A Novel Symmetric Skew-Hamiltonian Isotropic Lanczos Algorithm for Spectral Conformal Parameterizations

In the past decades, many methods for computing conformal mesh parameterizations have been developed in response to demand of numerous applications in the field of geometry processing. Spectral conformal parameterization (SCP) (Mullen et al. in Proceedings of the symposium on geometry processing, SGP ’08. Eurographics Association, Aire-la-Ville, Switzerland, pp 1487–1494, 2008) is one of these methods used to compute a quality conformal parameterization based on the spectral techniques. SCP focuses on a generalized eigenvalue problem (GEP) \(L_{C}{\mathbf {f}} = \lambda B{\mathbf {f}}\) whose eigenvector(s) associated with the smallest positive eigenvalue(s) provide the conformal parameterization result. This paper is devoted to studying a novel eigensolver for this GEP. Based on structures of the matrix pair \((L_{C},B)\) , we show that this GEP can be transformed into a small-scale compressed and deflated standard eigenvalue problem with a symmetric positive definite skew-Hamiltonian operator. We then propose a symmetric skew-Hamiltonian isotropic Lanczos algorithm ( \({\mathbb {S}}\) HILA) to solve the reduced problem. Numerical experiments show that our compressed deflating technique can exclude the impact of convergence from the kernel of \(L_{C}\) and transform the original problem to a more robust system. The novel \({\mathbb {S}}\) HILA method can effectively avoid the disturbance of duplicate eigenvalues. As a result, based on the spectral model of SCP, our numerical eigensolver can compute the conformal parameterization accurately and efficiently.     

Numerical solution of the kinematic dynamo problem for Beltrami flows in a sphere

The article discusses computational aspects of the kinematic problem of magnetic field generation by a Beltrami flow in a sphere. Galerkin's method is applied with a functional basis consisting of Laplace operator eigenfunctions. Dominant eigenvalues of the magnetic induction operator and associated magnetic eigenmodes are obtained numerically for a certain Beltrami flow for magnetic Reynolds numbers up to 100. The eigenvalue problem is solved by a highly optimized iterative procedure, which is quite general and can be applied to numerical treatment of arbitrary linear stability problems.     

Isospectral Domains for Discrete Elliptic Operators

Concerning the Laplace operator with homogeneous Dirichlet boundary conditions, the classical notion of isospectrality assumes that two domains are related when they give rise to the same spectrum. In two dimensions, non isometric, isospectral domains exist. It is not known however if all the eigenvalues relative to a specific domain can be preserved under suitable continuous deformation of its geometry. We show that this is possible when the 2D Laplacian is replaced by a finite dimensional version and the geometry is modified by respecting certain constraints. The analysis is carried out in a very small finite dimensional space, but it can be extended to more accurate finite-dimensional representations of the 2D Laplacian, with an increase of computational complexity. The aim of this paper is to introduce the preliminary steps in view of more serious generalizations.     

Pose analysis using spectral geometry

We propose a novel method to analyze a set of poses of 3D models that are represented with triangle meshes and unregistered. Different shapes of poses are transformed from the 3D spatial domain to a geometry spectrum domain that is defined by Laplace–Beltrami operator. During this space-spectrum transform, all near-isometric deformations, mesh triangulations and Euclidean transformations are filtered away. The different spatial poses from a 3D model are represented with near-isometric deformations; therefore, they have similar behaviors in the spectral domain. Semantic parts of that model are then determined based on the computed geometric properties of all the mapped vertices in the geometry spectrum domain. Semantic skeleton can be automatically built with joints detected as well. The Laplace–Beltrami operator is proved to be invariant to isometric deformations and Euclidean transformations such as translation and rotation. It also can be invariant to scaling with normalization. The discrete implementation also makes the Laplace–Beltrami operator straightforward to be applied on triangle meshes despite triangulations. Our method turns a rather difficult spatial problem into a spectral problem that is much easier to solve. The applications show that our 3D pose analysis method leads to a registration-free pose analysis and a high-level semantic part understanding of 3D shapes.     

方程-△u=λρu的Han元特征值渐近展开

有限元特征值的渐近展开式是科学工程计算中重要的问题.本文在林群等(Math.comput.77(2008):2061-2084)工作的基础上,对Laplace算子特征值问题,研究了它的Han元数值特征值渐近展开,并用数值试验验证了理论分析.     

