Algorithmic Differentiation in Python with Application Examples

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PART I.1: The Forward Mode of AD by Univariate Taylor Polynomial (UTP) Arithmetic Sebastian F. Walter, Humboldt-Universität zu Berlin Algorithmic () Differentiation in Python with Application Examples Wednesday, 10.07.2010 12 / 27
Univariate Taylor Polynomial Arithmetic (UTP) Basic Observation 1: Let f : R N → R, then d dt f (x + e it) ∣ = (∇ x f (x)) T · e i = ∂f t=0 ∂x i Basic Observation 2: Hessian d 2 f (x + e i t 1 + e j t 2 ) ∣ dt 1 dt 2 ∣ t1 =t 2 =0 = e T i ∇ 2 xf (x)e j = ∂2 f ∂x i ∂x j e i = (0, . . . , 1, . . . , 0) is the i’th cartesian basis vector. Sebastian F. Walter, Humboldt-Universität zu Berlin Algorithmic () Differentiation in Python with Application Examples Wednesday, 10.07.2010 13 / 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: Code Tracing with PYADOLC and ALGOP
Page 15 and 16: Univariate Taylor Polynomial Arithm
Page 17 and 18: Live Example: Directional Derivativ
Page 19 and 20: The Reverse Mode by Hand: Recall: y
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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