Modeling in computer science - what is it? Types and stages of modeling.

The concepts of “model”, “simulation”, various approaches to the classification of models. Modeling stages

Model (modelium)– about Latin measure, image, manner, etc.

Model- this is a new object, different from the original one, which has properties essential for modeling purposes and, within the framework of these goals, replaces the original object (the object is the original)

Or we can say in other words: a model is a simplified representation of a real object, process or phenomenon.

Conclusion. The model is needed in order to:

Understand how a specific object is structured - what are its structure, basic properties, laws of development and interaction with the outside world;

Learn to manage an object or process and determine the best management methods for given goals and criteria (optimization);

Predict direct and indirect consequences of implementing specified methods and forms of impact on the object;

Classification of models.

Signs by which models are classified:

1. Area of ​​use.

2. Taking into account the time factor and area of ​​use.

3. According to the method of presentation.

4. Branch of knowledge (biological, historical, sociological, etc.).

5. Area of ​​use

Educational: visual aids, training programs, various simulators;

Experienced: a ship model is tested in a pool to determine the stability of the ship when rocking;

Scientific and technical: an electron accelerator, a device that simulates a lightning discharge, a stand for testing a TV;

Gaming: military, economic, sports, business games;

Imitation: the experiment is either repeated many times in order to study and evaluate the consequences of any actions on a real situation, or is carried out simultaneously with many other similar objects, but placed under different conditions).

2. Taking into account the time factor and area of ​​use

Static model - it’s like a one-time slice through an object.

Example: You came to the dental clinic for an oral examination. The doctor examined me and wrote down all the information on the card. Entries in the card that give a picture of the state of the oral cavity at a given time (number of milk, permanent, filled, extracted teeth) will be a statistical model.

Dynamic model allows you to see changes in an object over time.

An example is the same card of a schoolchild, which reflects the changes occurring in his teeth at a certain point in time.

3. Classification by method of presentation

The first two large groups: material and informational. The names of these groups seem to indicate what the models are made of.

Material Models can otherwise be called objective, physical. They reproduce the geometric and physical properties of the original and always have a real embodiment.

Children's toys. From them the child gets his first impression of the world around him. A two-year-old child plays with a teddy bear. When, years later, a child sees a real bear in a zoo, he will easily recognize it.

School textbooks, physical and chemical experiments. They simulate processes, such as the reaction between hydrogen and oxygen. This experience is accompanied by a deafening bang. The model confirms the consequences of the emergence of an “explosive mixture” of harmless and widespread substances in nature.

Maps when studying history or geography, diagrams of the solar system and the starry sky in astronomy lessons and much more.

Conclusion. Material models implement a material (touch, smell, see, hear) approach to the study of an object, phenomenon or process.

Information models cannot be touched or seen with your own eyes; they have no material embodiment, because they are built only on information. This modeling method is based on an information approach to studying the surrounding reality.

Information models - a set of information that characterizes the properties and states of an object, process, phenomenon, as well as the relationship with the outside world.

Information characterizing an object or process can have different volumes and forms of presentation, and be expressed in different ways. This diversity is as limitless as the capabilities of each person and his imagination. Information models include symbolic and verbal.

Iconic model - an information model expressed by special signs, i.e., by means of any formal language.

Iconic models are all around us. These are drawings, texts, graphs and diagrams.

According to the method of implementation, iconic models can be divided into computer and non-computer ones.

Computer model - a model implemented by means of a software environment.

Verbal (from the Latin “verbalis” - oral) model - an information model in mental or spoken form.

These are models obtained as a result of reflection and inference. They can remain mental or be expressed verbally. An example of such a model would be our behavior when crossing the street.

The process of building a model is called modeling; in other words, modeling is the process of studying the structure and properties of the original using a model.

Planetariums" href="/text/category/planetarii/" rel="bookmark">planetarium, in architecture - building models, in aircraft manufacturing - aircraft models, etc.