An effective approach for parameter determination of the digital phase-locked loop in the z-domain

In digital communication systems, typical methodologies in determining loop parameters of the digital phase-locked loop (DPLL) are based on the mapping transformation from the analog domain to the digital domain. However, such transform based algorithms are relatively complicated and not straightforward, and they also cause the problem that loop parameters are affected by the pre-detection integration time greatly. To solve these issues, an effective direct method of determining loop parameters of the second-order DPLL in the z-domain is proposed in this paper. Through ascertaining specific positions of the closed-loop system function's poles inside the right-hand side of the z-plane's unit circle, unknown parameters are calculated directly and flexibly in this method, which enables the DPLL to acquire good low-pass filtering characteristic and system stability. This novel method not only reduces the complexity of solving the parameters, but also eliminates the effect of the pre-detection integration time on loop parameters. Simulation results are provided to confirm the feasibility of the proposed method and to show that the DPLL obtained by this method achieves the similar tracking performance to the discretized PLL.     

Stress concentration in an infinite plate containing an ovaloid hole

In this paper the stress concentration problem in the case of an infinite plate containing an ovaloid hole is discussed. Ling's conformal mapping is improved by us, and the improved conformal mapping is more suitable to meet the geometry of the ovaloid hole. Using the Muskhelishvili's method, we can easily solve the proposed problem. Several numerical results are given.     

Numerical methods for the QCD overlap operator IV: Hybrid Monte Carlo

The computational costs of calculating the matrix sign function of the overlap operator together with fundamental numerical problems related to the discontinuity of the sign function in the kernel eigenvalues are the major obstacle towards simulations with dynamical overlap fermions using the Hybrid Monte Carlo algorithm. In a previous paper of the present series we introduced optimal numerical approximation of the sign function and have developed highly advanced preconditioning and relaxation techniques which speed up the inversion of the overlap operator by nearly an order of magnitude.In this fourth paper of the series we construct an HMC algorithm for overlap fermions. We approximate the matrix sign function using the Zolotarev rational approximation, treating the smallest eigenvalues of the Wilson operator exactly within the fermionic force. Based on this we derive the fermionic force for the overlap operator. We explicitly solve the problem of the Dirac delta-function terms arising through zero crossings of eigenvalues of the Wilson operator. The main advantage of scheme is that its energy violations scale better than O(Δτ²) and thus are comparable with the violations of the standard leapfrog algorithm over the course of a trajectory. We explicitly prove that our algorithm satisfies reversibility and area conservation. We present test results from our algorithm on 4⁴, 6⁴, and 8⁴ lattices.     

Harmonic solutions of a mixed boundary problem arising in the modeling of macromolecular transport into vessel walls

The earliest events leading to atherosclerosis involve the transport of lpw density lipooprotein (LDL) cholesterol from the blood across endothelial cells that line the artery wall. Laplace’s equation describes the steady state diffusion profile of a tracer through the vessel wall. This gives rise to a boundary value problem with mixed Dirichlet and Robin conditions. We construct a linear system of integral equations that approximate the coefficients of the series expansion of the solution. We prove the existence of the solution to this problem analytically by using Gershgorin’s theorem on the location of the eigenvalues of the corresponding matrix. We give a uniqueness proof using Miranda’s theorem [C. Miranda, P.D.E. of Elliptic Type, Springer-Verlag, Berlin, 1970]. The analytical construction method forms the basis for a numerical calculation algorithm. We apply our results to the transport problem above, and use them to interpret experimental observations of the growth of localized tracer leakage spots with tracer circulation time.     

一种精确的袋装粮图像边缘检测算法

基于图像识别的国家储备粮仓袋装粮食数量自动监管与稽核系统的技术核心是智能识别各种粮仓场景图像中粮袋的数量,杜绝人为的弄虚作假。边缘检测是袋装粮图像识别的首要问题,在分析了经典的以及在其基础上进行各种改进的Laplace算子缺陷的基础上,提出了1种改进的Laplace算子,该算子通过设置合理的模板参数克服了原有算子的不足,提高了图像边缘检测的精度。实验结果证明,该算子检测效果优于其他模板,并且能够精确地检测出各种类型的边缘信息。     

Integrated fuzzy logic and genetic algorithmic approach for simultaneous localization and mapping of mobile robots

