Iterative methods for sparse systems of linear equations with application to guided-wave optics

2001 ◽  
Author(s):  
Gerard Sleijpen
Keyword(s):  
Iterative Methods ◽  
Linear Equations ◽  
Guided Wave ◽  
Wave Optics ◽  
Systems Of Linear Equations ◽  
Sparse Systems

Mutual Embeddings

2015 ◽  
Vol 15 (01n02) ◽  
pp. 1550001
Author(s):  
ILKER NADI BOZKURT ◽  
HAI HUANG ◽  
BRUCE MAGGS ◽  
ANDRÉA RICHA ◽  
MAVERICK WOO
Keyword(s):  
Iterative Methods ◽  
Lower Bounds ◽  
Linear Equations ◽  
Graph Embedding ◽  
Open Problems ◽  
Linear Arrays ◽  
Systems Of Linear Equations

This paper introduces a type of graph embedding called a mutual embedding. A mutual embedding between two n-node graphs [Formula: see text] and [Formula: see text] is an identification of the vertices of V1 and V2, i.e., a bijection [Formula: see text], together with an embedding of G1 into G2 and an embedding of G2 into G1 where in the embedding of G1 into G2, each node u of G1 is mapped to π(u) in G2 and in the embedding of G2 into G1 each node v of G2 is mapped to [Formula: see text] in G1. The identification of vertices in G1 and G2 constrains the two embeddings so that it is not always possible for both to exhibit small congestion and dilation, even if there are traditional one-way embeddings in both directions with small congestion and dilation. Mutual embeddings arise in the context of finding preconditioners for accelerating the convergence of iterative methods for solving systems of linear equations. We present mutual embeddings between several types of graphs such as linear arrays, cycles, trees, and meshes, prove lower bounds on mutual embeddings between several classes of graphs, and present some open problems related to optimal mutual embeddings.

scholarly journals PARALLEL APPROACH TO SOLVE OF THE DIRECT SOLUTION OF LARGE SPARSE SYSTEMS OF LINEAR EQUATIONS

2017 ◽  
Vol 13 ◽  
pp. 16
Author(s):  
Michal Bošanský ◽  
Bořek Patzák
Keyword(s):  
Finite Element ◽  
Numerical Solution ◽  
Distributed Memory ◽  
Linear Equations ◽  
Direct Solution ◽  
Parallel Solution ◽  
Finite Element Software ◽  
Systems Of Linear Equations ◽  
Sparse Systems ◽  
Symmetric Systems

The paper deals with parallel approach for the numerical solution of large, sparse, non-symmetric systems of linear equations, that can be part of any finite element software. In this contribution, the differences between the sequential and parallel solution are highlighted and the approach to efficiently interface with distributed memory version of SuperLU solver is described.

Iterative Methods for Systems of Linear Equations

2021 ◽  
pp. 235-339
Author(s):  
Peter Knabner ◽  
Lutz Angermann
Keyword(s):  
Iterative Methods ◽  
Linear Equations ◽  
Systems Of Linear Equations
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