Experiments with data perturbations to study condition numbers and numerical stability

Computing ◽  
1993 ◽  
Vol 51 (1) ◽  
pp. 29-44 ◽  
Author(s):  
J. Erhel
Keyword(s):  
Numerical Stability ◽  
Condition Numbers ◽  
Study Condition

scholarly journals Deterministic Sparse Sublinear FFT with Improved Numerical Stability

2021 ◽  
Vol 76 (2) ◽  
Author(s):  
Gerlind Plonka ◽  
Therese von Wulffen
Keyword(s):  
Numerical Stability ◽  
Numerical Experiments ◽  
A Priori ◽  
Condition Numbers ◽  
Vandermonde Matrices ◽  
Numerical Instabilities ◽  
Huge Impact ◽  
Fast Reconstruction ◽  
The Stability ◽  
Modified Algorithm

AbstractIn this paper we extend the deterministic sublinear FFT algorithm in Plonka et al. (Numer Algorithms 78:133–159, 2018. 10.1007/s11075-017-0370-5) for fast reconstruction of M-sparse vectors $${\mathbf{x}}$$ x of length $$N= 2^J$$ N = 2 J , where we assume that all components of the discrete Fourier transform $$\hat{\mathbf{x}}= {\mathbf{F}}_{N} {\mathbf{x}}$$ x ^ = F N x are available. The sparsity of $${\mathbf{x}}$$ x needs not to be known a priori, but is determined by the algorithm. If the sparsity M is larger than $$2^{J/2}$$ 2 J / 2 , then the algorithm turns into a usual FFT algorithm with runtime $${\mathcal O}(N \log N)$$ O ( N log N ) . For $$M^{2} < N$$ M 2 < N , the runtime of the algorithm is $${\mathcal O}(M^2 \, \log N)$$ O ( M 2 log N ) . The proposed modifications of the approach in Plonka et al. (2018) lead to a significant improvement of the condition numbers of the Vandermonde matrices which are employed in the iterative reconstruction. Our numerical experiments show that our modification has a huge impact on the stability of the algorithm. While the algorithm in Plonka et al. (2018) starts to be unreliable for $$M>20$$ M > 20 because of numerical instabilities, the modified algorithm is still numerically stable for $$M=200$$ M = 200 .

Treatment Fidelity Procedures for an Aphasia Intervention Within a Randomized Controlled Trial: Design, Feasibility, and Results

2020 ◽  
Vol 29 (1S) ◽  
pp. 412-424
Author(s):  
Elissa L. Conlon ◽  
Emily J. Braun ◽  
Edna M. Babbitt ◽  
Leora R. Cherney
Keyword(s):  
Randomized Controlled Trial ◽  
Controlled Trial ◽  
Treatment Protocol ◽  
Treatment Fidelity ◽  
Protocol Adherence ◽  
Study Condition ◽  
Randomized Controlled ◽  
Aphasia Therapy ◽  
Percent Accuracy ◽  
Over Time

Purpose This study reports on the treatment fidelity procedures implemented during a 5-year randomized controlled trial comparing intensive and distributed comprehensive aphasia therapy. Specifically, the results of 1 treatment, verb network strengthening treatment (VNeST), are examined. Method Eight participants were recruited for each of 7 consecutive cohorts for a total of 56 participants. Participants completed 60 hr of aphasia therapy, including 15 hr of VNeST. Two experienced speech-language pathologists delivered the treatment. To promote treatment fidelity, the study team developed a detailed manual of procedures and fidelity checklists, completed role plays to standardize treatment administration, and video-recorded all treatment sessions for review. To assess protocol adherence during treatment delivery, trained research assistants not involved in the treatment reviewed video recordings of a subset of randomly selected VNeST treatment sessions and completed the fidelity checklists. This process was completed for 32 participants representing 2 early cohorts and 2 later cohorts, which allowed for measurement of protocol adherence over time. Percent accuracy of protocol adherence was calculated across clinicians, cohorts, and study condition (intensive vs. distributed therapy). Results The fidelity procedures were sufficient to promote and verify a high level of adherence to the treatment protocol across clinicians, cohorts, and study condition. Conclusion Treatment fidelity strategies and monitoring are feasible when incorporated into the study design. Treatment fidelity monitoring should be completed at regular intervals during the course of a study to ensure that high levels of protocol adherence are maintained over time and across conditions.