Ideal modeling is fundamentally different from subject (material) modeling.

Perfect modeling is not based on a material analogy of an object and a model, but on an ideal, conceivable analogy.

Iconic modeling is modeling that uses symbolic transformations of any kind as models: diagrams, graphs, drawings, formulas, sets of symbols.

Mathematical modeling is modeling in which the study of an object is carried out through a model formulated in the language of mathematics: description and study of Newton's laws of mechanics using mathematical formulas.

The modeling process consists of the following stages:

The main task of the modeling process is to select the most adequate model to the original and transfer the research results to the original. There are quite general methods and methods of modeling.

Before building a model of an object (phenomenon, process), it is necessary to identify its constituent elements and the connections between them (conduct a system analysis) and “translate” (display) the resulting structure into some predetermined form - to formalize the information.

Formalization is the process of identifying and translating the internal structure of an object, phenomenon or process into a specific information structure - form.

Formalization is the reduction of essential properties and characteristics of a modeling object in the selected form (to the selected formal language).

Modeling stages

Before taking on any work, you need to clearly imagine the starting point and each point of the activity, as well as its approximate stages. The same can be said about modeling. The starting point here is a prototype. It can be an existing or designed object or process. The final stage of modeling is making a decision based on knowledge about the object.

The chain looks like this.

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STAGE I. STAGE TASKS.

A task is a problem that needs to be solved. At the stage of problem formulation, it is necessary to reflect three main points: description of the problem, determination of modeling goals and analysis of the object or process.

Description of the task

The problem is formulated in ordinary language, and the description should be clear. The main thing here is to define the modeling object and understand what the result should be.

Purpose of modeling

1) knowledge of the surrounding world

2) creation of objects with given properties (determined by posing the problem “how to do so that...”.

3) determining the consequences of impact on the object and making the right decision. The purpose of modeling problems like “what will happen if...” (what will happen if you increase the fare for transport, or what will happen if you bury nuclear waste in such and such an area?)

Object Analysis

At this stage, the modeled object and its main properties are clearly identified, what it consists of, and what connections exist between them.

A simple example of subordinate object connections is parsing a sentence. First, the main members (subject, predicate) are highlighted, then the minor members related to the main ones, then the words related to the secondary ones, etc.

STAGE II. MODEL DEVELOPMENT

1. Information model

At this stage, the properties, states, actions and other characteristics of elementary objects are clarified in any form: verbally, in the form of diagrams, tables. An idea is formed about the elementary objects that make up the original object, i.e., an information model.

Models must reflect the most essential features, properties, states and relationships of objects in the objective world. They provide complete information about the object.

2. Iconic model

Before starting the modeling process, a person makes preliminary sketches of drawings or diagrams on paper, derives calculation formulas, i.e., compiles an information model in one or another symbolic form, which can be either computer or non-computer.

3. Computer model

A computer model is a model implemented using a software environment.

There are many software packages that allow you to conduct research (modeling) of information models. Each software environment has its own tools and allows you to work with certain types of information objects.

The person already knows what the model will be and uses the computer to give it an iconic shape. For example, graphical environments are used to build geometric models and diagrams, and a text editor environment is used for verbal or tabular descriptions.

STAGE III. COMPUTER EXPERIMENT

With the development of computer technology, a new unique research method has emerged - a computer experiment. A computer experiment includes a sequence of working with a model, a set of targeted user actions on a computer model.

STAGE IV ANALYSIS OF MODELING RESULTS

The ultimate goal of modeling is making a decision, which should be made on the basis of a comprehensive analysis of the results obtained. This stage is decisive - either you continue the research or finish it. Perhaps you know the expected result, then you need to compare the obtained and expected results. If there is a match, you will be able to make a decision.

Modeling is the replacement of one object (original) with another (model) and fixation or study of the properties of the original by studying the properties of the model.

Model is a representation of an object, system or concept (idea) in some form that is different from the form of its real existence.

