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1 The adjoint method - Stanford University Computer Science

PDE-constrained optimization and the adjoint method1 Andrew M. BradleyOctober 15, 2019 (original November 16, 2010)PDE-constrained optimization and the adjoint method for solving these and re-lated problems appear in a wide range of application domains. Often the adjointmethod is used in an application without explanation. The purpose of this tuto-rial is to explain the method in detail in a general setting that is kept as simpleas use the following notation: the total derivative (gradient) is denoted dx(usually denoted d( )/dxor x); the partial derivative, x(usually, ( )/ x); thedifferential, d. We also use the notationfxfor both partial and total derivativeswhen we think the meaning is clear from context.

timization problem min pf or to assess the sensitivity of f to the elements of p. 1This document is licensed under CC BY 4.0. One method to approximate d pf is to compute n p nite di erences over the elements of p. Each nite di erence computation requires solving g(x;p) = 0. For moderate to large n p, this can be quite costly.

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