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What is ill-posed problem in model calibration (or inverse modeling)?

In the environmental modelling context the areal distribution of system properties must be inferred from scarce point measurements of some of them, and/or back-calculated from historical measurements of system state. Thus solution of an ill-posed inverse problem is a fundamental aspect of environmental model usage. The extreme heterogeneity of most earth systems makes it very unlikely that exact values of system properties can be inferred throughout the system.

The material model calibration process, also known as identification of the corneal material coefficients, is an inverse problem as these parameters are not known and should be derived from experimental data. An inverse problem is often ill-posed, defined as a problem having no unique solution (i.e. it has multiple solutions) or for which the solution is unstable (i.e. small variations in the experimental data could lead to large variations in the solutions). The unreliable predictions referred to earlier are a direct consequence of the ill-posed nature of the model calibration process.

The process of calculating a single parameter set with a special set of properties from a set of field measurement is often referred to as "inversion". The problem itself is often referred to as "the inverse problem". Inverse problems are often difficult to solve, the reason being that they are often characterized as being, in mathematical parlance, "ill-posed".

Inverse problem ill-posedness arises from the fact that if a model is endowed with parameterization complexity that reflects the complexity and heterogeneity of reality, then rarely, if ever, can these parameters be estimated uniquely on the basis of measurements of system state alone. Linear algebra provides a useful vehicle for analyzing this problem.

Restriction of the search for a solution to the inverse problem of model calibration to parameter combinations that lie within the solution space effectively restricts that search to the simplest set of parameters which allow the model to effectively reproduce historical system behaviour. This accords with the much-repeated precept offered by many calibration sages that the calibration process should pursue the principle of parsimony. It is important to note, however, that parsimony is desirable not as an end in itself, but because it is a means of achieving the only thing that is worth achieving through the calibration process, that is the solution to the inverse problem that is of minimum error variance.

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