The Mosaic Portrait method of mathematical modelling is designed to solve traditional problems in the construction of adequate mathematical models for large complex systems. Mosaic Portrait models are constructed on the basis of operational data from either the observation of a system’s normal activity (passive experiment) or from the observation of specially planned activities (active experiment).

Any numerical or descriptive data that can be arranged in a table of rows (data points) and columns (input and output parameters) is suitable for Mosaic Portrait modelling.

The Mosaic Portrait method depends on totally formal procedures, and is thus entirely objective in its methodology:

• Transfer of output parameters values to discrete scales.

• Transfer of input parameters values to discrete scales by sub-ranges.

• Search for combinations of sub-range values from various input parameters that are met only in "good" events and never met in "bad" events, and vice versa.

This results in the generation of a large number of patterns comprised of the various input parameters. Therefore, a mosaic portrait of the studied system is created from multiple independent patterns of input parameters sub-values. These patterns describe the relationship between the output parameter and the mutual effect of all input parameters. Each obtained pattern represents a formal correct expression, rather than a probabilistic approximation.

The meaningful specialist interpretation of the patterns obtained generates new knowledge about the systemic correlations of a studied system, which can be used to better understand and perfect the studied system. In practical applications, these patterns can be used to accurately recognise or predict new events as "bad" or "good", or to formally generate necessary conditions for exclusively "good" events.

The mathematical basis for the generation of the input conditions necessary for exclusively "good" outputs utilises a well-known logic algebra axiom: "A combined expression is true if and only if it is comprised of true simple expressions and does not contain any false expression". If combinations (patterns) of sub-range values in the "good" class of the system model are interpreted as simple true expressions, and in the "bad" class as simple false expressions, then possible solutions to the task can be obtained as the multitude of possible combined true expressions. In this method, combined expressions are created, which contain all possible combinations of "good" class expressions, but which do not form any "bad" class expression. These combined expressions represent possible optimal solutions to the task.

It is important to note that the transformation to discrete scales puts the model construction into the area of discrete programming tasks. It is recognised that such tasks can be solved only by the method of complete search, where time and labour are exponentially depended on the number of input and output parameters. Hitherto, tasks with a large dimensionality have been regarded as practically irresolvable. Mosaic Portrait successfully overcomes this problem. For example, Mosaic Portrait has solved the task of object identification with experimental data on 98 input parameters within 11 minutes. With traditional methods, the calculated time required for the solution of the same task by complete search would be 21*3^(98-10) sec or 7.4*10³⁵ years.

This represents a significant breakthrough in the area of mathematical modelling and makes it possible to solve many serious problems in many sectors of human activity.