# 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

## Synonyms

Adjoint differentiation; Back propagation; Time reversal

## Definition

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.

## Overview

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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## References

1. 1.
Arridge, S.R.: Optical tomography in medical imaging. Inverse Probl. 15, R41–R93 (1999)
2. 2.
Borup, D.T., Johnson, S.A., Kim, W.W., Berggren, M.J.: Nonperturbative diffraction tomography via Gauss-Newton iteration applied to the scattering integral equation. Ultrason. Imaging 14, 69–85 (1992)
3. 3.
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Dobson, D.: Convergence of a reconstruction method for the inverse conductivity problem. SIAM J. Appl. Math. 52, 442–458 (1992)
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Dierkes, T., Dorn, O., Natterer, F., Palamodov, V., Sielschott, H.: Fréchet derivatives for some bilinear inverse problems. Siam J. Appl. Math. 62, 2092–2113 (2002)
8. 8.
Hermann, G.: Image Reconstruction From Projections. Academic Press, San Francisco (1980)Google Scholar
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Natterer, F.: Numerical methods for bilinear inverse problems. Preprint, University of Münster, Department of Mathematics and Computer Science (1996)Google Scholar
10. 10.
Natterer, F., Wübbeling, F.: A propagation-backpropagation algorithm for ultrasound tomography. Inverse Probl. 11, 1225–1232 (1995)
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Natterer, F., Wübbeling, F.: Mathematical Methods of Image Reconstruction, p. 275. SIAM, Philadelphia (2001)
12. 12.
Vögeler, M.: Reconstruction of the three-dimensional refractive index in electromagnetic scattering by using a propagation-backpropagation method. Inverse Probl. 19, 739–753 (2003)