# Task 2

**JKU :**

Dedicated to the study of the formal **relationships** between field observation criteria *C(R) *and several parameter estimation criteria, denoted generically by *I(R)*. Our main goal is to establish the conditions under which optimal solutions to one of the problems can also be optimal (or nearly-optimal) for the other one. For instance, when is it possible to guarantee that

\[ if \quad R_i^* = arg \; max_R I(R), \quad R_c^* = arg \; min_R C(R), \rightarrow C(R_i^*) - C(R_c^*) \leq \epsilon \; ? \]

This conjecture, that optimal designs in terms of suitable estimation criteria *I(R)* (measuring the precision of the estimation of *b* and *g*)* *achieve good performance in terms of field reconstruction, i.e., for the

prediction criteria *C(R), *is* *strongly motivated by the KWET (see Section 1). We believe that search for this type of relations, at the core of the project, will enable us to develop algorithms for the sequential construction of designs that will simultaneously take into account the quality of the prediction task and the reduction of uncertainty in the estimation of the process parameters, as it happens for designs optimal in the sense of the Equivalence Theorem of (Kiefer & Wolfowitz, 1961).