention times of polychlorinated biphenyls:: from structural information to property characterization

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Lorentz Jaentschi et al.

INTERNATIONAL JOURNAL OF MOLECULAR SCIENCES (2007)

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Algorithm 826: A parallel eigenvalue routine for complex Hessenberg matrices

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