Browsing by Subject "Grouping"

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Open Access
Analyzing large sparse Markov chains of Kronecker products
(IEEE, 2009) Dayar, Tuğrul
Kronecker products are used to define the underlying Markov chain (MC) in various modeling formalisms, including compositional Markovian models, hierarchical Markovian models, and stochastic process algebras. The motivation behind using a Kronecker structured representation rather than a flat one is to alleviate the storage requirements associated with the MC. With this approach, systems that are an order of magnitude larger can be analyzed on the same platform. In the Kronecker based approach, the generator matrix underlying the MC is represented using Kronecker products [6] of smaller matrices and is never explicitly generated. The implementation of transient and steady-state solvers rests on this compact Kronecker representation, thanks to the existence of an efficient vector-Kronecker product multiplication algorithm known as the shuffle algorithm [6]. The transient distribution can be computed through uniformization using vector-Kronecker product multiplications. The steady-state distribution also needs to be computed using vector-Kronecker product multiplications, since direct methods based on complete factorizations, such as Gaussian elimination, normally introduce new nonzeros which cannot be accommodated. The two papers [2], [10] provide good overviews of iterative solution techniques for the analysis of MCs based on Kronecker products. Issues related to reachability analysis, vector-Kronecker product multiplication, hierarchical state space generation in Kronecker based matrix representations for large Markov models are surveyed in [5]. Throughout our discussion, we make the assumption that the MC at hand does not have unreachable states, meaning it is irreducible. And we take an algebraic view [7] to discuss recent results related to the analysis of MCs based on Kronecker products independently from modeling formalisms. We provide background material on the Kronecker representation of the generator matrix underlying a CTMC, show that it has a rich structure which is nested and recursive, and introduce a small CTMC whose generator matrix is expressed as a sum of Kronecker products; this CTMC is used as a running example throughout the discussion. We also consider preprocessing of the Kronecker representation so as to expedite numerical analysis. We discuss permuting the nonzero structure of the underlying CTMC symmetrically by reordering, changing the orders of the nested blocks by grouping, and reducing the size of the state space by lumping. The steady-state analysis of CTMCs based on Kronecker products is discussed for block iterative methods, multilevel methods, and preconditioned projection methods, respectively. The results can be extended to DTMCs based on Kronecker products with minor modifications. Areas that need further research are mentioned as they are discussed. Our contribution to this area over the years corresponds to work along iterative methods based on splittings and their block versions [11], associated preconditioners to be used with projection methods [4], near complete decomposability [8], a method based on iterative disaggregation for a class of lumpable MCs [9], a class of multilevel methods [3], and a recent method based on decomposition for weakly interacting subsystems [1]. © 2009 IEEE.
Open Access
The effects of the design and organization of learning environments on creativity : the case of two sixth grade art-rooms
(2000-05) Hasırcı, Deniz
This study analyzes the effects of the design and organization of learning evironments on creativity. Two types of organization -flexibility and grouping- inside the learning environment are influential on students' creativity, and this study aims to find which organization has more impact and in what ways. Furthermore, it dwells upon the physical characteristics that provoke students' interest and motivation, and that provide the ground for creativity to flourish on, in a learning environment. The four elements of creativity -the person, process, product, and environment- come together to clearly and completely define creativity; these four elements form the structure of the research. Two sixth grade art rooms have been chosen as the setting. one organized according to the idea of flexibility, and the other on grouping. Art rooms have been chosen because creativity can be more readily observed in art compared to other fields, and this age group has been chosen as children in this age group are at the peak point of creativity. After this age, creativity either stays at the same level or starts to diminish with the effects of social rules and regulations; thus results would be more informative. Questionnaires for the students and teachers; observations of each child, her/his creative process, product, and the characteristics of the art-room; and a 1/20 model of the desired art-room made by each student, were assessed in order to be able to form a complete picture of creativity in each art room. The main objective of the research was to obtain results that would define which characteristics -physical and social- enhance creativity in a learning environment. These characteristics are analyzed, the two schools are compared, and further research is proposed according to the findings.