Algorithm MGB to solve highly nonlinear elliptic PDEs in Õ(n) FLOPS

06/16/2023

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by   Sébastien Loisel, et al.

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We introduce Algorithm MGB (Multi Grid Barrier) for solving highly nonlinear convex Euler-Lagrange equations. This class of problems includes many highly nonlinear partial differential equations, such as p-Laplacians. We prove that, if certain regularity hypotheses are satisfied, then our algorithm converges in Õ(1) damped Newton iterations, or Õ(n) FLOPS, where the tilde indicates that we neglect some polylogarithmic terms. This the first algorithm whose running time is proven optimal in the big-Õ sense. Previous algorithms for the p-Laplacian required Õ(√(n)) damped Newton iterations or more.