This paper presents a novel method of integrating fuzzy logic (FL) and genetic algorithm (GA) to solve the simultaneous localization and mapping (SLAM) problem of mobile robots. The core of the proposed SLAM algorithm is based on an island model GA (IGA) which searches for the most probable map(s) such that the associated pose(s) provides the robot with the best localization information. Prior knowledge about the problem domain is transferred to GA in order to speed up the convergence. Fuzzy logic is employed to serve this purpose and allows the IGA to conduct the search starting from a potential region of the pose space. The underlying fuzzy mapping rules infer the uncertainty in the robot's location after executing a motion command and generate a sample-based prediction of its current position. This sample set is used as the initial population for the proposed IGA. Thus the GA-based search starts with adequate knowledge on the problem domain. The correspondence problem in SLAM is solved by exploiting the property of natural selection, which supports better performing individuals to survive in the competition. The proposed algorithm follows essentially no assumption about the environment and has the capacity to resolve the loop closure problem without maintaining explicit loop closure heuristics. The algorithm processes sensor data incrementally and therefore, has the capability of real time map generation. Experimental results in different indoor environments are presented to validate robustness of the algorithm.     

A quadratic eigenvalue problem involving Stokes equations

An existence theorem for the eigenvalues of a spectral problem is studied in this paper. The physical situation behind this mathematical problem is the determination of the eigenfrequencies and eigenmotions of a fluid-solid structure. The liquid part in this structure is represented by a viscous incompressible fluid, while the solid part is a set of parallel rigid tubes. The spectral problem governing this system is a quadratic eigenvalue problem which involves Stokes equations with a non-local boundary condition. The strategy for tackling the question of existence of eigenvalues consists of proving that the original problem is equivalent to that of determining the characteristic values of a linear (non-selfadjoint) compact operator. Sharp estimates for the eigenvalues give precise information about the region of ω where the eigenvalues are located. In particular, we prove that this problem admits a countable set of eigenvalues in which only a finite number of them have a non-zero imaginary part.     

A stability theorem of the direct Lyapunov's method for neutral-type systems in a critical case

A new stability theorem of the direct Lyapunov's method is proposed for neutral-type systems. The main contribution of the proposed theorem is to remove the condition that the 𝒟 operator is stable. In order to demonstrate the effectiveness, the proposed theorem is used to determine the stability of a neutral-type system in a critical case, i.e. the dominant eigenvalues of the principal neutral term (matrix D in Introduction) lie on the unit circle. This is difficult or infeasible in previous studies.     

A Level-Set Method for Computing the Eigenvalues of Elliptic Operators Defined on Compact Hypersurfaces

We demonstrate, through separation of variables and estimates from the semi-classical analysis of the Schrödinger operator, that the eigenvalues of an elliptic operator defined on a compact hypersurface in ? ⁿ can be found by solving an elliptic eigenvalue problem in a bounded domain Ω?? ⁿ . The latter problem is solved using standard finite element methods on the Cartesian grid. We also discuss the application of these ideas to solving evolution equations on surfaces, including a new proof of a result due to Greer (J. Sci. Comput. 29(3):321–351, 2006).     

A strategy for detecting extreme eigenvalues bounding gaps in the discrete spectrum of self-adjoint operators

For a self-adjoint linear operator with a discrete spectrum or a Hermitian matrix, the “extreme” eigenvalues define the boundaries of clusters in the spectrum of real eigenvalues. The outer extreme ones are the largest and the smallest eigenvalues. If there are extended intervals in the spectrum in which no eigenvalues are present, the eigenvalues bounding these gaps are the inner extreme eigenvalues.We will describe a procedure for detecting the extreme eigenvalues that relies on the relationship between the acceleration rate of polynomial acceleration iteration and the norm of the matrix via the spectral theorem, applicable to normal matrices. The strategy makes use of the fast growth rate of Chebyshev polynomials to distinguish ranges in the spectrum of the matrix which are devoid of eigenvalues.The method is numerically stable with regard to the dimension of the matrix problem and is thus capable of handling matrices of large dimension. The overall computational cost is quadratic in the size of a dense matrix; linear in the size of a sparse matrix. We verify computationally that the algorithm is accurate and efficient, even on large matrices.     

