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In research on inverse problems, it's common to construct a synthetic data set from a known set of parameters and then test whether the inversion technique can reconstruct those parameters. In doing so, it's important to add appropriate levels of random noise to the synthetic data. Furthermore, if the method used to compute the synthetic data is based on a finite difference or finite element grid, it's also important to not use that same grid in the inversion process. Otherwise, the inversion process is really inverting the approximate numerical forward model. The phrase "inverse crime" has been used to describe this.

This phrase was in common use when I first became interested in these problems. I'm aware that it appears in the book Inverse Acoustic and Electromagnetic Scattering Theory by Colton and Kress, published in 1992. I'd be interested in any earlier uses of the phrase.

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    $\begingroup$ The term is indeed commonly attributed to Rainer Kress. I believe he first used it in one of his talks, but the book seems to be the first time it was mentioned in print. $\endgroup$ – Christian Clason Nov 9 '17 at 19:33
  • $\begingroup$ @ChristianClason Why not post that as an answer? $\endgroup$ – Tommi Nov 10 '17 at 8:05
  • $\begingroup$ there is also a post on Math overflow about Strang using the term for abuses of the Galerkin method - termed Variational Crimes. mathoverflow.net/questions/26018/… $\endgroup$ – eigenjohnson Nov 11 '17 at 14:15
  • $\begingroup$ @eigenjohnson That's interesting, but something completely different (Strang never talked about inverse crimes, nor is that term ever used in this context). $\endgroup$ – Christian Clason Nov 11 '17 at 15:45
  • $\begingroup$ I think that the "Variational Crimes" phrase might well have inspired "Inverse Crimes" $\endgroup$ – Brian Borchers Nov 12 '17 at 15:13
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The term inverse crime for a numerical test of a parameter identification method that uses data contained in the range of the discrete(!) forward operator used for the inversion (thus essentially reducing the problem to a well-posed finite-dimensional one that behaves fundamentally different from the original infinite-dimensional one -- it is important to stress that being in the range is the problem here, not the finite-dimensionality) is indeed commonly attributed to Rainer Kress. From what I have heard (this was before my time), he coined this term in one of his talks; the first time it is found in print seems to be indeed in his book [1] (on page 154 in the current, third, edition). This is in fact the usual reference given when people feel they need to give one for this concept.

I've also sometimes seen citations to [2], where the term is used frequently albeit in a slightly different context (but with the same general meaning); the authors also attribute it to Rainer Kress.

[1] Colton, David; Kress, Rainer, Inverse acoustic and electromagnetic scattering theory, Applied Mathematical Sciences. 93. Berlin: Springer-Verlag. x, 305 p. (1992). ZBL0760.35053.

[2] Kaipio, Jari; Somersalo, Erkki, Statistical and computational inverse problems., Applied Mathematical Sciences 160. New York, NY: Springer (ISBN 0-387-22073-9/hbk). xvi, 339 p. (2005). ZBL1068.65022.

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