Conceptual Understanding
The forward problem can be conceptually formulated as follows:
- Data → Model parameters
The inverse problem is considered the "inverse" to the forward problem which relates the model parameters to the data that we observe:
- Model parameters → Data
The transformation from data to model parameters (or vice versa) is a result of the interaction of a physical system with the object that we wish to infer properties about. In other words, the transformation is the physics that relates the physical quantity (i.e. the model parameters) to the observed data.
The table below shows some examples of physical systems, the governing physics, the physical quantity that we are interested, and what we actually observe.
| Physical system | Governing equations | Physical quantity | Observed data |
|---|---|---|---|
| Earth's gravitational field | Newton's law of gravity | Density | Gravitational field |
| Earth's magnetic field (at the surface) | Maxwell's equations | Magnetic susceptibility | Magnetic field |
| Seismic waves (from earthquakes) | Wave equation | Wave-speed (density) | Particle velocity |
Linear algebra is useful in understanding the physical and mathematical construction of inverse problems, because of the presence of the transformation or "mapping" of data to the model parameters.
Read more about this topic: Inverse Problems
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