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Katsikis et al. presented a computational method in order to calculate the Moore-Penrose inverse of an arbitrary matrix (including singular and rectangular) (2011). In this paper, an improved version of this method is presented for computing the pseudo inverse of an
m
×
n
real matrix A with rank
r
>
0
. Numerical experiments show that the resul...

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## Citations

... (27) to obtain W = O. Setting W = O in Eq. (29) then results in the approximate nodal displacements to be ...

... We calculate the pseudo-inverse of K using the improved qrginv method developed by Ataei [29] which is shown to be much faster than the traditional method of calculating the pseudo-inverse of a matrix using the singular value decomposition (or SVD) method [30]. The improved qrginv method is described in Section 3.3. ...

... We calculate the Moore-Penrose pseudoinverse of the stiffness matrix using the method developed by Ataei [29] which is determined to be faster than the usage of the traditionally-used singular value decomposition method [30] which is also very popularly used in the machine learning community [33][34][35], and the qrginv method developed by Katsikis et al. [36]. The improved qrginv method is also particularly attractive since it has been shown to obtain very fast and accurate results when tested for sparse matrices [29], and the SVD method is slower because it does not respect matrix sparseness. ...

In this work, we develop a pseudoinverse-based static finite-element solver to model the elastic deformation and non-local brittle fracture of solids. The pseudoinverse of the finite-element stiffness matrix is calculated using a QR decomposition-based method which is faster than using the traditional approach based on the singular value decomposition method. This new finite-element framework has two advantages: (1) the finite-element equations can still be solved even when the finite-element stiffness matrix is singular, and (2) there is no need for introducing an artificial elastic rest energy or viscous regularization for the sole purpose of keeping the finite-element stiffness matrix non-singular in order to solve the finite-element equations. We also show that the proposed method is robust in solving chosen boundary value problems involving the elastic deformation and abrupt non-local mode I and mixed-mode brittle fracture of solids when compared to the traditional element kill method where a small but finite residual stiffness is maintained at a material in order to prevent the finite-element stiffness matrix from being singular. Hence, this proposed method allows the modeling of separation/fragmentation of solids when fracture occurs.
Finally, we use the new computational framework to model the fracture of PMMA beam samples at room temperature. By calibrating the material parameters in the constitutive theory using analytical methods and fitting to a Mode I fracture experiment force–displacement response, we show that our newly-proposed computational method is able to predict the experimental fracture loci and crack propagation characteristics in PMMA beam samples undergoing mixed-mode fracture conditions to good accord.

... Table 2 lists the computation time of six different algorithms and our method to obtain the tiptoe force with a tolerance of 10 −13 . For the Moore-Penrose inverse computation problem of small rectangular matrix that occurs during buffer landings, the methods of generalized inverse (Ginv) [33], tensor product matrix (TPM) [38], and improved Qrginv (IMqrg) [39], have the fastest solution speed. The methods of singular value decomposition (SVD), QR generalized inverse (Qrg) [36], and VTLSA have medium speed, and generalized inverse (Geninv) [34] method has the slowest solution speed. ...

The prober with an immovable lander and a movable rover is commonly used to explore the Moon’s surface. The rover can complete the detection on relatively flat terrain of the lunar surface well, but its detection efficiency on deep craters and mountains is relatively low due to the difficulties of reaching such places. A lightweight four-legged landing and walking robot called “FLLWR” is designed in this study. It can take off and land repeatedly between any two sites wherever on deep craters, mountains or other challenging landforms that are difficult to reach by direct ground movement. The robot integrates the functions of a lander and a rover, including folding, deploying, repetitive landing, and walking. A landing control method via compliance control is proposed to solve the critical problem of impact energy dissipation to realize buffer landing. Repetitive landing experiments on a five-degree-of-freedom lunar gravity testing platform are performed. Under the landing conditions with a vertical velocity of 2.1 m/s and a loading weight of 140 kg, the torque safety margin is 10.3% and 16.7%, and the height safety margin is 36.4% and 50.1% for the cases with or without an additional horizontal disturbance velocity of 0.4 m/s, respectively. The study provides a novel insight into the next-generation lunar exploration equipment.

... Schulz [40] presents an iterative method to approximate the pseudoinverse based on the classical Newton-Rapshon method. Other related works are presented in [32,[41][42][43][44]. ...

This paper proposes a new method to compute generalized low-rank matrix approximation (GLRMA). The GLRMA is a general case of the well-known low-rank approximation problem proposed by Eckart-Young in 1936. This new method, so-called the fast-GLRMA method, is based on tensor product and Tikhonov’s regularization to approximate the pseudoinverse and bilateral random projections to estimate, in turn, the low-rank approximation. The fast-GLRMA method significantly reduces the execution time to compute the optimal solution, while preserving the accuracy of the classical method of solving the GLRMA. Computational experiments to measure execution time and speedup confirmed the efficiency of the proposed method.

... where B denotes the design matrix related to the network [34]; ∆ĥ denotes the estimated differential height of all arcs. (·) + denotes the pseudo-inverse and is solved by a fast algorithm [35]. Q ∆ĥ is the Covariance (CV) matrix related to the quality of the arc solutions, which is defined as ...

