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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.

The expression f x g 1 is a row vector times an n x n x matrix and may be understood in terms of linear algebra as the solution to the linear equation gT x = fT x; (2) where Tis the matrix transpose. The matrix conjugate transpose (just the trans-pose when working with reals) is also called the matrix adjoint, and for this reason,

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