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 |

OR | ||

Direct: Physical Reality
Measurements |
/ | Inverse: Measurements
Physical Reality |