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Curvilinear linesearch for tensor methods
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DOI：10.1137/S1064827502406658.png)
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This paper presents a curvilinear linesearch for use when solving nonlinear equations with tensor methods. Standard tensor methods use a combination of the tensor step and Newton step in a linesearch, but they are handled separately and in an ad hoc manner. Our curvilinear linesearch combines the two directions in a single parametric step, which guarantees a monotonic decrease on the tensor model and also asymptotically approaches the Newton direction as the step length shrinks to zero, thus guaranteeing descent on the nonlinear equations. Numerical experiments on a set of 35 small-scale problems, drawn primarily from the constrained and unconstrained testing environment, show an 18 - 23% improvement ( in terms of function evaluations) over previous tensor linesearches when using quadratic backtracking and a 41 - 83% improvement when halving lambda before each trial. Our results also suggest that the curvilinear linesearch is more robust than linesearch-based Newton's method or the standard tensor linesearch, producing fewer failures than either method. Experiments on two large-scale fluid flow problems complement the small-scale results and give preliminary indication of the applicability of the tensor method and the efficiency of the curvilinear linesearch. The theoretical properties, coupled with the better performance, make this a desirable improvement over previous tensor method linesearches.
Keyword:
nonlinear systems
tensor methods
Newton's method
linesearch
curvilinear linesearch
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