Encyclopedia of Applied and Computational Mathematics

2015 Edition
| Editors: Björn Engquist

Adjoint Methods as Applied to Inverse Problems

  • Frank Natterer
Reference work entry
DOI: https://doi.org/10.1007/978-3-540-70529-1_2


Adjoint differentiation; Back propagation; Time reversal


Adjoint methods are iterative methods for inverse problems of partial differential equations. They make use of the adjoint of the Fréchet derivative of the forward map. Applying this adjoint to the residual can be viewed as time reversal, back propagation, or adjoint differentiation.


Inverse problems for linear partial differential equations are nonlinear problems, but they often have a bilinear structure; see [9]. This structure can be used for iterative methods. As an introduction, see [11].

Wave Equation Imaging

As a typical example that has all the relevant features, we consider an inverse problem for the wave equation. Let Ω be a domain in R n, n > 1 and T = [0, t 1]. Let u j be the solution of
$$\displaystyle\begin{array}{rcl} \frac{\partial ^{2}u_{j}} {\partial ^{2}t} = f\Delta u_{j}\mbox{ in }\varOmega \times T,& &{}\end{array}$$
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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Frank Natterer
    • 1
  1. 1.Department of Mathematics and Computer Science, Institute of Computational Mathematics and InstrumentalUniversity of MünsterMünsterGermany