scholarly journals EA/AE-Eigenvectors of Interval Max-Min Matrices

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 882 ◽  
Author(s):  
Martin Gavalec ◽  
Ján Plavka ◽  
Daniela Ponce
Keyword(s):  
Discrete Event ◽  
Interval Data ◽  
Transition Matrices ◽  
Binary Operations ◽  
Data Errors ◽  
Fuzzy Algebra ◽  
Interval Approach ◽  
The Matrix ◽  
Interval Coefficients ◽  
Matrix Vector

Systems working in discrete time (discrete event systems, in short: DES)—based on binary operations: the maximum and the minimum—are studied in so-called max–min (fuzzy) algebra. The steady states of a DES correspond to eigenvectors of its transition matrix. In reality, the matrix (vector) entries are usually not exact numbers and they can instead be considered as values in some intervals. The aim of this paper is to investigate the eigenvectors for max–min matrices (vectors) with interval coefficients. This topic is closely related to the research of fuzzy DES in which the entries of state vectors and transition matrices are kept between 0 and 1, in order to describe uncertain and vague values. Such approach has many various applications, especially for decision-making support in biomedical research. On the other side, the interval data obtained as a result of impreciseness, or data errors, play important role in practise, and allow to model similar concepts. The interval approach in this paper is applied in combination with forall–exists quantification of the values. It is assumed that the set of indices is divided into two disjoint subsets: the E-indices correspond to those components of a DES, in which the existence of one entry in the assigned interval is only required, while the A-indices correspond to the universal quantifier, where all entries in the corresponding interval must be considered. In this paper, the properties of EA/AE-interval eigenvectors have been studied and characterized by equivalent conditions. Furthermore, numerical recognition algorithms working in polynomial time have been described. Finally, the results are illustrated by numerical examples.

Selecting optimal SpMV realizations for GPUs via machine learning

2021 ◽  
pp. 109434202199073
Author(s):  
Ernesto Dufrechou ◽  
Pablo Ezzatti ◽  
Enrique S Quintana-Ortí
Keyword(s):  
Machine Learning ◽  
Sparse Matrix ◽  
Machine Learning Techniques ◽  
Optimal Method ◽  
Learning Techniques ◽  
General Rules ◽  
Machine Learning Approach ◽  
The Matrix ◽  
Time And Energy ◽  
Matrix Vector

More than 10 years of research related to the development of efficient GPU routines for the sparse matrix-vector product (SpMV) have led to several realizations, each with its own strengths and weaknesses. In this work, we review some of the most relevant efforts on the subject, evaluate a few prominent routines that are publicly available using more than 3000 matrices from different applications, and apply machine learning techniques to anticipate which SpMV realization will perform best for each sparse matrix on a given parallel platform. Our numerical experiments confirm the methods offer such varied behaviors depending on the matrix structure that the identification of general rules to select the optimal method for a given matrix becomes extremely difficult, though some useful strategies (heuristics) can be defined. Using a machine learning approach, we show that it is possible to obtain unexpensive classifiers that predict the best method for a given sparse matrix with over 80% accuracy, demonstrating that this approach can deliver important reductions in both execution time and energy consumption.

scholarly journals Matrix forecasting to investigate the capital efficiency of the insurance market: Case of Italy

2020 ◽  
Vol 9 (3) ◽  
pp. 72-83
Author(s):  
Stefano Dell’Atti ◽  
Stefania Sylos Labini ◽  
Iryna Nyenno
Keyword(s):  
Initial Position ◽  
Development Strategies ◽  
Insurance Market ◽  
Transition Matrices ◽  
Policy Makers ◽  
Insurance Sector ◽  
Forecasting Method ◽  
The Matrix ◽  
Capital Efficiency ◽  
Scientific Results

The article aims to verify whether the matrix forecasting method is valid for predicting the insurance market development trends. The methodology is based on the application of the Franchon & Romanet matrix to the insurance market. Our results indicate that the Franchon & Romanet matrix could be usefully employed in the insurance market to identify the initial market position (calculated as financial development potential distributed through the structure of the capital funds available for insurance and financial activities) and the possible future development strategies. The core limits are concerned with the small use of the matrix methods for performance measurement of the insurance market. No empirical study has been conducted. The application of the matrix is concerned with risk management or rating transition matrices. Despite this circumstance gives originality to our paper, it poses a problem of data collection and limits the possibility to conduct the comparison with other scientific results. The construction of the matrix allows identifying the initial position of a country’s insurance market, to evaluate the possible development strategies and to choose the preferable ones. The originality of the paper consists firstly, in the innovativeness of applying to the insurance sector the tool of matrix forecasting; secondly, in providing a supporting tool to policy-makers decisions.

