Fast error estimates for indirect measurements: applications to pavement engineering

In many real-life situations, we have the following problem: we want to know the value of some characteristicy that is difficult to measure directly (e.g., lifetime of a pavement, efficiency of an engine, etc.). To estimatey, we must know the relationship betweeny and some directly measurable physical quantitiesx ₁,...,x _n . From this relationship, we extract an algorithmf that allows us, givenx _i , to computey: y=f(x ₁, ...,x _n ). So, we measurex _i , apply an algorithmf, and get the desired estimate. Existing algorithms for error estimate (interval mathematics, Monte-Carlo methods, numerical differentiation, etc.) require computation time that is several times larger than the time necessary to computey=f(x ₁, ...,x _n ). So, if an algorithmf is already time-consuming, error estimates will take too long. In many cases, this algorithmf consists of two parts: first, we usex _i to determine the parametersz _k of a model that describes the measured object, and second, we use these parameters to estimatey. The most time-consuming part is findingz _k ; this is done by solving a system of non-linear equations; usually least squares method is used. We show that for suchf, one can estimate errors repeating this time-consuming part off only once. So, we can compute bothy and an error estimate fory with practically no increase in total computation time. As an example of this methodology, we give pavement lifetime estimates.