APPLICATION OF PROJECTION ALGORITHMS TO DIFFERENTIAL EQUATIONS: BOUNDARY VALUE PROBLEMS

The Douglas–Rachford method has been employed successfully to solve many kinds of nonconvex feasibility problems. In particular, recent research has shown surprising stability for the method when it is applied to finding the intersections of hypersurfaces. Motivated by these discoveries, we reformul...

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Bibliographic Details
Published in:The ANZIAM journal 2019-01, Vol.61 (1), p.23-46
Main Authors: LAMICHHANE, BISHNU P., LINDSTROM, SCOTT B., SIMS, BRAILEY
Format: Article
Language:English
Subjects:
Algorithms
Approximation
Boundary value problems
Differential equations
Feasibility
Hyperspaces
Mathematical analysis
Methods
Ordinary differential equations
Stability
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Summary:The Douglas–Rachford method has been employed successfully to solve many kinds of nonconvex feasibility problems. In particular, recent research has shown surprising stability for the method when it is applied to finding the intersections of hypersurfaces. Motivated by these discoveries, we reformulate a second order boundary value problem (BVP) as a feasibility problem where the sets are hypersurfaces. We show that such a problem may always be reformulated as a feasibility problem on no more than three sets and is well suited to parallelization. We explore the stability of the method by applying it to several BVPs, including cases where the traditional Newton’s method fails.
ISSN:1446-1811
1446-8735
DOI:10.1017/S1446181118000391