Application of Spline Collocation Method in Analysis of Beam and Continuous Beam

2003 ◽  
Vol 19 (2) ◽  
pp. 319-326 ◽  
Author(s):  
Lai-Yun Wu ◽  
Yang-Tzung Chen
Keyword(s):  
Collocation Method ◽  
Differential Quadrature Method ◽  
Spline Functions ◽  
Continuous Beam ◽  
Spline Collocation ◽  
Spline Collocation Method ◽  
The Matrix ◽  
Weighting Coefficients ◽  
Matrix Vector ◽  
Excellent Accuracy

ABSTRACTIn this paper, spline collocation method (SCM) is successfully extended to solve the generalized problems of beam structures. The spline functions in SCM are re-formulated by finite difference method in a systematical way that is easily understood by engineers. The manipulation of SCM is further simplified by the introduction of quintic table so that the matrix-vector governing equation can be easily formulated to solve the weighting coefficients. SCM is first examined by the problems of a generalized single-span beam undergoing various types of loadings and boundary conditions, and it is then extended to the problems of continuous beam with multiple spans. By comparing with available analytical results, differential quadrature method (DQM), if any, excellent accuracy in deflection is achieved.

Advanced three-dimensional electromagnetic modeling using a nested integral equation approach

2021 ◽  
Author(s):  
Chaojian Chen ◽  
Mikhail Kruglyakov ◽  
Alexey Kuvshinov
Keyword(s):  
Integral Equation ◽  
Three Dimensional ◽  
Region Of Interest ◽  
Electromagnetic Modeling ◽  
Multi Scale ◽  
Scale Modeling ◽  
Uniform Grids ◽  
The Matrix ◽  
Integral Equation Approach ◽  
Matrix Vector

Summary Most of the existing three-dimensional (3-D) electromagnetic (EM) modeling solvers based on the integral equation (IE) method exploit fast Fourier transform (FFT) to accelerate the matrix-vector multiplications. This in turn requires a laterally-uniform discretization of the modeling domain. However, there is often a need for multi-scale modeling and inversion, for instance, to properly account for the effects of non-uniform distant structures, and at the same time, to accurately model the effects from local anomalies. In such scenarios, the usage of laterally-uniform grids leads to excessive computational loads, both in terms of memory and time. To alleviate this problem, we developed an efficient 3-D EM modeling tool based on a multi-nested IE approach. Within this approach, the IE modeling is first performed at a large domain and on a (laterally-uniform) coarse grid, and then the results are refined in the region of interest by performing modeling at a smaller domain and on a (laterally-uniform) denser grid. At the latter stage, the modeling results obtained at the previous stage are exploited. The lateral uniformity of the grids at each stage allows us to keep using the FFT for the matrix-vector multiplications. An important novelty of the paper is a development of a “rim domain” concept which further improves the performance of the multi-nested IE approach. We verify the developed tool on both idealized and realistic 3-D conductivity models, and demonstrate its efficiency and accuracy.

scholarly journals Improved Generalized Single-Source Tangential Equivalence Principle Algorithm with Contact-Region Modeling Method for Array Structures

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Yao Han ◽  
Hanru Shao ◽  
Jianfeng Dong
Keyword(s):  
Equivalence Principle ◽  
Contact Region ◽  
Near Field ◽  
Single Source ◽  
Matrix Vector Multiplication ◽  
The Matrix ◽  
Array Structures ◽  
Fast Multipole Algorithm ◽  
Matrix Vector ◽  
Equivalence Principle Algorithm

