Inverse problem

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A mathematical problem is called an inverse problem when one tries to deduce the cause of the effect from an observed or desired effect of a system . Inverse problems are usually very difficult or even impossible to solve . The opposite of an inverse problem is a direct problem (sometimes also called a forward problem ), in which one would like to derive the effect of the system based on the known cause .

A distinction is made between well-posed and poorly posed inverse problems. In addition to questions of mathematical-physical stability , numerical stability (for example in large systems of normal equations ) must also be taken into account. Regularization methods are used to improve the numerical stability .

Explanatory examples

This abstract explanation and the difficulty of inverse problems can be illustrated with an example: A submarine is at a point x at a depth t in the sea.
The drive emits sound waves (engine and propeller noises).
If one knows the properties of these sound waves ( strength , frequency ) and the medium being transmitted ( water ), one can easily calculate how loud a microphone at a distant point y can hear the submarine. This is a straightforward problem that is easy to solve. One inferred from the cause (noise at location x at depth t ) on the effect (acoustic signal on the microphone). Conversely, in the context of submarine tracking, one would like to know from the engine noise measured at location y where and at what depth the submarine is.
This is the associated inverse problem, in which one would like to deduce the cause from the effect. The location problem is much more difficult to solve. If you have no information about the direction from which the sound is coming from the received signal, the problem is insoluble, because even if you know the properties of the sound waves emitted by the submarine, you can only deduce the distance between the submarine and the receiver, not but direction and depth.

Further examples of inverse problems:

  1. In computed tomography , one would like to draw conclusions about the local course of the X-ray absorption inside the body (cause) from the measurements of an X-ray beam weakened when shining through a body (effect). Many inverse problems arise in connection with tomographic issues.
  2. Recordings of astronomical objects are sometimes reduced in quality by the properties of the recording devices or by the refraction of the earth's atmosphere . One would like to deduce the unadulterated image of the object (cause) from a bad image (observed effect).
  3. From the measured signals of an earthquake (effect) one would like to derive properties of the earth's interior (cause of the earthquake).
  4. From vertical deviations or gravity anomalies , conclusions should be drawn about the mass distribution in the interior of the earth ( reversal problem of potential theory )
  5. The spectral data of the IR spectroscopy or Raman spectroscopy of a mixture of gases or liquids represent an overlay of the spectra of the pure components contained in the mixture. With knowledge of the pure substance spectra , one would like to use the different intense peaks in the mixture spectrum (effect) on the concentrations ( Cause) of the individual components in the mixture.
  6. The calibration of parameters of financial mathematical models using market prices of traded derivative instruments (swaptions, caps, floors, etc.) is also an inverse problem.

For inverse scattering problems of the Sturm-Liouville type there is the Gelfand-Levitan theory (1951), after Israel Gelfand and Boris Levitan . These include, for example, wave equations with scatter potential and the stationary Schrödinger equation with potential. The task is to reconstruct the potential from the scatter data.

Some inverse problems also lead to integral equations of the Abelian type .

literature

General inverse problems

  • Alfred Louis: Inverse and Badly Asked Problems . Teubner, Stuttgart 2001, ISBN 3-519-02084-X .
  • Andreas Rieder: No problems with inverse problems . Vieweg, Wiesbaden 2003, ISBN 3-528-03198-0 .
  • Albert Tarantola: Inverse Problem Theory . ( as PDF ), Society for Industrial and Applied Mathematics, Philadelphia 2005, ISBN 0-89871-572-5 .
  • Heinz W. Engl, Martin Hanke, Andreas Neubauer: Regularization of inverse problems . Springer Netherland, Berlin 1996, ISBN 0-7923-4157-0 .

Inverse Problems in Medical Imaging

  • Frank Natterer : The Mathematics of Computerized Tomography . Society for Industrial and Applied Mathematics, Philadelphia 2001. ISBN 0-89871-493-1 .
  • Frank Natterer and Frank Wübbeling: Mathematical Methods in Image Reconstruction . Society for Industrial and Applied Mathematics, Philadelphia 2001, ISBN 0-89871-472-9 .

See also