The Hierarchical Subspace Iteration Method for Laplace–Beltrami Eigenproblems

Comparison of Lanczos and subspace iterations for hyperspherical reaction path calculations

Chapter 4 - Self-Consistent Field using Direct Inversion in Iterative Subspace Method and Quasi-Newton Vectors

Construct validity of pixel-by-pixel DCE-MRI: Correlation with conventional MRI scores in juvenile idiopathic arthritis

In this paper, a new wideband direction-of-arrival (DOA) estimation algorithm, which is an iterative coherent subspace method based on two-dimensional discrete Fourier transform (2D-DFT), is proposed. Two stages of focusing are accomplished in the algorithm. Firstly, a modified 2D-DFT, which has accomplished a preliminary focusing process, is exploited on the received data to achieve initial estimates that are more accurate than the results of beamforming methods. More importantly, in the first stage, the real array manifold is projected to a new virtual array manifold that is more suitable for subsequent focusing. Secondly, based on the transformed data, an iterative process in which each step works on the estimates available from previous iterations is applied to the transformed data to estimate the DOA. Because of the implementation of 2D-DFT, the proposed algorithm can achieve a higher resolution and a better convergence performance than conventional methods, especially when the incoming sources have irregular distributions in the frequency domain. Performance analysis and simulation results show that the proposed algorithm works properly and effectively.

Three algorithms for efficient solution of optimal control problems for high-dimensional systems are presented. Each bypasses the intermediate (and, unnecessary) step of open-loop model reduction. Each also bypasses the solution of the full Riccati equation corresponding to the LQR problem, which is numerically intractable for large n. Motivation for this effort comes from the field of model-based flow control, where open-loop model reduction often fails to capture the dynamics of interest (governed by the Navier–Stokes equation). Our minimum control energy method is a simplified expression for the well-known minimum-energy stabilizing control feedback that depends only on the left eigenvectors corresponding to the unstable eigenvalues of the system matrix A. Our Adjoint of the Direct-Adjoint method is based on the repeated iterative computation of the adjoint of a forward problem, itself defined to be the direct-adjoint vector pair associated with the LQR problem. Our oppositely-shifted subspace iteration (OSSI, the main new result of the present paper) method is based on our new subspace iteration method for computing the Schur vectors corresponding, notably, to the \(m\ll n\) central eigenvalues (near the imaginary axis) of the Hamiltonian matrix related to the Riccati equation of interest. Prototype OSSI implementations are tested on a low-order control problem to illustrate its behavior.

The direct inversion in the iterative subspace (DIIS) method is applied to several simple SCF wave functions in an effective Fock matrix formulation. The following cases are treated: high-spin-restricted open shell, open-shell singlet, and two-configuration wave functions. Open-shell singlet states are described by a three-determinant 2×2 CAS expansion which is equivalent to Davidson’s nonorthogonal SCF method in the case of the first open-shell singlet. Very sharp convergence is usually obtained in less than 20 cycles. The method is applicable to slowly convergent or even inherently divergent cases, and able to enforce convergence to excited states not the lowest of their symmetry. For these simple wave functions, the present first order method is asymptotically more efficient than second-order methods. Examples are presented for H2O, H2O2, C2H4, F2, several states of NO2, C2H5, formaldehyde, and ketene.

Representing complex shapes with simple primitives in high accuracy is\nimportant for a variety of applications in computer graphics and geometry\nprocessing. Existing solutions may produce suboptimal samples or are complex to\nimplement. We present methods to approximate given shapes with user-tunable\nnumber of spheres to balance between accuracy and simplicity: touching\nmedial/scale-axis polar balls and k-means smallest enclosing circles. Our\nmethods are easy to implement, run efficiently, and can approach quality\nsimilar to manual construction.\n

Geometric feature learning for 3-D surfaces is critical for many applications in computer graphics and 3-D vision. However, deep learning currently lags in hierarchical modeling of 3-D surfaces due to the lack of required operations and/or their efficient implementations. In this article, we propose a series of modular operations for effective geometric feature learning from 3-D triangle meshes. These operations include novel mesh convolutions, efficient mesh decimation, and associated mesh (un)poolings. Our mesh convolutions exploit spherical harmonics as orthonormal bases to create continuous convolutional filters. The mesh decimation module is graphics processing unit (GPU)-accelerated and able to process batched meshes on-the-fly, while the (un)pooling operations compute features for upsampled/downsampled meshes. We provide an open-source implementation of these operations, collectively termed Picasso. Picasso supports heterogeneous mesh batching and processing. Leveraging its modular operations, we further contribute a novel hierarchical neural network for perceptual parsing of 3-D surfaces, named PicassoNet++. It achieves highly competitive performance for shape analysis and scene segmentation on prominent 3-D benchmarks. The code, data, and trained models are available at https://github.com/EnyaHermite/Picasso.

