The Mathematical Institute, University of Oxford, Eprints Archive

Automatic Frechet differentiation for the numerical solution of boundary-value problems

Birkisson, Asgeir and Driscoll, Tobin A. (2010) Automatic Frechet differentiation for the numerical solution of boundary-value problems. Technical Report. ACM. (Submitted)

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Abstract

A new solver for nonlinear boundary-value problems (BVPs) in Matlab is presented, based on the Chebfun software system for representing functions and operators automatically as numerical objects. The solver implements Newton's method in function space, where instead of the usual Jacobian matrices, the derivatives involved are Frechet derivatives. A major novelty of this approach is the application of automatic differentiation (AD) techniques to compute the operator-valued Frechet derivatives in the continuous context. Other novelties include the use of anonymous functions and numbering of each variable to enable a recursive, delayed evaluation of derivatives with forward mode AD. The AD techniques are applied within a new Chebfun class called chebop which allows users to set up and solve nonlinear BVPs in a few lines of code, using the "nonlinear backslash" operator (\). This framework enables one to study the behaviour of Newton's method in function space.

Item Type:Technical Report (Technical Report)
Subjects:H - N > Numerical analysis
Research Groups:Numerical Analysis Group
ID Code:1023
Deposited By:Lotti Ekert
Deposited On:30 Nov 2010 07:13
Last Modified:30 Nov 2010 07:13

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