What question did this study set out to answer?

The aim is to create locally adapted hierarchical basis functions that improve preconditioning for large optimization problems in computer graphics.

January 1, 2006

Locally adapted hierarchical basis preconditioning

Key Points

The aim is to create locally adapted hierarchical basis functions that improve preconditioning for large optimization problems in computer graphics.
Developed a method for creating basis functions based on local coefficient matrix structure.
Implemented a recursive variable elimination process for efficiency.
Achieved simplification of coarse-level problems to handle inhomogeneous data effectively.
The new approach significantly outperforms existing methods in speed and efficiency.
It eliminates the need for heuristic adjustments in preconditioning levels.
Successfully integrates with modern data-parallel architectures, enhancing performance on GPUs.

Abstract

This paper develops locally adapted hierarchical basis functions for effectively preconditioning large optimization problems that arise in computer graphics applications such as tone mapping, gradient-domain blending, colorization, and scattered data interpolation. By looking at the local structure of the coefficient matrix and performing a recursive set of variable eliminations, combined with a simplification of the resulting coarse level problems, we obtain bases better suited for problems with inhomogeneous (spatially varying) data, smoothness, and boundary constraints. Our approach removes the need to heuristically adjust the optimal number of preconditioning levels, significantly outperforms previously proposed approaches, and also maps cleanly onto data-parallel architectures such as modern GPUs.

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