The benefits of modeling can only be achieved if the following fairly obvious conditions are met:

The model adequately reflects the properties of the original that are significant from the point of view of the purpose of the study;

The model allows you to eliminate the problems inherent in taking measurements on real objects.

Approaches (methods) to modeling.

1) Classic (inductive) examines the system by moving from the particular to the general, i.e. The system model is built from the bottom up and synthesized by merging the element models of the component systems, developed separately.

2) System. Transition from general to specific. The model is based on the purpose of the study. It is from this that they start when creating a model. The goal is what we want to know about the object.

Let's consider the basic principles of modeling.

1) The principle of information sufficiency. It is necessary to collect information that will provide a sufficient level of information.

2) The principle of feasibility. The model must ensure achievement of the goal within a realistically specified time.

3) Aggregation principle. A complex system consists of subsystems (units), for which You can build independent models and combine them into a common model. The model turns out to be flexible. When changing the goal, a number of component modules can be used. The model is feasible if

And
.

Classification of modeling methods.

1) By the nature of the processes being studied

Deterministic - during the functioning of the modeled object, random factors are not taken into account (everything is predetermined).

Stochastic – the impact of various factors on existing real systems is taken into account

2) Based on development over time

Static – the behavior of an object is described at a certain time

Dynamic – for a certain period of time

3) According to the presentation of information in the model

Discrete - if events leading to changes in states occur at a certain point in time.

Continuous, discrete-continuous.

4) According to the form of presentation of the modeling object

mental- if the modeling object does not exist, or exists outside the conditions for its physical creation.

A) Symbolic. Creating a logical object that replaces the real one.

B) Mathematical

Analytical. An object is described using functional relationships, followed by an attempt to obtain an explicit solution.

Imitation. The algorithm that describes the functioning of the system reproduces the process of the object’s operation over time. This method is also called statistical, because statistics of simulated phenomena are collected. (based on the Monte Carlo method - static test method)

B) Visual

Real- there is an object.

A) Natural. The experiment is carried out on the modeling object itself. The most common form is testing.

B) Physical. Research is carried out on a special basis. Installations, processes in the cat. They have a physical similarity with processes in real objects.

The analytical model can be studied using the following methods:

A) analytical: an attempt to obtain solutions explicitly (general);

b) numerical: obtain a numerical solution under given initial conditions (partial nature of the solutions);

V) quality: Without having an explicit solution, you can find the properties of the solution in explicit form.

In simulation modeling, the algorithm that describes the functioning of the system reproduces the process of the object’s operation over time. This method is also called statistical, because statistics of simulated phenomena are collected. (based on the Monte Carlo method)

Simulation method the most promising research method requires a certain level of mathematical training from the psychologist. Here, mental phenomena are studied on the basis of an approximate image of reality - its model. The model makes it possible to focus the psychologist’s attention only on the main, most significant features of the psyche. A model is an authorized representative of the object being studied (mental phenomenon, thinking process, etc.). Of course, it is better to immediately get a holistic understanding of the phenomenon being studied. But this is usually impossible due to the complexity of psychological objects.

The model is related to its original by a similarity relationship.

Cognition of the original from the standpoint of psychology occurs through complex processes of mental reflection. The original and its psychic reflection are related like an object and its shadow. Complete cognition of an object is carried out sequentially, asymptotically, through a long chain of cognition of approximate images. These approximate images are models of the cognizable original.

The need for modeling arises in psychology when:
- the systemic complexity of an object is an insurmountable obstacle to creating its holistic image at all levels of detail;
- rapid study of a psychological object is required to the detriment of the detail of the original;
- mental processes with a high level of uncertainty are subject to study and the patterns to which they obey are unknown;
- optimization of the object under study is required by varying input factors.

Modeling tasks:
- description and analysis of mental phenomena at various levels of their structural organization;
- forecasting the development of mental phenomena;
- identification of mental phenomena, i.e. establishing their similarities and differences;
- optimization of conditions for the occurrence of mental processes.

