Algorithmic Differentiation in Python with Application Examples

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PART I.2: The Reverse Mode of AD Sebastian F. Walter, Humboldt-Universität zu Berlin Algorithmic () Differentiation in Python with Application Examples Wednesday, 10.07.2010 18 / 27
The Reverse Mode by Hand: Recall: y = f (x) = sin(x 1 + cos(x 2 )x 1 ) independent v −1 = x 1 = 3 independent v 0 = x 2 = 7 v 1 = φ 1 (v 0 ) = cos(v 0 ) v 2 = φ 2 (v 1 , v −1 ) = v 1 v −1 v 3 = φ 3 (v −1 , v 2 ) = v −1 + v 2 v 4 = φ 4 (v 3 ) = sin(v 3 ) dependent y = v 4 Reverse Mode by Hand: Successive Pullbacks dy = dφ 4 (v 3 ) = ∂φ 4(z) ∂z ˛ dv 3 = cos(v 3 ) ˛z=v3 = ¯v 3 dφ 3 (v −1 , v 2 ) = ¯v 3 dv −1 + ¯v 3 |{z} |{z} =¯v −1 = (¯v −1 + ¯v 2 v 1 ) dv −1 + ¯v 2 v −1 dv 1 | {z } | {z } =¯v −1 =¯v 1 = ¯v −1 dv −1 + (−¯v 1 sin(v 0 )) dv 0 | {z } =¯v 0 Interpretation: ¯v −1 ≡ df and ¯v dx 0 ≡ df 1 dx 2 Need to store v 0 , v 1 , v 3 , v 4 for the reverse mode! dv 3 | {z } =¯v 3 =¯v 2 dv 2 Sebastian F. Walter, Humboldt-Universität zu Berlin Algorithmic () Differentiation in Python with Application Examples Wednesday, 10.07.2010 19 / 27
Page 1 and 2: Algorithmic Differentiation in Pyth
Page 3 and 4: What is Algorithmic Differentiation
Page 5 and 6: Why not Symbolic Differentiation A
Page 7 and 8: Why not Finite Differences Problem:
Page 9 and 10: Part I: Intro Algorithmic Different
Page 11 and 12: Code Tracing with PYADOLC and ALGOP
Page 13 and 14: Univariate Taylor Polynomial Arithm
Page 15 and 16: Univariate Taylor Polynomial Arithm
Page 17: Live Example: Directional Derivativ
Page 21 and 22: Forward Mode vs Reverse Mode Task:
Page 23 and 24: A Minimal Surface Problem part of u
Page 25 and 26: Optimum Experimental Design in Chem
Page 27 and 28: Algorithm: Forward UTPM of the Rect
Page 29 and 30: Sebastian F. Walter, Humboldt-Unive

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