Inverse problems

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Inverse Problems
S. R. Moghadasi Sharif University of Technology
Department of mathematics
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First examples
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Computer Tomography
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Computer Tomography
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Computer Tomography
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Computer Tomography
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Inverse Gravity
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Inverse Gravity
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Inverse Gravity
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Inverse Gravity
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Inverse Gravity
The direct problem is to compute gravity
potential and force when the density is known.
The inverse problem is to determine the density
by measuring the force on the given surface.
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Inverse Conductivity
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Inverse Conductivity
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Inverse Conductivity
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Inverse Conductivity
The direct problem is to determine electrical
current for a given potential on the surface.
The inverse problem is to determine a(x) by
measuring current induced by a given potential on
the surface.
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Inverse Conductivity
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Inverse Conductivity
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Inverse scattering
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Inverse scattering
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Inverse scattering
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Inverse scattering
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Image Recovery
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Given g in Y, we are looking for f in X such
that A( f ) g.

• Existence For each g in Y there is at
  least one f in X such that A( f ) g
• Uniqueness For each g in Y there is at most
  one f in X such that A( f ) g
• Stability f depends continuously on g

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• All norms of a finite dimensional vector space
  are equivalent. Therefore they induce the same
  topology on the space. So unit ball is compact in
  these spaces.
• All linear maps between finite dimensional spaces
  are continuous (with respect to the topology
  induced by norms).

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• Topology of infinite dimensional vector spaces
  are more complicated. There are linear maps which
  are not continuous. Unit sphere is not compact.

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• Since inverse of the above map is not continuous
  we can not use the above computational method for
  inverse map. There are some regularization
  methods for this aim.

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