Numerical Stability of GFEM Evaluation for Free Vibration Analysis in Trussed Structures

2017 ◽  
Author(s):  
Thamara Petroli ◽  
Marcos Arndt ◽  
Paulo de Oliveira Weinhardt ◽  
ROBERTO Dalledone Machado
Keyword(s):  
Free Vibration ◽  
Numerical Stability ◽  
Vibration Analysis ◽  
Free Vibration Analysis

An Adaptive Combined Preconditioner with Applications in Radiation Diffusion Equations

2015 ◽  
Vol 18 (5) ◽  
pp. 1313-1335 ◽  
Author(s):  
Xiaoqiang Yue ◽  
Shi Shu ◽  
Xiao wen Xu ◽  
Zhiyang Zhou
Keyword(s):  
Convergence Analysis ◽  
Estimation Theory ◽  
Numerical Results ◽  
Diffusion Equations ◽  
Condition Numbers ◽  
Radiation Diffusion ◽  
Discrete Problems ◽  
Preconditioned Gmres

AbstractThe paper aims to develop an effective preconditioner and conduct the convergence analysis of the corresponding preconditioned GMRES for the solution of discrete problems originating from multi-group radiation diffusion equations. We firstly investigate the performances of the most widely used preconditioners (ILU(k) and AMG) and their combinations (Bco and Bco), and provide drawbacks on their feasibilities. Secondly, we reveal the underlying complementarity of ILU(k) and AMG by analyzing the features suitable for AMG using more detailed measurements on multiscale nature of matrices and the effect of ILU(k) on multiscale nature. Moreover, we present an adaptive combined preconditioner Bcoα involving an improved ILU(0) along with its convergence constraints. Numerical results demonstrate that Bcoα-GMRES holds the best robustness and efficiency. At last, we analyze the convergence of GMRES with combined preconditioning which not only provides a persuasive support for our proposed algorithms, but also updates the existing estimation theory on condition numbers of combined preconditioned systems.

Stable Calibrations of Six-DOF Serial Robots by Using Identification Models with Equalized Singular Values

Robotica ◽  
2021 ◽  
pp. 1-22
Author(s):  
Zhouxiang Jiang ◽  
Min Huang
Keyword(s):  
Industrial Robot ◽  
Singular Values ◽  
Calibration Method ◽  
Identification Accuracy ◽  
Condition Numbers ◽  
Measurement Instruments ◽  
Unknown Parameters ◽  
Calibration Methods ◽  
Six Dof ◽  
The Stability

SUMMARY In typical calibration methods (kinematic or non-kinematic) for serial industrial robot, though measurement instruments with high resolutions are adopted, measurement configurations are optimized, and redundant parameters are eliminated from identification model, calibration accuracy is still limited under measurement noise. This might be because huge gaps still exist among the singular values of typical identification Jacobians, thereby causing the identification models ill conditioned. This paper addresses such problem by using new identification models established in two steps. First, the typical models are divided into the submodels with truncated singular values. In this way, the unknown parameters corresponding to the abnormal singular values are removed, thereby reducing the condition numbers of the new submodels. However, these models might still be ill conditioned. Therefore, the second step is to further centralize the singular values of each submodel by using a matrix balance method. Afterward, all submodels are well conditioned and obtain much higher observability indices compared with those of typical models. Simulation results indicate that significant improvements in the stability of identification results and the identifiability of unknown parameters are acquired by using the new identification submodels. Experimental results indicate that the proposed calibration method increases the identification accuracy without incurring additional hardware setup costs to the typical calibration method.

scholarly journals Quiescent Gap Solitons in Coupled Nonuniform Bragg Gratings with Cubic-Quintic Nonlinearity

2021 ◽  
Vol 11 (11) ◽  
pp. 4833
Author(s):  
Afroja Akter ◽  
Md. Jahedul Islam ◽  
Javid Atai
Keyword(s):  
Numerical Stability ◽  
Bragg Gratings ◽  
Stability Boundaries ◽  
Quintic Nonlinearity ◽  
Numerical Stability Analysis ◽  
Gap Solitons ◽  
The Stability ◽  
Dual Core

We study the stability characteristics of zero-velocity gap solitons in dual-core Bragg gratings with cubic-quintic nonlinearity and dispersive reflectivity. The model supports two disjointed families of gap solitons (Type 1 and Type 2). Additionally, asymmetric and symmetric solitons exist in both Type 1 and Type 2 families. A comprehensive numerical stability analysis is performed to analyze the stability of solitons. It is found that dispersive reflectivity improves the stability of both types of solitons. Nontrivial stability boundaries have been identified within the bandgap for each family of solitons. The effects and interplay of dispersive reflectivity and the coupling coefficient on the stability regions are also analyzed.

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