Unstructured Primal-Dual Mesh Improvement: Target Matrix Optimization Paradigm (Final Technical Report)

A New Hybrid Preconditioner for the Interior Point Method

TEMA (São Carlos) ◽

10.5540/tema.2019.020.02.359 ◽

2019 ◽

Vol 20 (2) ◽

pp. 359

Author(s):

Manolo Rodriguez Heredia ◽

Cecilia Orellana Castro ◽

Aurélio Ribeiro Leitte Oliveira

Keyword(s):

Interior Point ◽

Interior Point Method ◽

Large Scale ◽

Hybrid Approach ◽

Original Data ◽

Point Method ◽

Cholesky Factorization ◽

Normal Equation ◽

Matrix Optimization ◽

Primal Dual

This study aims to improve the computation of the search direction in the primal-dual Interior Point Method through preconditioned iterative methods. It is about a hybrid approach that combines the Controlled Cholesky Factorization preconditioner and the Splitting preconditioner. This approach has shown good results, however, in these preconditioners there are factors that reduce their efficiency, such as faults on the diagonal when performing the Cholesky factorization, as well as a demand for excessive memory, among others. Thus, some modifications are proposed in these preconditioners, as well as a new phase change, in order toimprove the performance of the hybrid preconditioner. In the Controlled Cholesky Factorization, the parameters that control the filling and the correction of the faults which occur on the diagonal are modified. It considers the relationship between the components from Controlled Cholesky Factorization obtained before and after the fault on the diagonal. In the Splitting preconditioner, in turn, a sparse base is constructed through an appropriate ordering of the columns from constrained matrix optimization problem. In addition, a theoretical result is presented, which shows that, with the proposed ordering, the condition number of the preconditioned Normal Equation matrix with the Splitting preconditioner is uniformly limited by an amount that depends only on the original data of the problem and not on the iteration of the Interior Point Method. Numerical experiments with large scale problems, corroborate the robustness and computational efficiency from this approach.

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Primal/Dual Mesh with Application to Triangular/Simplex Mesh and Delaunay/Voronoi

The Insight Journal ◽

10.54294/9lt70e ◽

2012 ◽

Author(s):

Humayun Irshad ◽

Stephane Rigaud ◽

Alexandre Gouaillard

Keyword(s):

Data Structure ◽

Source Code ◽

Primal Dual ◽

Primal And Dual ◽

Dual Mesh ◽

As If

This document describes an extension of ITK to handle both primal and dual meshes simultaneously. This paper describe in particular the data structure, an extension of itk::QuadEdgeMesh, a filter to compute and add to the the structure the dual of an existing mesh, and an adaptor which let a down- ward pipeline process the dual mesh as if it was a native itk::QuadEdgeMesh. The new data structure, itk::QuadEdgeMeshWithDual, is an extension of the already existing itk::QuadEdgeMesh, which already included by default the due topology, to handle dual geometry as well. Two types of primal meshes have been specifically illustrated: triangular / simplex meshes and Voronoi / Delaunay. A functor mechanism has been implemented to allow for different kind of computation of the dual geometry. This paper is accompanied with the source code and examples.

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