Abstract. We apply an optimized method to the adjoint generation of a time-evolving land ice model through algorithmic differentiation (AD). The optimization involves a special treatment of the fixed-point iteration required to solve the nonlinear stress balance, which differs from a straightforward application of AD software, and leads to smaller memory requirements and in some cases shorter computation times of the adjoint. The optimization is done via implementation of the algorithm of Christianson (1994) for reverse accumulation of fixed-point problems, with the AD tool OpenAD. For test problems, the optimized adjoint is shown to have far lower memory requirements, potentially enabling larger problem sizes on memory-limited machines. In the case of the land ice model, implementation of the algorithm allows further optimization by having the adjoint model solve a sequence of linear systems with identical (as opposed to varying) matrices, greatly improving performance. The methods introduced here will be of value to other efforts applying AD tools to ice models, particularly ones which solve a hybrid shallow ice/shallow shelf approximation to the Stokes equations.

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One of the critical problems associated with emerging chip multiprocessors (CMPs) is the management of on-chip shared cache space. Unfortunately, single processor centric data locality optimization schemes may not work well in the CMP case as data accesses from multiple cores can create conflicts in the shared cache space. The main contribution of this paper is a compiler directed code restructuring scheme for enhancing locality of shared data in CMPs. The proposed scheme targets the last level shared cache that exist in many commercial CMPs and has two components, namely, allocation, which determines the set of loop iterations assigned to each core, and scheduling, which determines the order in which the iterations assigned to a core are executed. Our scheme restructures the application code such that the different cores operate on shared data blocks at the same time, to the extent allowed by data dependencies. This helps to reduce reuse distances for the shared data and improves onchip cache performance. We evaluated our approach using the Splash-2 and Parsec applications through both simulations and experiments on two commercial multi-core machines. Our experimental evaluation indicates that the proposed data locality optimization scheme improves inter-core conflict misses in the shared cache by 67% on average when both allocation and scheduling are used. Also, the execution time improvements we achieve (29% on average) are very close to the optimal savings that could be achieved using a hypothetical scheme.

Stencil loops are a common motif in computations including convolutional neural networks, structured-mesh solvers for partial differential equations, and image processing. Stencil loops are easy to parallelise, and their fast execution is aided by compilers, libraries, and domain-specific languages. Reverse-mode automatic differentiation, also known as algorithmic differentiation, autodiff, adjoint differentiation, or back-propagation, is sometimes used to obtain gradients of programs that contain stencil loops. Unfortunately, conventional automatic differentiation results in a memory access pattern that is not stencil-like and not easily parallelisable.In this paper we present a novel combination of automatic differentiation and loop transformations that preserves the structure and memory access pattern of stencil loops, while computing fully consistent derivatives. The generated loops can be parallelised and optimised for performance in the same way and using the same tools as the original computation. We have implemented this new technique in the Python tool PerforAD, which we release with this paper along with test cases derived from seismic imaging and computational fluid dynamics applications. Figure 1: Stencils with gather access occur in convolutions and many other applications and are easy to parallelise. Automatic differentiation or back-propagation transforms this operation into a scatter update, which is hard to parallelise without race conditions. With adjoint stencils, backpropagation can be performed by using only easily parallelisable stencils with gather access.

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