Briefly about the classification of models in psychology. There are object and symbolic models. Subject ones have a physical nature and, in turn, are divided into natural and artificial. The basis of natural models is made up of representatives of living nature: people, animals, insects. Let's remember man's faithful friend - the dog, which served as a model for studying the functioning of human physiological mechanisms. Artificial models are based on elements of “second nature” created by human labor. As an example, we can cite F. Gorbov’s homeostat and N. Obozov’s cybernometer, which are used to study group activity.

Sign models are created on the basis of a system of signs of very different nature. This:
- alphanumeric models, where letters and numbers act as signs (such, for example, is the model for regulating joint activities of N. N. Obozov);
- models of special symbols (for example, algorithmic models of the activities of A. I. Gubinsky and G. V. Sukhodolsky in engineering psychology or musical notation for an orchestral piece of music, which contains all the necessary elements that synchronize the complex joint work of performers);
- graphic models that describe an object in the form of circles and lines of communication between them (the former can express, for example, the states of a psychological object, the latter - possible transitions from one state to another);
- mathematical models that use a diverse language of mathematical symbols and have their own classification scheme;
- cybernetic models are built on the basis of the theory of automatic control and simulation systems, information theory, etc.

Sometimes models are written in programming languages, but this is a long and expensive process. Mathematical packages can be used for modeling, but experience shows that they usually lack many engineering tools. It is optimal to use a simulation environment.

In our course, we chose . The labs and demos you will encounter in the course should be run as projects in the Stratum-2000 environment.

The model, made taking into account the possibility of its modernization, of course, has disadvantages, for example, low speed of code execution. But there are also undeniable advantages. The model structure, connections, elements, subsystems are visible and saved. You can always go back and redo something. A trace in the history of model design is preserved (but when the model is debugged, it makes sense to remove service information from the project). In the end, the model that is handed over to the customer can be designed in the form of a specialized automated workstation (AWS), written in a programming language, in which attention is mainly paid to the interface, speed parameters and other consumer properties that are important for customer. The workstation is, of course, an expensive thing, so it is released only when the customer has fully tested the project in the modeling environment, made all the comments and undertakes not to change his requirements anymore.

Modeling is an engineering science, a technology for solving problems. This remark is very important. Since technology is a way to achieve a result with a quality known in advance and guaranteed costs and deadlines, then modeling as a discipline:

• studies ways to solve problems, that is, it is an engineering science;
• is a universal tool that guarantees the solution of any problems, regardless of the subject area.

Subjects related to modeling are: programming, mathematics, operations research.

Programming because the model is often implemented on an artificial medium (plasticine, water, bricks, mathematical expressions), and the computer is one of the most universal media of information and, moreover, active (simulates plasticine, water, bricks, calculates mathematical expressions, etc.). Programming is a way of expressing an algorithm in a language form. Algorithm is one of the ways of representing (reflecting) a thought, process, phenomenon in an artificial computing environment, which is a computer (von Neumann architecture). The specificity of the algorithm is to reflect the sequence of actions. Modeling can use programming if the object being modeled is easy to describe in terms of its behavior. If it is easier to describe the properties of an object, then it is difficult to use programming. If the simulation environment is not built on the basis of von Neumann architecture, programming is practically useless.

What is the difference between an algorithm and a model?

An algorithm is a process of solving a problem by implementing a sequence of steps, while a model is a set of potential properties of an object. If you pose a question to the model and add additional conditions in the form of initial data (connection with other objects, initial conditions, restrictions), then it can be resolved by the researcher regarding unknowns. The process of solving a problem can be represented by an algorithm (but other solution methods are also known). In general, examples of algorithms in nature are unknown; they are the product of the human brain, the mind, capable of establishing a plan. Actually, the algorithm is a plan, developed into a sequence of actions. It is necessary to distinguish between the behavior of objects associated with natural causes and the providence of the mind, controlling the course of movement, predicting the result based on knowledge and choosing the appropriate behavior.

model + question + additional conditions = task.

