Automatic Frechet Differentiation for the Numerical Solution of Boundary-Value Problems

作者：Birkisson Asgeir^*; Driscoll Tobin A

来源：ACM Transactions on Mathematical Software, 2012, 38(4): 26.

DOI：10.1145/2331130.2331134

• Algorithms
• Design
• Performance
• Chebfun
• Newton%26apos
• s method in function space
• linearization of boundary-value problems
• object-oriented MATLAB

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%26apos;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, both scalar and systems of coupled equations, in a few lines of code, using the %26quot;nonlinear backslash%26quot; operator (\). This framework enables one to study the behaviour of Newton%26apos;s method in function space.

• 出版日期2012-8