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Principles and Techniques of Automatic Differentiation

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Computing accurate derivatives of a numerical model is a crucial task in many domains of Scientific Computing, in particular for gradient-based optimization and inverse problems. Automatic Differentiation (AD) is a software technique to obtain derivatives of functions provided as programs. Given a numerical model F implemented as a program P, AD adapts or transforms P into a new program that computes derivatives of F. We show the mathematical formalization that both justifies AD and explains its limitations. We shortly describe the software analyses that allow AD tools to produce more efficient code. We focus on the adjoint mode of AD, arguably the only way to obtain gradients at a reasonable cost, and show two real Scientific Computing applications. We give a brief panorama of current AD tools and conclude on research directions.

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