A Higher-Order Compact Method in Space and Time Based on Parallel Implementation of the Thomas Algorithm

2000 ◽  
Vol 161 (1) ◽  
pp. 182-203 ◽  
Author(s):  
Alex Povitsky ◽  
Philip J. Morris
2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Qing Yao ◽  
Bingsheng Chen ◽  
Tim S. Evans ◽  
Kim Christensen

AbstractWe study the evolution of networks through ‘triplets’—three-node graphlets. We develop a method to compute a transition matrix to describe the evolution of triplets in temporal networks. To identify the importance of higher-order interactions in the evolution of networks, we compare both artificial and real-world data to a model based on pairwise interactions only. The significant differences between the computed matrix and the calculated matrix from the fitted parameters demonstrate that non-pairwise interactions exist for various real-world systems in space and time, such as our data sets. Furthermore, this also reveals that different patterns of higher-order interaction are involved in different real-world situations. To test our approach, we then use these transition matrices as the basis of a link prediction algorithm. We investigate our algorithm’s performance on four temporal networks, comparing our approach against ten other link prediction methods. Our results show that higher-order interactions in both space and time play a crucial role in the evolution of networks as we find our method, along with two other methods based on non-local interactions, give the best overall performance. The results also confirm the concept that the higher-order interaction patterns, i.e., triplet dynamics, can help us understand and predict the evolution of different real-world systems.


2011 ◽  
Vol 9 (4) ◽  
pp. 897-916 ◽  
Author(s):  
Y. V. S. S. Sanyasiraju ◽  
Nachiketa Mishra

AbstractThis paper presents an exponential compact higher order scheme for Convection-Diffusion Equations (CDE) with variable and nonlinear convection coefficients. The scheme is for one-dimensional problems and produces a tri-diagonal system of equations which can be solved efficiently using Thomas algorithm. For two-dimensional problems, the scheme produces an accuracy over a compact nine point stencil which can be solved using any line iterative approach with alternate direction implicit procedure. The convergence of the iterative procedure is guaranteed as the coefficient matrix of the developed scheme satisfies the conditions required to be positive. Wave number analysis has been carried out to establish that the scheme is comparable in accuracy with spectral methods. The higher order accuracy and better rate of convergence of the developed scheme have been demonstrated by solving numerous model problems for one and two-dimensional CDE, where the solutions have the sharp gradient at the solution boundary.


2000 ◽  
Vol 10 (02n03) ◽  
pp. 239-250 ◽  
Author(s):  
CHRISTOPH A. HERRMANN ◽  
CHRISTIAN LENGAUER

We propose the higher-order functional style for the parallel programming of algorithms. The functional language [Formula: see text], a subset of the language Haskell, facilitates the clean integration of skeletons into a functional program. Skeletons are predefined programming schemata with an efficient parallel implementation. We report on our compiler, which translates [Formula: see text] programs into C+MPI, especially on the design decisions we made. Two small examples, the n queens problem and Karatsuba's polynomial multiplication, are presented to demonstrate the programming comfort and the speedup one can obtain.


2021 ◽  
pp. 57-64
Author(s):  
Mila Mojsilović

By redefining the notion of fragmentarity and existing theoretical conceptions, from romantic fascination with ruins to the contemporary position of variability, the paper places incompleteness as the essential potentiality of form, imagination and contingency in a way that opens new spacetime categories. In the paper, fragmentation is understood as a model for interpreting reality and for examining the capacity of architectural incompleteness. Setting complexity as the context, change as the method and variability as the model for understanding the architectural contemporaries within its reality, spatial and temporal uncertainty become characteristics of the fragmentation and destabilization of relations - their reflection. This way of structuring order out of chaos, or destabilizing order for the purpose of new structures, is the complexity of a higher order. The uncaptured nature of all things, in distracting the new, transports its own limitations, thematized through places of change, separation and path - specific singularities, allowing flexibility through imperfection. The elusive nature of all things, in opening to the new, transcends its own limitations, thematizing itself through places of change, separation and cracking. These are specific singularities that allow flexibility through incompleteness, thus opening up towards new forms of reality, between uncertainty and indeterminacy - in the zone of their overlap, space and time become fragmented. This true spontaneity is the greatest complexity that carries within itself the power of change and essential potentiality - a meaning that is always just emerging. The question of the degree of incompleteness is in the core of the concept of openness, in which the alteration of form and geometry take place.


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