Balanced incomplete factorization preconditioner with pivoting

Efficient use of global navigation satellite system (GNSS) observations improves when applying rational satellite selection algorithms. By combining the Sherman–Morrison formula and singular value decomposition, a smaller-GDOP (geometric dilution of precision)-value method is proven for an increasing number of visible satellites. By combining this smaller-GDOP-value method with the maximum-volume-tetrahedron method, a new rapid satellite selection algorithm based on the Sherman–Morrison formula for GNSS multi-systems is proposed. The basic idea of the algorithm is as follows: First, the maximum-volume-tetrahedron method is used to obtain four initial visible satellites. Then, the other visible satellites are selected by using the smaller-GDOP-value method to reduce the GDOP value and improve the accuracy of the overall algorithm. When the number of included satellites reaches a certain value, the rate of GDOP decrease tends to approach zero. Considering the algorithm precision and the computation efficiency, reasonable thresholds and end of calculation condition equation are given, which can make the proposed algorithm autonomous. The reasonable thresholds and the end of calculation parameters are suggested by means of experiments. Under the thresholds and the end of calculation parameters, the algorithm has an adaptive functionality. Furthermore, the GDOP values of the algorithm are less than 2, indicating that this algorithm can meet one of the requirements of high-precision navigation. Moreover, compared with the computation complexity values of the optimal GDOP estimation method, which includes all visible satellites, the values of the new algorithm are about half, indicating that this algorithm has a rapid performance. These findings verify that the proposed satellite selection algorithm based on the Sherman–Morrison formula provides autonomous functionality, high-performance computing, and high-accuracy results.

In this paper we present a new incomplete factorization of a square matrix into triangular factors in which we get standard $LU$ or $LDL^T$ factors (direct factors) and their inverses (inverse factors) at the same time. Algorithmically, we derive this method from the approach based on the Sherman–Morrison formula [R. Bru, J. Cerdán, J. Marín, and J. Mas, SIAM J. Sci. Comput., 25 (2003), pp. 701–715]. In contrast to the robust incomplete decomposition (RIF) algorithm [M. Benzi and M. Tůma, Numer. Linear Algebra Appl., 10 (2003), pp. 385–400] the direct and inverse factors here directly influence each other throughout the computation. Consequently, the algorithm to compute the approximate factors may mutually balance dropping in the factors and control their conditioning in this way. For the symmetric positive definite case, we derive the theory and present an algorithm for computing the incomplete $LDL^T$ factorization, and we discuss experimental results. We call this new approximate $LDL^T$ factorization the balanced incomplete factorization (BIF). Our experimental results confirm that this factorization is very robust and may be useful in solving difficult ill conditioned problems by preconditioned iterative methods. Moreover, the internal coupling of the computation of direct and inverse factors results in much shorter setup times (times to compute approximate decomposition) than RIF, a method of a similar and very high level of robustness. We also derive and present the theory for the general nonsymmetric case, but do not discuss its implementation.

The solution of linear systems by using the Sherman–Morrison formula

An approximate inverse LU preconditioner is constructed based on the Sherman–Morrison formula. Applying recursively that inversion formula a multiplicative decomposition of the inverse of a matrix is obtained. This recursion in compact form is the base to build the proposed preconditioner that we call V–AISM. For nonsingular M -matrices and H -matrices of the invertible class the stability of the preconditioner is proved. Numerical results show that V–AISM is robust and competitive compared with other preconditioners.

Model updating plays an important role in dynamics modeling with high accuracy, which is widely used in mechanical engineering. In this paper, a model updating method using frequency response function (FRF) is proposed based on Sherman–Morrison formula, in which only the initial FRFs and parameter perturbations are employed to calculate the sensitivity avoiding repeated finite element (FE) analyses and improving the computational efficiency. Firstly, the sensitivity of FRFs to the design parameters is calculated by Sherman–Morrison formula based on the QR decomposition of the system dynamic stiffness matrix variation after parameter perturbations, then the influence of damping on the amplitude of FRFs is considered to select an appropriate frequency range, and finally conduct the model updating according to the sensitivity of the FRFs. By employing simulation examples of a truss and a solar wing and the experiment of an aluminum frame, the updating error is still within ±1.00% in the condition of 5% random white noise, which shows the proposed method has high accuracy and a certain anti-noise capability. When only a few numbers of frequency points are selected near the resonance peak of the FRFs, the result shows that selecting the appropriate frequency range and points can reduce the computational cost. The results of the experiment study show that the proposed method can effectively identify the structural parameters. The above results verify the feasibility and effectiveness of proposed model updating method using FRFs.

