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What is an Inverse Problem? 

An inverse problem is determining unknown causes based on observation of their effects.

If the following two questions are both answered ‘yes,’ then you have an inverse problem:

  • Am I using a mathematical model?
  • Am I using data?

Inverse problems include both parameter estimation and function estimation. Both of these inverse problems are “bridges” that bring together experimental work and data analysis.

Inverse problems have a wide range of applications, such as making clear a blurred photo, medical imaging (MRI), oil drilling, and echolocation (SONAR, bats, and dolphins).   A common characteristic is that we attempt to infer causes from measured effects.  A forward, or direct problem has known causes that produce effects or results defined by the mathematical model.  Because the measured data is often noisy or indistinct, the solution to the inverse problem may be difficult to obtain accurately.

An example of an inverse problem is estimating the thermal diffusivity of a solid from transient temperature measurements, using the heat diffusion PDE. The corresponding forward problem is to compute the solid temperature using a known thermal diffusivity. Commercial finite-element programs are typically forward-problem solvers, and require the parameter(s) to be known.

Estimating heat flux at a solid surface from measured temperatures within the solid is an example of function estimation.