Multi-temporal interferometric synthetic aperture radar (MT-InSAR) can be applied to monitor the structural health of infrastructure such as railways, bridges, and highways. However, for the successful interpretation of the observed deformation within a structure, or between structures, it is imperative to associate a radar scatterer unambiguously with an actual physical object. Unfortunately, the limited positioning accuracy of the radar scatterers hampers this attribution, which limits the applicability of MT-InSAR. In this study, we propose an approach for health monitoring of railway system combining MT-InSAR and LiDAR (laser scanning) data. An amplitude-augmented interferometric processing approach is applied to extract continuously coherent scatterers (CCS) and temporary coherent scatterers (TCS), and estimate the parameters of interest. Based on the 3D confidence ellipsoid and a decorrelation transformation, all radar scatterers are linked to points in the point cloud and their coordinates are corrected as well. Additionally, several quality metrics defined using both the covariance matrix and the radar geometry are introduced to evaluate the results. Experimental results show that most radar scatterers match well with laser points and that LiDAR data are valuable as auxiliary data to classify the radar scatterers.

... where B 1 denotes the design matrix related to the network; v 1 denotes the estimated differential deformation rate of all arcs; (·) + denotes the pseudoinverse and is solved by a faster algorithm [43]. Q −1 1 is the weight matrix related to the quality of the arc solutions which is defined as ...

Multi-temporal interferometric synthetic aperture radar (MT-InSAR) is used for many applications in earth observation. Most MT-InSAR methods select scatterers with high coherence throughout the entire time series. However, as time series lengthen, inevitable changes in surface scattering lead to decorrelation, which systematically decreases the number of coherent scatterers. Here, we propose a novel method to detect and process temporary coherent scatterers (TCS) by subsequently analyzing the amplitude and the interferometric phase. Two hypothesis tests are developed for amplitude analysis in order to identify the moments of appearing and/or disappearing coherent scatterers. Based on the amplitude analysis, the parameters of interest are then estimated using the interferometric phase. An optimized adaptive temporal subset approach is proposed to improve the precision of the estimated parameters. If the scatterers are not evenly distributed over the area, a secondary (support) network is designed to improve the spatial point distribution. The main advantage of this method is the reliable extraction of a subset of time series without using any contextual information. Experimental results show that the TCSs significantly increase the number of observations for displacement monitoring and improve the change detection capability in urban construction areas.

... Specifically, the function pinv(A, tol) returns the Moore-Penrose pseudoinverse, obtained by SVD decomposition where the values above tolerance (tol) are set to zero; this may be adapted to an ill-conditioned problem (A is not of full rank). Another option to obtain the Moore-Penrose pseudoinverse is proposed in [37], which makes use of QR-factorization and an algorithm based on a reverse order law for generalized inverse matrices; this method was later refined in [38]. An iterative solution to obtain the Moore-Penrose pseudoinverse was published in [39]. ...

Background
The response of many biomedical systems can be modelled using a linear combination of damped exponential functions. The approximation parameters, based on equally spaced samples, can be obtained using Prony’s method and its variants (e.g. the matrix pencil method). This paper provides a tutorial on the main polynomial Prony and matrix pencil methods and their implementation in MATLAB and analyses how they perform with synthetic and multifocal visual-evoked potential (mfVEP) signals.
This paper briefly describes the theoretical basis of four polynomial Prony approximation methods: classic, least squares (LS), total least squares (TLS) and matrix pencil method (MPM). In each of these cases, implementation uses general MATLAB functions. The features of the various options are tested by approximating a set of synthetic mathematical functions and evaluating filtering performance in the Prony domain when applied to mfVEP signals to improve diagnosis of patients with multiple sclerosis (MS).
Results
The code implemented does not achieve 100%-correct signal approximation and, of the methods tested, LS and MPM perform best. When filtering mfVEP records in the Prony domain, the value of the area under the receiver-operating-characteristic (ROC) curve is 0.7055 compared with 0.6538 obtained with the usual filtering method used for this type of signal (discrete Fourier transform low-pass filter with a cut-off frequency of 35 Hz).
Conclusions
This paper reviews Prony’s method in relation to signal filtering and approximation, provides the MATLAB code needed to implement the classic, LS, TLS and MPM methods, and tests their performance in biomedical signal filtering and function approximation. It emphasizes the importance of improving the computational methods used to implement the various methods described above.

We propose a route choice model in which traveler behavior is represented as a utility maximizing assignment of flow across an entire network under a flow conservation constraint. Substitution between routes depends on how much they overlap. The model is estimated considering the full set of route alternatives, and no choice set generation is required. Nevertheless, estimation requires only linear regression and is very fast. Predictions from the model can be computed using convex optimization, and computation is straightforward even for large networks. We estimate and validate the model using a large dataset comprising 1,337,096 GPS traces of trips in the Greater Copenhagen road network.

During the data processing and interpretation of mine transient electromagnetic method (MTEM), in order to improve the identification of the geo-electrical interface between low resistivity abnormal body and surrounding rock in advanced detection, transient electromagnetic field was transformed to pseudo-seismic wave-field based on the function relationship between spreading electromagnetic field and seismic wave-field in whole-space. Meanwhile, the signal of pseudo-seismic after being transformed was processed with correlative stack to strengthen the amplitude and improve signal-noise-ratio (SNR) using the synthetic aperture imaging (SAI). The data of different detection directions on one surveying point in MTEM were imaged as aperture data, which would highlight weak abnormality and improve SNR. In addition, the application effect of SAI was testified with the in-site advanced detection in roadway using MTEM. The results showed that the SAI of the MTEM could improve SNR of the electrical interface information and highlight the geometrical resolution to increase exploration accuracy. The technique is significant theoretically and practically in accurate advanced detection of aquifer structure by MTEM.