Inverse Problems
by scientific-computing.info
Inverse problems are concerned with the determination of causes for a desired or an
observed effect.
In the mathematical sense, inverse problems can be viewed as lying outside the
typical classical problem such as initial-value and boundary value problems.
Inverse problems most often do not fulfill Hadamard's
postulates of well-posedness; they might not have a solution in
the strict sense, solutions might not be unique and/or might not depend
continuously on the data.
Books on Inverse Problems
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Some examples of inverse problems
- Inverse heat conduction or inverse diffusion problems: For example
the determination of
an unknown boundary heat flux which leads to a desired temperature field or
a desired boundary between solid and liquid phase or the determination of an
initial thermal/material distribution from the distribution at a time T.
This latter problem is considered in Inverse Diffusion
by the Operator-Splitting Method
- Inverse scattering problems: From measurements of waves scattered by an
obstacle, one wants to determine e.g. the shape or the location of this
obstacle. [Direct acoustic scattering and radiation problems are considered
in boundary-element-method.com.
- Computerized tomography: Here, one wants to reconstruct the spatially
varying absorption coefficients within the human body from measurements of
intensity decays of X-rays sent through the body. Mathematically, this leads
to the problem of inverting the Radon transform. Similar problems
appear in non-destructive testing.
- Parameter identification: Here, spatially and/or temporally parameters
appearing in e.g. partial differential equations have to be determined from
measurements of the solution, either in the whole domain, or on the boundary
only. The latter case has important applications in medicine and
non-destructive testing: an electrical conductivity can be determined from
measurements of current and voltage on the boundary (impedance
tomography).
All these problems are ill-posed. A consequence is that arbitrarily
small changes in the data may lead to arbitrarily large changes in the
solution. As in the numerical treatment of inverse problems data errors are
inevitable, one has to use stabilizing procedures for successfully dealing with
ill-posed problems, so-called regularization methods.
Books on Inverse Problems