Affine Invariant Adaptive Newton Codes for Discretized PDEs

Please always quote using this URN: urn:nbn:de:0297-zib-7005
  • The paper deals with three different Newton algorithms that have recently been worked out in the general frame of affine invariance. Of particular interest is their performance in the numerical solution of discretized boundary value problems (BVPs) for nonlinear partial differential equations (PDEs). Exact Newton methods, where the arising linear systems are solved by direct elimination, and inexact Newton methods, where an inner iteration is used instead, are synoptically presented, both in affine invariant convergence theory and in numerical experiments. The three types of algorithms are: (a) affine covariant (formerly just called affine invariant) Newton algorithms, oriented toward the iterative errors, (b) affine contravariant Newton algorithms, based on iterative residual norms, and (c) affine conjugate Newton algorithms for convex optimization problems and discrete nonlinear elliptic PDEs.

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Author:Peter Deuflhard, Ulrich Nowak, Martin WeiserORCiD
Document Type:ZIB-Report
Tag:Affine invariant Newton methods; adaptive trust region methods; global Newton methods; inexact Newton methods; nonlinear partial differential equa
MSC-Classification:65-XX NUMERICAL ANALYSIS / 65Hxx Nonlinear algebraic or transcendental equations / 65H10 Systems of equations
65-XX NUMERICAL ANALYSIS / 65Hxx Nonlinear algebraic or transcendental equations / 65H20 Global methods, including homotopy approaches [See also 58C30, 90C30]
Date of first Publication:2002/10/24
Series (Serial Number):ZIB-Report (02-33)
Published in:Appeared in: P. Deuflhard: Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms. Springer 2004. Series Computational Mathemtics, 35