Mathematics is a science that provides the possibility of calculating models that can be reduced to a standard (canonical) form. The science of finding solutions to analytical models (analysis) using formal transformations.

Operations Research a discipline that implements methods for studying models from the point of view of finding the best control actions on models (synthesis). Mostly deals with analytical models. Helps make decisions using built models.

Design the process of creating an object and its model; modeling a way to evaluate the design result; There is no modeling without design.

Related disciplines for modeling include electrical engineering, economics, biology, geography, and others in the sense that they use modeling methods to study their own applied object (for example, a landscape model, an electrical circuit model, a cash flow model, etc.).

As an example, let's look at how a pattern can be detected and then described.

Let’s say that we need to solve the “Cutting Problem”, that is, we need to predict how many cuts in the form of straight lines will be required to divide the figure (Fig. 1.16) into a given number of pieces (for example, it is enough that the figure is convex).

Let's try to solve this problem manually.

From Fig. 1.16 it can be seen that with 0 cuts 1 piece is formed, with 1 cut 2 pieces are formed, with two 4, with three 7, with four 11. Can you now tell in advance how many cuts will be required to form, for example, 821 pieces ? In my opinion, no! Why are you having trouble? You do not know the pattern K = f(P) , Where K number of pieces, P number of cuts. How to spot a pattern?

Let's make a table connecting the known numbers of pieces and cuts.

The pattern is not yet clear. Therefore, let's look at the differences between individual experiments, let's see how the result of one experiment differs from another. Having understood the difference, we will find a way to move from one result to another, that is, a law connecting K And P .

A certain pattern has already emerged, hasn’t it?

Let's calculate the second differences.

Now everything is simple. Function f called generating function. If it is linear, then the first differences are equal. If it is quadratic, then the second differences are equal to each other. And so on.

Function f There is a special case of Newton's formula:

Odds a , b , c , d , e for our quadratic functions f are in the first cells of the rows of experimental table 1.5.

So, there is a pattern, and it is this:

K = a + b · p + c · p · ( p 1)/2 = 1 + p + p · ( p 1)/2 = 0.5 · p 2 + 0.5 p + 1 .

Now that the pattern has been determined, we can solve the inverse problem and answer the question posed: how many cuts must be made to get 821 pieces? K = 821 , K= 0.5 · p 2 + 0.5 p + 1 , p = ?

Solving a quadratic equation 821 = 0.5 · p 2 + 0.5 p + 1 , we find the roots: p = 40 .

Let's summarize (pay attention to this!).

We couldn't guess the solution right away. Conducting the experiment turned out to be difficult. I had to build a model, that is, find a pattern between the variables. The model was obtained in the form of an equation. By adding a question and an equation reflecting a known condition to the equation, a problem was formed. Since the problem turned out to be of a typical type (canonical), it was solved using one of the well-known methods. Therefore, the problem was solved.

And it is also very important to note that the model reflects cause-and-effect relationships. There is indeed a strong connection between the variables of the constructed model. A change in one variable entails a change in another. We said earlier that “the model plays a system-forming and meaning-forming role in scientific knowledge, it allows us to understand the phenomenon, the structure of the object under study, and establish the connection between cause and effect.” This means that the model allows us to determine the causes of phenomena and the nature of the interaction of its components. The model relates causes and effects through laws, that is, variables are related to each other through equations or expressions.

But!!! Mathematics itself does not make it possible to derive any laws or models from the results of experiments, as it may seem after the example just considered. Mathematics is only a way of studying an object, a phenomenon, and, moreover, one of several possible ways of thinking. There is also, for example, a religious method or a method that artists use, an emotional-intuitive one, with the help of these methods they also learn about the world, nature, people, themselves.

