Application of projection algorithms to differential equations: boundary value problems
This work offers a novel approach for solving boundary value problems, potentially benefiting researchers in numerical analysis and differential equations, but the results are preliminary and lack concrete metrics.
The paper reformulates second-order boundary value problems as feasibility problems on hypersurfaces, enabling the use of the Douglas-Rachford method. The method shows stability on examples where Newton's method fails, though no quantitative performance numbers are provided.
The Douglas-Rachford method has been employed successfully to solve many kinds of non-convex 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 valued 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 examples of BVPs, including cases where the traditional Newton's method fails.