Total Least Squares Fitting of k-Spheres in n-D Euclidean Space Using an (n+2)-D Isometric Representation

We fit k-spheres optimally to n-D point data, in a geometrically total least squares sense. A specific practical instance is the optimal fitting of 2D-circles to a 3D point set. Among the optimal fitting methods for 2D-circles based on 2D (!) point data compared in Al-Sharadqah and Chernov (Electron. J. Stat. 3:886–911, 2009), there is one with an algebraic form that permits its extension to optimally fitting k-spheres in n-D. We embed this ‘Pratt 2D circle fit’ into the framework of conformal geometric algebra (CGA), and doing so naturally enables the generalization. The procedure involves a representation of the points in n-D as vectors in an (n+2)-D space with attractive metric properties. The hypersphere fit then becomes an eigenproblem of a specific symmetric linear operator determined by the data. The eigenvectors of this operator form an orthonormal basis representing perpendicular hyperspheres. The intersection of these are the optimal k-spheres; in CGA the intersection is a straightforward outer product of vectors. The resulting optimal fitting procedure can easily be implemented using a standard linear algebra package; we show this for the 3D case of fitting spheres, circles and point pairs. The fits are optimal (in the sense of achieving the KCR lower bound on the variance). We use the framework to show how the hyperaccurate fit hypersphere of Al-Sharadqah and Chernov (Electron. J. Stat. 3:886–911, 2009) is a minor rescaling of the Pratt fit hypersphere.     

Applying particle swarm optimization algorithm to roundness measurement

Measuring the roundness of a circular workpiece is a common problem of quality control and inspection. In this area, maximum inscribed circle (MIC) and maximum circumscribing circle (MCC), minimum zone circle (MZC) and least square circle (LSC) are four commonly used methods. In particular, MIC, MCC, and MZC, which are nonlinear constrained optimization problems, have not been thoroughly discussed lately. This study proposes a machine vision-based roundness measuring method that applies the particle swarm optimization algorithm (PSO) to compute MIC, MCC and MZC. To facilitate the PSO process, five different PSO’s were encoded using a radius (R) and circle center (x, y) and extensively evaluated using an experimental design, in which the impact of inertia weight, maximum velocity and the number of particles on the performance of the particle swarm optimizer was analyzed. The proposed method was verified with a set of testing images and benchmarked with the GA-based (genetic algorithm) method [Chen, M. C. (2000). Roundness inspection strategies for machine visions using non-linear programs and genetic algorithms. International Journal of Production Research, 38, 2967–2988]. The experimental results reveal that the PSO-based method effectively solved the MIC, MCC, and MZC problems and outperforms GA-based method in both accuracy and the efficiency. As a finals, several industrial applications are presented to explore the effectiveness and efficiency of the proposed method.     

Discrete heat kernel determines discrete Riemannian metric

The Laplace–Beltrami operator of a smooth Riemannian manifold is determined by the Riemannian metric. Conversely, the heat kernel constructed from the eigenvalues and eigenfunctions of the Laplace–Beltrami operator determines the Riemannian metric. This work proves the analogy on Euclidean polyhedral surfaces (triangle meshes), that the discrete heat kernel and the discrete Riemannian metric (unique up to a scaling) are mutually determined by each other. Given a Euclidean polyhedral surface, its Riemannian metric is represented as edge lengths, satisfying triangle inequalities on all faces. The Laplace–Beltrami operator is formulated using the cotangent formula, where the edge weight is defined as the sum of the cotangent of angles against the edge. We prove that the edge lengths can be determined by the edge weights unique up to a scaling using the variational approach.The constructive proof leads to a computational algorithm that finds the unique metric on a triangle mesh from a discrete Laplace–Beltrami operator matrix.     

A modified boundary integral evolution formulation for the wave equation

We apply a modified boundary integral formulation otherwise known as the Green element method (GEM) to the solution of the two-dimensional scalar wave equation.GEM essentially combines three techniques namely: (a) finite difference approximation of the time term (b) finite element discretization of the problem domain and (c) boundary integral replication of the governing equation. These unique and advantageous characteristics of GEM facilitates a direct numerical approximation of the governing equation and obviate the need for converting the governing partial differential equation to a Helmholtz-type Laplace operator equation for an easier boundary element manipulation. C₁ continuity of the computed solutions is established by using Overhauser elements. Numerical tests show a reasonably close agreement with analytical results. Though in the case of the Overhauser GEM solutions, the level of accuracy obtained does not in all cases justify the extra numerical rigor.