Large Scale Matrix Problems.

1983 ◽  
Vol 40 (161) ◽  
pp. 413
Author(s):  
Ake Bjorck ◽  
Robert J. Plemmons ◽  
Hans Schneider
Keyword(s):  
Large Scale ◽  
Matrix Problems ◽  
Scale Matrix

scholarly journals On Randomized Trace Estimates for Indefinite Matrices with an Application to Determinants

2021 ◽  
Author(s):  
Alice Cortinovis ◽  
Daniel Kressner
Keyword(s):  
Large Scale ◽  
Gaussian Process Regression ◽  
Likelihood Estimation ◽  
Random Vectors ◽  
Symmetric Positive Definite ◽  
Matrix Logarithm ◽  
The Matrix ◽  
Trace Estimation ◽  
Scale Matrix ◽  
Existing Result

AbstractRandomized trace estimation is a popular and well-studied technique that approximates the trace of a large-scale matrix B by computing the average of $$x^T Bx$$ x T B x for many samples of a random vector X. Often, B is symmetric positive definite (SPD) but a number of applications give rise to indefinite B. Most notably, this is the case for log-determinant estimation, a task that features prominently in statistical learning, for instance in maximum likelihood estimation for Gaussian process regression. The analysis of randomized trace estimates, including tail bounds, has mostly focused on the SPD case. In this work, we derive new tail bounds for randomized trace estimates applied to indefinite B with Rademacher or Gaussian random vectors. These bounds significantly improve existing results for indefinite B, reducing the number of required samples by a factor n or even more, where n is the size of B. Even for an SPD matrix, our work improves an existing result by Roosta-Khorasani and Ascher (Found Comput Math, 15(5):1187–1212, 2015) for Rademacher vectors. This work also analyzes the combination of randomized trace estimates with the Lanczos method for approximating the trace of f(B). Particular attention is paid to the matrix logarithm, which is needed for log-determinant estimation. We improve and extend an existing result, to not only cover Rademacher but also Gaussian random vectors.

Improved square-root cubature information filter

2016 ◽  
Vol 39 (4) ◽  
pp. 579-588 ◽  
Author(s):  
Yulong Huang ◽  
Yonggang Zhang ◽  
Ning Li ◽  
Lin Zhao
Keyword(s):  
Large Scale ◽  
Estimation Error ◽  
Sampling Interval ◽  
Prediction Errors ◽  
Square Root ◽  
Sigma Point ◽  
Theoretical Comparison ◽  
Error Covariance ◽  
Information Filter ◽  
Scale Matrix

In this paper, a theoretical comparison between existing the sigma-point information filter (SPIF) framework and the unscented information filter (UIF) framework is presented. It is shown that the SPIF framework is identical to the sigma-point Kalman filter (SPKF). However, the UIF framework is not identical to the classical SPKF due to the neglect of one-step prediction errors of measurements in the calculation of state estimation error covariance matrix. Thus SPIF framework is more reasonable as compared with UIF framework. According to the theoretical comparison, an improved cubature information filter (CIF) is derived based on the superior SPIF framework. Square-root CIF (SRCIF) is also developed to improve the numerical accuracy and stability of the proposed CIF. The proposed SRCIF is applied to a target tracking problem with large sampling interval and high turn rate, and its performance is compared with the existing SRCIF. The results show that the proposed SRCIF is more reliable and stable as compared with the existing SRCIF. Note that it is impractical for information filters in large-scale applications due to the enormous computational complexity of large-scale matrix inversion, and advanced techniques need to be further considered.

Bacterial Multicellularity: The Biology of Escherichia coli Building Large-Scale Biofilm Communities

2021 ◽  
Vol 75 (1) ◽  
Author(s):  
Diego O. Serra ◽  
Regine Hengge
Keyword(s):  
Escherichia Coli ◽  
Large Scale ◽  
Second Messengers ◽  
Emergent Properties ◽  
Annual Review ◽  
Publication Date ◽  
Growth And Survival ◽  
Chemical Gradients ◽  
Life Itself ◽  
Scale Matrix

Biofilms are a widespread multicellular form of bacterial life. The spatial structure and emergent properties of these communities depend on a polymeric extracellular matrix architecture that is orders of magnitude larger than the cells that build it. Using as a model the wrinkly macrocolony biofilms of Escherichia coli, which contain amyloid curli fibers and phosphoethanolamine (pEtN)-modified cellulose as matrix components, we summarize here the structure, building, and function of this large-scale matrix architecture. Based on different sigma and other transcription factors as well as second messengers, the underlying regulatory network reflects the fundamental trade-off between growth and survival. It controls matrix production spatially in response to long-range chemical gradients, but it also generates distinct patterns of short-range matrix heterogeneity that are crucial for tissue-like elasticity and macroscopic morphogenesis. Overall, these biofilms confer protection and a potential for homeostasis, thereby reducing maintenance energy, which makes multicellularity an emergent property of life itself. Expected final online publication date for the Annual Review of Microbiology, Volume 75 is October 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

scholarly journals An approach to compute and design the delay margin of a large-scale matrix delay equation

2018 ◽  
Vol 29 (4) ◽  
pp. 1101-1121 ◽  
Author(s):  
Adrián Ramírez ◽  
Min Hyong Koh ◽  
Rifat Sipahi
Keyword(s):  
Large Scale ◽  
Delay Equation ◽  
Scale Matrix ◽  
Delay Margin
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