So, the hypothesis about the connection between variables A and B must be introduced by the researcher himself, from the outside, in addition. How does a person do this? It’s easy to advise introducing a hypothesis, but how to teach this, explain this action, and therefore, again, how to formalize it? We will show this in detail in the future course “Modeling Artificial Intelligence Systems”.

But why this must be done from the outside, separately, additionally and in addition, we will explain now. This reasoning bears the name of Gödel, who proved the incompleteness theorem: it is impossible to prove the correctness of a certain theory (model) within the framework of the same theory (model). Look again at Fig. 1.12. The higher level model transforms equivalent lower level model from one species to another. Or it generates a lower-level model based on its equivalent description. But she cannot transform herself. The model builds the model. And this pyramid of models (theories) is endless.

In the meantime, in order to “not get blown up by nonsense,” you need to be on your guard and check everything with common sense. Let's give an example, an old well-known joke from the folklore of physicists.

According to this feature, models are divided into two broad classes:

• abstract (mental) models;
• material models.

Rice. 1.1.

Often in modeling practice there are mixed, abstract-material models.

Abstract models represent certain designs of generally accepted signs on paper or other material media or in the form of a computer program.

Abstract models, without going into excessive detail, can be divided into:

• symbolic;
• mathematical.

Symbolic model is a logical object that replaces a real process and expresses the basic properties of its relationships using a certain system of signs or symbols. These are either words of natural language or words of the corresponding thesaurus, graphs, diagrams, etc.

A symbolic model can have independent significance, but, as a rule, its construction is the initial stage of any other modeling.

Mathematical modeling is the process of establishing correspondence between a modeled object and some mathematical structure, called a mathematical model, and the study of this model, which allows one to obtain the characteristics of the modeled object.

Mathematical modeling is the main goal and main content of the discipline being studied.

Mathematical models can be:

• analytical;
• imitation;
• mixed (analytical and simulation).

Analytical models- these are functional relationships: systems of algebraic, differential, integro-differential equations, logical conditions. Maxwell's equations are an analytical model of the electromagnetic field. Ohm's law is a model of an electrical circuit.

Transformation of mathematical models according to known laws and rules can be considered as experiments. A solution based on analytical models can be obtained as a result of a one-time calculation, regardless of the specific values ​​of the characteristics (“in general terms”). This is visual and convenient for identifying patterns. However, for complex systems it is not always possible to construct an analytical model that sufficiently fully reflects the real process. However, there are processes, for example, Markov processes, the relevance of modeling which with analytical models has been proven in practice.

Simulation modeling. The creation of computers led to the development of a new subclass of mathematical models - simulation ones.

Simulation modeling involves representing the model in the form of some algorithm - a computer program - the execution of which simulates the sequence of changes in states in the system and thus represents the behavior of the simulated system.

The process of creating and testing such models is called simulation, and the algorithm itself is called a simulation model.

What is the difference between simulation and analytical models?

In the case of analytical modeling, the computer is a powerful calculator, an adding machine. Analytical model is being decided on a computer.

In the case of simulation modeling, the simulation model - program - being implemented on a computer.

Simulation models quite simply take into account the influence of random factors. This is a serious problem for analytical models. In the presence of random factors, the necessary characteristics of the simulated processes are obtained by repeated runs (implementations) of the simulation model and further statistical processing of the accumulated information. Therefore, simulation modeling of processes with random factors is often called statistical modeling.

If the study of an object is difficult using only analytical or simulation modeling, then mixed (combined), analytical and simulation modeling is used. When constructing such models, the processes of an object’s functioning are decomposed into component subprocesses, for which analytical models are possibly used, and simulation models are built for the remaining subprocesses.

Material Modeling based on the use of models representing real technical structures. This can be the object itself or its elements (full-scale modeling). This can be a special device - a model that has either a physical or geometric similarity to the original. This may be a device of a different physical nature than the original, but the processes in which are described by similar mathematical relationships. This is the so-called analogue modeling. This analogy is observed, for example, between the vibrations of a satellite communication antenna under wind load and the fluctuations of electric current in a specially selected electrical circuit.

