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Efficient Non‐linear Optimization via Multi‐scale Gradient Filtering
Author(s) -
Martin Tobias,
Joshi Pushkar,
Bergou Miklós,
Carr Nathan
Publication year - 2013
Publication title -
computer graphics forum
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.578
H-Index - 120
eISSN - 1467-8659
pISSN - 0167-7055
DOI - 10.1111/cgf.12019
Subject(s) - broyden–fletcher–goldfarb–shanno algorithm , gradient descent , conjugate gradient method , computer science , nonlinear conjugate gradient method , convergence (economics) , mathematical optimization , algorithm , gradient method , rate of convergence , scale (ratio) , manifold (fluid mechanics) , mathematics , artificial intelligence , artificial neural network , key (lock) , mechanical engineering , computer network , physics , asynchronous communication , computer security , quantum mechanics , engineering , economics , economic growth
We present a method for accelerating the convergence of continuous non‐linear shape optimization algorithms. We start with a general method for constructing gradient vector fields on a manifold, and we analyse this method from a signal processing viewpoint. This analysis reveals that we can construct various filters using the Laplace–Beltrami operator of the shape that can effectively separate the components of the gradient at different scales. We use this idea to adaptively change the scale of features being optimized to arrive at a solution that is optimal across multiple scales. This is in contrast to traditional descent‐based methods, for which the rate of convergence often stalls early once the high frequency components have been optimized. We demonstrate how our method can be easily integrated into existing non‐linear optimization frameworks such as gradient descent, Broyden–Fletcher–Goldfarb–Shanno (BFGS) and the non‐linear conjugate gradient method. We show significant performance improvement for shape optimization in variational shape modelling and parameterization, and we also demonstrate the use of our method for efficient physical simulation.