Classification and recognition of objects for images is an important issue in many scientific fields such as computer vision, biometrics and medical image analysis. An important feature of many objects is shape, so shape analysis has become an important part of classification. One method of shape analysis is to estimate boundaries and analyze the shape of these boundaries to determining the characteristics of the original object. However, many literature studies on point cloud shape analysis are based on existing shapes. This paper mainly refers to Ripley’s K-function in spatial point analysis, through this judgment on spatial distribution of point cloud data to determine the existence of shape in point cloud data, through the spatial distribution of 2D point cloud data and 3D point cloud data. Judging by the randomness of the experiment, K-function has a considerable effect on judging existence of point cloud data shape through relevant experimental verification analysis.

Shape analysis finds many important applications in shape understanding, matching and retrieval. Among the various shape analysis methods, spectral shape analysis aims to study the spectrum of the Laplace–Beltrami operator of some well‐designed shape‐dependent equations and obtain a spectral shape descriptor that can in turn be used for shape analysis purposes. The success of such approaches depends greatly on the discriminating power of a shape descriptor. On the other hand, the medial axis of a shape is widely known for its complete shape representation. It is sensitive to small perturbation of the boundary of a shape which often poses difficulty in its effective use for shape analysis. In this paper, we propose a new spectral shape descriptor, called the medial axis spectrum for 2D shapes, which directly applies spectral analysis to the medial axes of the shapes. We extend the Laplace–Beltrami operator onto the medial axis, and take the solution to an extended Laplacian eigenvalue problem defined on the axis as the medial axis spectrum. The medial axis spectrum is robust in the presence of shape boundary noise, and is invariant under rigid transformations, uniform scaling and isometry of the medial axis. We demonstrate these benefits of such a medial axis spectrum representation through extensive experiments. The medial axis spectrum is further used for 2D shape retrieval, and its superiority over previous work is shown by comparison.

We report how imaginary time wavepacket propagation may be used to efficiently calculate the lowest-lying eigenstates of the electronic Hamiltonian. This approach, known as the relaxation method in the quantum dynamics community, represents a fundamentally different approach to the solution of the electronic eigenvalue problem in comparison to traditional iterative subspace diagonalization schemes such as the Davidson and Lanczos methods. In order to render the relaxation method computationally competitive with existing iterative subspace methods, an extended short iterative Lanczos wavepacket propagation scheme is proposed and implemented. In the examples presented here, we show that by using an efficient wavepacket propagation algorithm the relaxation method is, at worst, as computationally expensive as the commonly used block Davidson-Liu algorithm, and in certain cases, significantly less so.

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und MechanikVolume 86, Issue 2 p. 150-150 Book Review Book Review: Helmut Ratschek and Jon Rokne, Geometric Computations with Interval and New Robust Methods. Applications in Computer Graphics, GIS and Computational Geometry G. Alefeld, G. Alefeld KarlsruheSearch for more papers by this author G. Alefeld, G. Alefeld KarlsruheSearch for more papers by this author First published: 20 January 2006 https://doi.org/10.1002/zamm.200690009AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume86, Issue2February 2006Pages 150-150 RelatedInformation

The subspace iteration method – Revisited

Various geometry optimization techniques are systematically investigated. The rational function (RF) and direct inversion in the iterative subspace (DIIS) methods are compared and optimized for the purpose of geometry optimization. Various step restriction and line search procedures are tested. The model Hessian recently proposed by Lindh et al. has been used in conjunction with different Hessian update procedures. Optimization for over 30 molecules have been performed in Z-matrix coordinates, local normal coordinates, and curvilinear natural internal coordinates, using the same approximations for the Hessian in all cases. The most effective and stable procedure for optimization of equilibrium structures was found to be the DIIS minimization in natural internal coordinates using the BFGS update of the model Hessian. Our method shows faster overall convergence than all previously published methods for the same test suite of molecules. © 1997 John Wiley & Sons, Inc. J Comput Chem 18: 1473–1483, 1997

This study considers four means of defining differential operators for extracting local aspects of shape in ill-specified environments: fuzzy differentiation as kernel smoothing; differentiation in the sense of weak or generalized derivatives; differentiation for fuzzy functions between normed spaces; and fuzzy differentiation for mappings between fuzzy manifolds. More consideration is given to the last, norm-free approach, which involves the notions of an abstract fuzzy topological vector space, fuzzy differentiation between fuzzy topological vector spaces, fuzzy atlases, and tangent vectors of fuzzy manifolds.