SummaryA numerical algorithm for the solution of advection–diffusion equation on the surface of a sphere is suggested. The velocity field on a sphere is assumed to be known and non‐divergent. The discretization of advection–diffusion equation in space is carried out with the help of the finite volume method, and the Gauss theorem is applied to each grid cell. For the discretization in time, the symmetrized double‐cycle componentwise splitting method and the Crank–Nicolson scheme are used. The numerical scheme is of second order approximation in space and time, correctly describes the balance of mass of substance in the forced and dissipative discrete system and is unconditionally stable. In the absence of external forcing and dissipation, the total mass and L2‐norm of solution of discrete system is conserved in time. The one‐dimensional periodic problems arising at splitting in the longitudinal direction are solved with Sherman–Morrison's formula and Thomas's algorithm. The one‐dimensional problems arising at splitting in the latitudinal direction are solved by the bordering method that requires a prior determination of the solution at the poles. The resulting linear systems have tridiagonal matrices and are solved by Thomas's algorithm. The suggested method is direct (without iterations) and rapid in realization. It can also be applied to linear and nonlinear diffusion problems, some elliptic problems and adjoint advection–diffusion problems on a sphere. Copyright © 2015 John Wiley & Sons, Ltd.

A class of preconditioners based on matrix splitting for nonsymmetric linear systems

Structural modifications in the form of mass, stiffness and damping to a structure change the dynamic properties of that structure. However, in some cases, after modifications are made to the structures, certain specified natural frequencies of the structure are desired to remain unchanged. This study is interested in the determination of necessary stiffness modifications in order to keep a certain number of natural frequencies of the system unchanged despite mass attachments. In particular, two methods based on the Sherman–Morrison formula are developed in order to determine the spring coefficients needed to keep one and more than one natural frequency of the structures unchanged. The developed methods directly use the Frequency Response Functions of the unmodified system relating the modification coordinates only and they need neither a physical model nor a modal model. The numerical simulations show that they are very effective. However, due to the nature of the inverse problem, any solution or practical realistic solution may be not found. The existence of the solution depends on also the modification coordinates chosen. A simple sensitivity approach demonstrated by a 3D graph is proposed to be able to choose a suitable modification.

Aeroelastic systems with softening nonlinearity are not fully understood. The literature is not definitive, contains partial explanations, and even conflicting results. This problem is addressed for the case of a simple pitch-flap wing, with softening cubic nonlinearity in the pitch stiffness. Complex dynamical behavior is revealed. The study is carried out in three parts, the first of which is the identification and stability analysis of limit cycles in the frequency domain using describing functions combined with the Sherman–Morrison formula. Numerical integration of the nondimensionalized governing equations in the time domain is then carried out and, in addition to confirming the frequency domain results, new behavior is revealed, including asymmetric limit cycles and chaos. Finally, bifurcation analysis is undertaken using numerical continuation methods to reveal Hopf, symmetry breaking (pitchfork), fold and period doubling (flip) bifurcations. The effects of initial conditions and the proximity of limit cycle oscillation and chaos to static divergence are considered. Sections of the basin of attraction are presented for different wing configurations to show that the boundaries separating the different regions of dynamic behavior may be simple or nonsimple. It is found that some regions may appear where predicted stable limit cycle oscillation is free from the destabilizing effect of softening nonlinearity.

Recommendation systems are successful personalizing tools and information filtering in web. One of the most important recommendation methods is matrix factorization method. In matrix factorization method, the latent features of users and items are determined in such a way that the inner product of the latent features of a user with the latent features of an item is equal to that user's rating on that item. This model is solved using alternate optimization algorithm. The solution and the prediction error of this algorithm depend on the initial values of the latent features of users which are usually set to small random values. The purpose of this paper is to propose a fast alternate optimization algorithm for matrix factorization which converges to a good solution. To do so, firstly, we show experimentally that if the latent feature vector of each user is initialized by a vector of which elements are equal, we can also obtain a proper solution using the alternate optimization algorithm. Then, we prove that if our proposed initialization method is used, the alternate optimization algorithm for matrix factorization can be simplified using Sherman–Morrison formula. Experimental results on 5 real datasets show that the runtime of our proposed algorithm is 2–45 times less than the traditional method.

Hyperspectral image contains various wavelength channels and the corresponding imagery processing requires a computation platform with high performance. Target and anomaly detection on hyperspectral image has been concerned because of its practicality in many real-time detection fields while wider applicability is limited by the computing condition and low processing speed. The field programmable gate arrays (FPGAs) offer the possibility of on-board hyperspectral data processing with high speed, low-power consumption, reconfigurability and radiation tolerance. In this paper, we develop a novel FPGA-based technique for efficient real-time target detection algorithm in hyperspectral images. The collaborative representation is an efficient target detection (CRD) algorithm in hyperspectral imagery, which is directly based on the concept that the target pixels can be approximately represented by its spectral signatures, while the other cannot. To achieve high processing speed on FPGAs platform, the CRD algorithm reduces the dimensionality of hyperspectral image first. The Sherman–Morrison formula is utilized to calculate the matrix inversion to reduce the complexity of overall CRD algorithm. The achieved results demonstrate that the proposed system may obtains shorter processing time of the CRD algorithm than that on 3.40 GHz CPU.

Clever eye algorithm for target detection of remote sensing imagery

Cloud/web mapping and geoprocessing services – Intelligently linking geoinformation

This note corrects an error in the formula to obtain the Whittle index using the Sherman–Morrison formula in Akbarzadeh and Mahajan (2022). Also, some other minor typos are highlighted.

A Sherman–Morrison approach to the solution of linear systems