Advances in Automatic Differentiation (Lecture Notes in by Christian H. Bischof, H. Martin Bücker, Paul Hovland, Uwe PDF

By Christian H. Bischof, H. Martin Bücker, Paul Hovland, Uwe Naumann, Jean Utke

ISBN-10: 3540689354

ISBN-13: 9783540689355

ISBN-10: 3540689427

ISBN-13: 9783540689423

This assortment covers advances in automated differentiation conception and perform. computing device scientists and mathematicians will know about fresh advancements in automated differentiation conception in addition to mechanisms for the development of strong and robust computerized differentiation instruments. Computational scientists and engineers will enjoy the dialogue of varied functions, which supply perception into potent suggestions for utilizing computerized differentiation for inverse difficulties and layout optimization.

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Extra resources for Advances in Automatic Differentiation (Lecture Notes in Computational Science and Engineering)

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To ensure trust in the AD process, we propose to shift the burden of proof from the AD client to the AD producer by using the proof-carrying code paradigm [12]: an AD software must provide a machine-checkable proof for the correctness of an AD generated code or a counter-example demonstrating for example that the input code is not piecewise differentiable; an AD user can check the correctness proof using a simple program that is polynomial in the size of the given proof. In a more foundational (as opposed to applied) perspective, we show that at least, in some simple cases, one can establish the correctness of a mechanical AD transformation and certain static analyses used to that end by using a variant of Hoare logic [10, Chap.

The assignment rule (asgn), expresses that if the lhs z is active in a post-state, then there exists a variable t of the rhs e, which becomes active because it is usefully used by the calculation of z. If z is passive in a post-state, then the pre-state is the same. The sequence rule (seq) tells us that if we have proved {φ }s1 {φ0 } and {φ0 }s2 {ψ }, then we have {φ }s1 ; s2 {ψ }. This rule enables us to compose the proofs of individual components of a sequence of statements by using intermediate conditions.

Determinant If we define A to be the matrix of co-factors of A, then det A = ∑ Ai, j Ai, j , A−1 = (det A)−1 AT . j for any fixed choice of i. If C = det A, it follows that ∂C = Ai, j ∂ Ai, j =⇒ dC = ∑ Ai, j dAi, j = C Tr(A−1 dA). i, j Hence, in forward mode we have ˙ C˙ = C Tr(A−1 A), while in reverse mode C and C are both scalars and so we have C dC = Tr(CC A−1 dA) and therefore A = CC A−T . Note: in a paper in 1994 [10], Kubota states that the result for the determinant is well known, and explains how reverse mode differentiation can therefore be used to compute the matrix inverse.

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Advances in Automatic Differentiation (Lecture Notes in Computational Science and Engineering) by Christian H. Bischof, H. Martin Bücker, Paul Hovland, Uwe Naumann, Jean Utke


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