By Adrian Sandu (auth.), Christian H. Bischof, H. Martin Bücker, Paul Hovland, Uwe Naumann, Jean Utke (eds.)

This assortment covers advances in computerized differentiation conception and perform. laptop scientists and mathematicians will find out about fresh advancements in computerized differentiation idea in addition to mechanisms for the development of sturdy and robust automated differentiation instruments. Computational scientists and engineers will enjoy the dialogue of assorted functions, which supply perception into powerful suggestions for utilizing automated differentiation for inverse difficulties and layout optimization.

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Is there a data-flow reversal with cost n + p that uses k ≤ K memory units? Theorem 1. FCDR is NP-complete. Proof. The proof is by reduction from V ERTEX C OVER as described in [13]. 2 DAG R EVERSAL Given are a DAG G and integers K and C such that n ≤ K ≤ n + p and K ≤ C. Is there a data-flow reversal that uses at most K memory units and costs c ≤ C? Theorem 2. DAGR is NP-complete. Proof. The idea behind the proof in [13] is the following. An algorithm for DAGR can be used to solve FCDR as follows: For K = n + p “store-all” is a solution of DAGR for C = n + p.

Dummy calls can be performed in any of the other seven modes. Figure 2 illustrates the reversal in split (b), classical joint (c), and joint with result checkpointing (d) modes for the call tree in (a). The order of the calls is from left to right and depth-first. Call Tree Reversal 15 For the purpose of conceptual illustration we assume that the sizes of the tapes of all three subroutine calls in Fig. 2 (a) as well as the corresponding computational complexities are identically equal to 2 (memory units/floating-point operation (flop) units).

X0 = x0 · sin(x0 · x1 ); x1 = x0 /x1 ; x0 = cos(x0 ); x0 = sin(x0 ); x1 = cos(x1 ). (2) A representation as in (1) is obtained easily by mapping the physical memory space (x0 , x1 ) onto the single-assignment memory space (v−1 , . . , v7 ). The problem faced by all developers of adjoint code compiler technology is to generate the code such that for a given amount of persistent memory the values required for a correct evaluation of the adjoints can be recovered efficiently by combinations of storing and recomputing [6, 10, 11].