Automatic first- and second-order adjoints for truncated Newton

The analysis and modification of numerical programs in the context of generating and optimizing adjoint code automatically probably ranges among the technically and theoretically most challenging source transformation problems known today. A complete compiler for the target language (Fortran in our case) is needed to cover the technical side. This amounts to a mathematically motivated semantic transformation of the source code that involves the reversal of the flow of data through the program. Both the arithmetic complexity and the memory requirement can be substantial for large-scale numerical simulations. Finding the optimal data-flow reversal schedule turns out to be an NP-complete problem. The same complexity result applies to other domain-specific peephole optimizations. In this paper we present a first research prototype of the NAGWare Fortran compiler with the ability to generate adjoint code automatically.Moreover, we discuss an approach to generating second-order adjoint code for use in Newton-type algorithms for unconstrained nonlinear optimization. While the focus of this paper is mostly on the compiler issues some information on the mathematical background will be found helpful for motivational purposes