scholarly journals On non-negative solutions to large systems of random linear equations

2020 ◽  
Vol 552 ◽  
pp. 122544 ◽  
Author(s):  
Stefan Landmann ◽  
Andreas Engel
Keyword(s):  
Linear Equations ◽  
Large Systems

scholarly journals A method of accelerating stationary iterative methods for solving linear systems

1988 ◽  
Vol 30 (1) ◽  
pp. 1-23
Author(s):  
G. K. Robinson
Keyword(s):  
Quadratic Forms ◽  
Simple Technique ◽  
Linear Equations ◽  
Full Rank ◽  
Systems Of Equations ◽  
Linear Combinations ◽  
Simultaneous Linear Equations ◽  
Large Systems ◽  
Stationary Iterative Methods ◽  
Refined Version

AbstractThe speed of convergence of stationary iterative techniques for solving simultaneous linear equations may be increased by using a method similar to conjugate gradients but which does not require the stationary iterative technique to be symmetrisable. The method of refinement is to find linear combinations of iterates from a stationary technique which minimise a quadratic form. This basic method may be used in several ways to construct refined versions of the simple technique. In particular, quadratic forms of much less than full rank may be used. It is suggested that the method is likely to be competitive with other techniques when the number of linear equations is very large and little is known about the properties of the system of equations. A refined version of the Gauss-Seidel technique was found to converge satisfactorily for two large systems of equations arising in the estimation of genetic merit of dairy cattle.

scholarly journals On the stability of ADI methods

2017 ◽  
Vol 42 (598) ◽  
Author(s):  
Ole Østerby
Keyword(s):  
Linear Equations ◽  
Implicit Methods ◽  
Stability Properties ◽  
Require Time ◽  
Jump Discontinuities ◽  
Systems Of Linear Equations ◽  
Two Space Dimensions ◽  
Large Systems ◽  
Adi Methods ◽  
The Stability

When solving parabolic equations in two space dimensions implicit methods are preferred to the explicit method because of their better stability properties. Straightforward implementation of implicit methods require time-consuming solution of large systems of linear equations, and ADI methods are preferred instead. We expect the ADI methods to inherit the stability properties of the implicit methods they are derived from, and we demonstrate that this is partly true. The Douglas-Rachford and Peaceman-Rachford methods are absolutely stable in the sense that their growth factors are ≤ 1 in absolute value. Near jump discontinuities, however, there are differences w.r.t. how the ADI methods react to the situation: do they produce oscillations and how effectively do they damp them. We demonstrate the behaviour on two simple examples.

A global random walk on grid algorithm for second order elliptic equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Karl K. Sabelfeld ◽  
Dmitry Smirnov ◽  
Ivan Dimov ◽  
Venelin Todorov
Keyword(s):  
Random Walk ◽  
Elliptic Equations ◽  
Linear Equations ◽  
Computational Cost ◽  
Transition Matrices ◽  
Systems Of Linear Equations ◽  
Systems Of Elliptic Equations ◽  
Grid Algorithm ◽  
Large Systems ◽  
Second Order Elliptic Equations

Abstract In this paper we develop stochastic simulation methods for solving large systems of linear equations, and focus on two issues: (1) construction of global random walk algorithms (GRW), in particular, for solving systems of elliptic equations on a grid, and (2) development of local stochastic algorithms based on transforms to balanced transition matrix. The GRW method calculates the solution in any desired family of prescribed points of the gird in contrast to the classical stochastic differential equation based Feynman–Kac formula. The use in local random walk methods of balanced transition matrices considerably decreases the variance of the random estimators and hence decreases the computational cost in comparison with the conventional random walk on grids algorithms.

PARALLEL GAUSS-SEIDEL ON A TORUS NETWORK-ON-CHIP ARCHITECTURE

2012 ◽  
Vol 13 (01n02) ◽  
pp. 1250001 ◽  
Author(s):  
MOHAMMAD H. AL-TOWAIQ ◽  
KHALED DAY
Keyword(s):  
Linear Equations ◽  
Network On Chip ◽  
Multicore Architectures ◽  
Processing Elements ◽  
Recent Developments ◽  
Systems Of Linear Equations ◽  
Large Systems ◽  
On Chip ◽  
Large Linear Systems ◽  
Torus Network

Network-on-chip multicore architectures with a large number of processing elements are becoming a reality with the recent developments in technology. In these modern systems the processing elements are interconnected with regular network-on-chip (NoC) topologies such as meshes and trees. In this paper we propose a parallel Gauss-Seidel (GS) iterative algorithm for solving large systems of linear equations on a torus NoC architecture. The proposed parallel algorithm is O (Nn2/k2) time complexity for solving a system with matrix of order n on a k × k torus NoC architecture with N iterations assuming n and N are large compared to k (i.e. for large linear systems that require a large number of iterations). We show that under these conditions the proposed parallel GS algorithm has near optimal speedup.

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