|input f||black box A||output g|
The cause furnishes the input f, symbolized by the left-hand arrow. The effect is the output g, symbolized by the right arrow. The "black box" designates the mechanism A that transforms the cause into the effect, symbolically:
(1) g = A f
If we have a linear system, than f and g may be vectors, and A may denote a matrix. If A is a regular square matrix, then the solution of (1) is
(2) f = A* g
A* denoting the matrix inverse to A.
Generally, A may be any (linear or nonlinear) "operator" which acts on the input f to give the output g. Thus, A is the operator of the direct problem. Depending on the context, A is alternatively called "mapping", ["projection" (of "nature" f onto the "observation space" g)] or "function", also writing g = A ( f ). Then (2) denotes the inverse problem (mapping, function), or rather its solution.
The simple model (1) and its inverse (2) are extremely general. A may be a computer program with input f and output g. It may be an elementary function g = A ( f ) = sin ( f ), like y = sin ( x ). However, f may also denote the universe at time 2000.0, A its evolution, and g the universe at the present moment.
"Interesting" inverse problems are frequently extremly difficult by not being "well-posed" in the sense of Hadamard.
Let us summarize:
Structure of inverse problems
|Direct: Cause Effect||/||Inverse: Effect Cause|
|Direct: Physical Reality Measurements||/||Inverse: Measurements Physical Reality|
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