Download PDF by V.P. Maslov: Asymptotics of operator and pseudo-differential equations

By V.P. Maslov

ISBN-10: 0306110148

ISBN-13: 9780306110146


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4 as well as in Chapter 4. Rather than choosing an input variable and calculating the sensitivity of every intermediate variable with respect to that input, we choose instead an output variable and calculate the sensitivity of that output with respect to each of the intermediate variables. , starting from the output variables. We use the term "reverse mode" for this technique because the label "backward differentiation" is well established for certain methods to integrate stiff ODEs [HNW96]. For our example there is only one output variable, namely, y.

I Tangents and Gradients This page intentionally left blank Chapter 2 A Framework for Evaluating Functions To read this book only a basic knowledge of differential calculus and Fortran is needed. , in [KK+86] The remark above is certainly appropriate for the first part, Chapters 2-5, of this book, if not for the rest of it. In this second chapter we address the question, What can be differentiated by our computational techniques? The formal answer is that we wish to differentiate a more or less arbitrary vectorvalued Naturally, if all we are told about a function F is that its domain T> is an open subset of R n and its range F(V) a subset of R m , then differentiation cannot proceed beyond the formal act of appending the symbol F with one or more primes.

3. Once again, the augmented evaluation trace is generated by a program for calculating derivatives. This time the augmented trace is called the reverse trace. Note that the adjoint statements are lined up vertically underneath the original statements that spawned them. 01188 v-1 As with the forward propagation method, the total floating point operation count of the added lines is a small multiple of that for the underlying code to evaluate y. 0118 obtained for x\ = dy/dx\ agrees to four digits with the value obtained for dyjdx\ = y with x\ = 1 in the forward mode.

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Asymptotics of operator and pseudo-differential equations by V.P. Maslov

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