projection method
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2021 ◽  
Vol 9 (12) ◽  
pp. 1461
Author(s):  
Jose M. Gonzalez-Ondina ◽  
Lewis Sampson ◽  
Georgy I. Shapiro

Data assimilation methods are an invaluable tool for operational ocean models. These methods are often based on a variational approach and require the knowledge of the spatial covariances of the background errors (differences between the numerical model and the true values) and the observation errors (differences between true and measured values). Since the true values are never known in practice, the error covariance matrices containing values of the covariance functions at different locations, are estimated approximately. Several methods have been devised to compute these matrices, one of the most widely used is the one developed by Hollingsworth and Lönnberg (H-L). This method requires to bin (combine) the data points separated by similar distances, compute covariances in each bin and then to find a best fit covariance function. While being a helpful tool, the H-L method has its limitations. We have developed a new mathematical method for computing the background and observation error covariance functions and therefore the error covariance matrices. The method uses functional analysis which allows to overcome some shortcomings of the H-L method, for example, the assumption of statistical isotropy. It also eliminates the intermediate steps used in the H-L method such as binning the innovations (differences between observations and the model), and the computation of innovation covariances for each bin, before the best-fit curve can be found. We show that the new method works in situations where the standard H-L method experiences difficulties, especially when observations are scarce. It gives a better estimate than the H-L in a synthetic idealised case where the true covariance function is known. We also demonstrate that in many cases the new method allows to use the separable convolution mathematical algorithm to increase the computational speed significantly, up to an order of magnitude. The Projection Method (PROM) also allows computing 2D and 3D covariance functions in addition to the standard 1D case.


2021 ◽  
Vol 2 (2) ◽  
pp. 71-77
Author(s):  
Yi-Han Yang ◽  
Ying Wang ◽  
Jing Wang

City branding is the most concentrated embodiment of city image and the most valuable intangible asset of a city. City branding can attract investors, talents, tourists, and the public’s attention to the city, as well as enhance the competitiveness of the city and bring greater economic benefits and growth potential to the city. Past studies in the context of city branding systematically focus on the development, shaping and communication. Little is known on the combination of city branding and brand archetypes. The projection technique is applied for this research. The objective of this research is to explore the archetypes of city brand based on the Twelve Chinese Cultural Archetypes theory, and takes Xiamen, Zhangzhou, and Quanzhou as the examples. The research reveals that Xiamen belongs to the archetype of “Beauty”, Zhangzhou is close to the archetype of “Neighborhood”, and Quanzhou belongs to the archetype of “Benevolent”.


2021 ◽  
Vol 31 (12) ◽  
pp. 123124
Author(s):  
Alessandro Bongarzone ◽  
Francesco Viola ◽  
François Gallaire

2021 ◽  
pp. 25-34
Author(s):  
O.G. Revunova ◽  
◽  
A.V. Tyshcuk ◽  
О.О. Desiateryk ◽  
◽  
...  

Introduction. In technical systems, there is a common situation when transformation input-output is described by the integral equation of convolution type. This situation accurses if the object signal is recovered by the results of remote measurements. For example, in spectrometric tasks, for an image deblurring, etc. Matrices of the discrete representation for the output signal and the kernel of convolution are known. We need to find a matrix of the discrete representation of a signal of the object. The well known approach for solving this problem includes the next steps. First, the kernel matrix has to be represented as the Kroneker product. Second, the input-output transformation has to be presented with the usage of Kroneker product matrices. Third, the matrix of the discrete representation of the object has to be found. The object signal matrix estimation obtained with the help of pseudo inverting of Kroneker decomposition matrices is unstable. The instability of the object signal estimation in the case of usage of Kroneker decomposition matrices is caused by their discrete ill posed matrix properties (condition number is big and the series of the singular numbers smoothly decrease to zero). To find solutions of discrete ill-posed problems we developed methods based on the random projection and the random projection with an averaging by the random matrices. These methods provide a stable solutions with a small computational complexity. We consider the problem of object signals recovering in the systems where an input-output transformation is described by the integral equation of a convolution. To find a solution for these problems we need to build a generalization for two-dimensional signals case of the random projection method. Purpose. To develop a stable method of the recovery of object signal for the case in which an input-output transformation is described by the integral equation of a convolution. Results and conclusions. We developed the method of a stable recovery of object signal for the case in which an input-output transformation is described by the integral equation of a convolution. The stable estimation of the object signal is provided by Kroneker decomposition of the kernel matrix of convolution, computation of random projections for Kroneker factorization matrices, and a selection of the optimal dimension of a projector matrix. The method is illustrated by its application in technical problems.


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