Those two are inherently not different in many cases. Especially when estimation of discrete parameter set has to be done based on the experimental data, they are identical. However, we call it inverse problem in the following cases.

Think about the inverse problem:

Find a1, a2, a3,...,an an that meets f(x_k;a1,a2,...an)=f_exp_k for many k

1. The unknown a1, a2 .. is a continuous function.

2. The solution process for f requires complex PDE solution or stochastic process. Usually, f is not in a form of algebraic equation.

3. The estimation is undetermined that extra treatment is required based on the inverse theory.

4. The determination procedure is unstable that the sensitivity of f to ai is low.

An inverse problem is a mathematical procedure that starts from a set of observations and calculates the causal factors that produced them by minimizing an objective function (e.g. the difference between the measured/observed data and the modelled ones). A forward problem (or direct problem) is the use of a model in order to simulate an outcome. The causal factors are a set of user-defined parameters. The modeller have to change them by a trial-and-error approach in order to minimize the difference between the observed and the modelled data. The inversion automatically does what the modeller would do manually (calibration). In fact, in the inversion process a large number of direct models are solved as long as the objective function is minimized. Usually, it takes a number of iterations.

Dear Karan

As claimed before by Sun and Gola calibration could be considered-intended as a part of an inversion procedure or all entire procedure.

Integrating the previous answers, in the inversion it should distinguish between the model precision and model exactness. The union of precision and exactness gives the model accuracy. The first is the capacity of your model to predict the observables (its measure is a function of difference between the observables and the predicted observables in the data-observable space), the second one is the capacity of the model to estimate the model parameters with an error (its measure is a difference of the estimated parameters of the most precise models in the model space, i.e. covariance model for linearized approaches or a metrics in the space model).

The term calibration sometime is considered as the procedure to assess the model exactness and/or the model stability exploring the different responses of models solving the direct problem, as claimed by Gola, and other times as the procedure to assess the model accuracy (as claimed by Sun). Finally, the calibration is intended as the procedure to predict with a model

an observable not included in the observable-data space which was used in the inversion. Obviously in the last case the model parameters must be suitable to calculate the model prediction of the new observable. An example in my research field, inverting seismic refraction time-distance data I obtain the seismic velocity parameters in each layer, then with this obtained model I predict the layer resonance which is measured with ambient noise measurement, not included in the data space of the time-distance model inversion.

FEM modelling, the only technique in which calibration has a meaning, is used both for : a) prediction before construction (in this case the calibration of n parameters should follow a stochastic algorithm : example if model has 5 constant the number of run is 1x2x3x4x5 =120 runs)

b) back analysi from field measures in this case the Reality is only one , it is called sensitività

c) open system with energy input , if dynamic the history is not real, if chemical energy is add , the opinion is amatter of fact, the repetition of monitoring is the right wY