Functions for determining the value of information in constructive theory of optimal aggregated models
A topical problem in mathematical modeling theory is the development of methods that produce adequate and realizable working models as a source of new knowledge and new theories in the areas of optimal prediction and ``control'' under real-life conditions of incomplete knowledge and incomplete data. The notion of ``working model'' includes ordinary mathematical models, computer models (computer ``simulators'' used to investigate causal relationships), expert systems, as well as complex information systems that integrate various mathematical models (with different levels of aggregation and different resolving power) with special-purpose expert systems and simulators in order to improve the quality of prediction of possible consequences of alternative management strategies and to ensure efficient management decisions. Construction of optimal working models can be based on the new measures of adequacy and realizability. In these studies we have also defined the optimality criterion for models and developed methods for construction of optimal models through identification and optimal use of the most valuable information. Useful information for a working model is generally supplied from a number of diverse sources. These sources include theoretical results of basic research, previously developed mathematical models of various aggregation levels, empirical generalizations of observed regularities, and data files containing the results of natural and computer experiments and observations. Other sources of useful information include knowledge bases of expert systems that reflect the accumulated experience of practitioners and analyticians. Searching for useful information in these diverse sources requires a ``function for evaluating the usefulness of information'' (a solving operator which is a generalized solution of the functional equation for the value of information. This operator combines the concepts of pseudoinverse operator and Green's function for nonlinear and nonsmooth operator equations. Its first-order asymptotic approximation generalizes the sensitivity function of modeling and control theory to the case of nonsmooth parametric systems and the Pontryagin influence function to the case of nonsmooth dynamic systems. Its higher order asymptotic approximations define local asymptotic approximations of corresponding order for the inverse operator, and thus lead to accelerated convergence algorithms for solving nonlinear systems and computing singular solutions (fast convergence analogues of Newton's methods that do not require solving the complete system of linearized operator equations). Under incomplete data, the function for evaluating the usefulness of information enables us to obtain optimal state estimates, and also design optimal control and optimal decision strategy that generalize the Kalman-Bucy filter and the Bellman-function optimal control synthesis to the case of nonlinear nonsmooth systems. Optimal models produce the best guaranteed state estimates of real processes, and this result is responsible for the objective relationship of optimal mathematical models with real processes. The construction of solving operators is the topic of this paper.
scientific article; zbMATH DE number 1051090