An improved generalized single-source tangential equivalence principle algorithm (GSST-EPA) is proposed for analyzing array structures with connected elements. In order to use the advantages of GSST-EPA, the connected array elements are decomposed and computed by a contact-region modeling (CRM) method, which makes that each element has the same meshes. The unknowns of elements can be transferred onto the equivalence surfaces by GSST-EPA. The scattering matrix in GSST-EPA needs to be solved and stored only once due to the same meshes for each element. The shift invariant of translation matrices is also used to reduce the computation of near-field interaction. Furthermore, the multilevel fast multipole algorithm (MLFMA) is used to accelerate the matrix-vector multiplication in the GSST-EPA. Numerical results are shown to demonstrate the accuracy and efficiency of the proposed method.

scholarly journals Solvability of a Bounded Parametric System in Max-Łukasiewicz Algebra

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1026 ◽  
Author(s):  
Martin Gavalec ◽  
Zuzana Němcová
Keyword(s):  
Linear System ◽  
Fuzzy Systems ◽  
Polynomial Algorithm ◽  
Sufficient Conditions ◽  
Recognition Algorithm ◽  
Triangular Norm ◽  
Binary Operations ◽  
Interval Coefficients ◽  
Necessary And Sufficient ◽  
The Given

The max-Łukasiewicz algebra describes fuzzy systems working in discrete time which are based on two binary operations: the maximum and the Łukasiewicz triangular norm. The behavior of such a system in time depends on the solvability of the corresponding bounded parametric max-linear system. The aim of this study is to describe an algorithm recognizing for which values of the parameter the given bounded parametric max-linear system has a solution—represented by an appropriate state of the fuzzy system in consideration. Necessary and sufficient conditions of the solvability have been found and a polynomial recognition algorithm has been described. The correctness of the algorithm has been verified. The presented polynomial algorithm consists of three parts depending on the entries of the transition matrix and the required state vector. The results are illustrated by numerical examples. The presented results can be also applied in the study of the max-Łukasiewicz systems with interval coefficients. Furthermore, Łukasiewicz arithmetical conjunction can be used in various types of models, for example, in cash-flow system.

scholarly journals Casselman’s Basis of Iwahori Vectors and Kazhdan–Lusztig Polynomials

2019 ◽  
Vol 71 (6) ◽  
pp. 1351-1366
Author(s):  
Daniel Bump ◽  
Maki Nakasuji
Keyword(s):  
Representation Theory ◽  
Functional Equations ◽  
Transition Matrix ◽  
Principal Series ◽  
Surprising Result ◽  
Transition Matrices ◽  
Natural Basis ◽  
Series Representations ◽  
The Matrix ◽  
Lusztig Polynomials

AbstractA problem in representation theory of $p$-adic groups is the computation of the Casselman basis of Iwahori fixed vectors in the spherical principal series representations, which are dual to the intertwining integrals. We shall express the transition matrix $(m_{u,v})$ of the Casselman basis to another natural basis in terms of certain polynomials that are deformations of the Kazhdan–Lusztig R-polynomials. As an application we will obtain certain new functional equations for these transition matrices under the algebraic involution sending the residue cardinality $q$ to $q^{-1}$. We will also obtain a new proof of a surprising result of Nakasuji and Naruse that relates the matrix $(m_{u,v})$ to its inverse.

Duality results for block-structured transition matrices

1999 ◽  
Vol 36 (4) ◽  
pp. 1045-1057 ◽  
Author(s):  
Yiqiang Q. Zhao ◽  
Wei Li ◽  
Attahiru Sule Alfa
Keyword(s):  
Transition Matrix ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Transition Kernel ◽  
Transition Matrices ◽  
Duality Results ◽  
Markov Renewal Processes ◽  
The Matrix ◽  
Probabilistic Measures ◽  
Block Structured

In this paper, we consider a certain class of Markov renewal processes where the matrix of the transition kernel governing the Markov renewal process possesses some block-structured property, including repeating rows. Duality conditions and properties are obtained on two probabilistic measures which often play a key role in the analysis and computations of such a block-structured process. The method used here unifies two different concepts of duality. Applications of duality are also provided, including a characteristic theorem concerning recurrence and transience of a transition matrix with repeating rows and a batch arrival queueing model.

Sign in / Sign up

Export Citation Format

Share Document