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1 The adjoint method

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.

The two equations di er in form only by the adjoint. If g(x;p) = 0 is a nonlinear equation, then software that solves the system for xgiven a particular value for pquite likely solves, at least approximately, a sequence of linear equations of the form @ xg(x;p) x= g(x;p): (3) @ xg= g x is the Jacobian matrix for the function g(x;p), and (3) is ...

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