Algorithmic Differentiation in Python with Application Examples

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Objective Function of Opt. Exp. Design Part I: Computation of J 1 and J 2 J 1 [n mts, :] = √ wmts d (h(tnmts, x(tnmts; s, u(tnmts; q), p))) σ nmts (x(t nmts ; s, u(t nmts ; q), q) d(p, s) J 2 = d r(q, p, s) d(p, s) Part II: Numerical Linear Algebra „ J T C(J 1 , J 2 ) = (I, 0) 1 J 1 J2 T « −1 „ « I J 2 0 0 ” = “Q T 2 (Q 2J1 T J 1Q T 2 )−1 Q 2 Φ = λ 1 (C) , max. eigenvalue where J2 T = (QT 1 , QT 2 )(L, 0)T Computational Graph [p] [h], [r] [J 1 ], [J 2 ] [C] [Φ] [q] [s] [x 0 ] [x 1 ] [x 2 ] [x 3 ] [x 4 ] . . . [x N mts−1] [x N mts ] statex(t) atmeasurementtimes (mts) independent/dependent variables N mts Number measurement times, w measurement weight, σ std of a measurement, q controls, p nature Sebastian givenF. Walter, parameter, Humboldt-Universität s pseudo-Parameter zu Berlin Algorithmic ()(e.g. initial Differentiation values), in uPython control with functions Application Examples Wednesday, 10.07.2010 26 / 27
Algorithm: Forward UTPM of the Rectangular QR Decomposition input : [A] D = [A 0 , . . . , A D− 1], where A d ∈ R M×N , d = 0, . . . , D − 1, M ≥ N. output: [Q] D = [Q 0 , . . . , Q D− 1] matrix with orthonormal column vectors, where Q d ∈ R M×N , d = 0, . . . , D − 1 output: [R] D = [R 0 , . . . , R D− 1] upper triangular, where R d ∈ R N×N , d = 0, . . . , D − 1 Q 0 , R 0 = qr (A 0 ) for d = 1 to D − 1 do ∆F = A d − P d−1 k=1 Q d−kR k S = − 1 P d−1 2 k=1 QT d−k Q k P L ◦ X = P L ◦ (Q T 0 ∆FR−1 0 − S) X = P L ◦ X − (P L ◦ X) T R d = Q T 0 ∆F − (S + X)R 0 Q d = (∆F − Q 0 R d )R −1 0 end Sebastian F. Walter, Humboldt-Universität zu Berlin Algorithmic () Differentiation in Python with Application Examples Wednesday, 10.07.2010 27 / 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 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: Optimum Experimental Design in Chem
Page 29 and 30: Sebastian F. Walter, Humboldt-Unive

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