Often created material-abstract models. That part of the operation that cannot be described mathematically is modeled materially, the rest - abstractly. These are, for example, command and staff exercises, when the work of the headquarters is a full-scale experiment, and the actions of the troops are reflected in documents.

Classification according to the considered characteristic - the method of implementing the model - is shown in Fig. 1.2.

Rice. 1.2.

1.3. Modeling stages

Mathematical modeling like anything else, it is considered an art and a science. A well-known specialist in the field of simulation modeling, Robert Shannon, titled his book, widely known in the scientific and engineering world: " Simulation modeling- art and science." Therefore, in engineering practice there are no formalized instructions on how to create models. And, nevertheless, an analysis of the techniques that model developers use allows us to see a fairly transparent stage-by-stage modeling.

First stage: understanding the goals of modeling. In fact, this is the main stage of any activity. The goal significantly determines the content of the remaining stages of modeling. Note that the difference between a simple system and a complex one is generated not so much by their essence, but also by the goals set by the researcher.

Typically the goals of modeling are:

• forecasting the behavior of an object under new modes, combinations of factors, etc.;
• selection of combinations and values ​​of factors that ensure the optimal value of process efficiency indicators;
• analysis of the system’s sensitivity to changes in certain factors;
• testing various kinds of hypotheses about the characteristics of random parameters of the process under study;
• determination of functional relationships between the behavior ("response") of the system and influencing factors, which can contribute to the prediction of behavior or sensitivity analysis;
• understanding the essence, a better understanding of the object of study, as well as the formation of the first skills for operating a simulated or operating system.

Second stage: building a conceptual model. Conceptual model(from lat. conception) - a model at the level of the defining plan, which is formed during the study of the modeled object. At this stage, the object is examined and the necessary simplifications and approximations are established. Essential aspects are identified and minor ones are excluded. Units of measurement and ranges of variation of model variables are established. If possible, then conceptual model is presented in the form of well-known and well-developed systems: queuing, control, auto-regulation, various types of automatic machines, etc. Conceptual model fully summarizes the study of design documentation or experimental examination of the modeled object.

The result of the second stage is a generalized model diagram, fully prepared for mathematical description - construction of a mathematical model.

Third stage: choosing a programming or modeling language, developing an algorithm and model program. The model can be analytical or simulation, or a combination of both. In the case of an analytical model, the researcher must be proficient in solution methods.

In the history of mathematics (and this, incidentally, is the history of mathematical modeling) there are many examples of when the need to model various kinds of processes led to new discoveries. For example, the need to model motion led to the discovery and development of differential calculus (Leibniz and Newton) and related solution methods. Problems of analytical modeling of ship stability led Academician A. N. Krylov to the creation of the theory of approximate calculations and an analog computer.

The result of the third stage of modeling is a program compiled in the most convenient language for modeling and research - universal or special.

Fourth stage: experiment planning. Mathematical model is the object of the experiment. The experiment must be as informative as possible, satisfy the limitations, and provide data with the required accuracy and reliability. There is a theory of experimental planning; we will study the elements of this theory we need in the appropriate place in the discipline. GPSS World, AnyLogic, etc.) and can be applied automatically. It is possible that during the analysis of the results obtained, the model can be refined, supplemented, or even completely revised.

After analyzing the modeling results, their interpretation is carried out, that is, the results are translated into terms subject area. This is necessary because usually subject matter specialist(the one who needs research results) does not have the terminology of mathematics and modeling and can perform his tasks using only concepts that are well known to him.

This concludes our consideration of the modeling sequence, having made a very important conclusion about the need to document the results of each stage. This is necessary for the following reasons.

Firstly, modeling is an iterative process, that is, from each stage a return can be made to any of the previous stages to clarify the information needed at this stage, and documentation can save the results obtained at the previous iteration.

Secondly, in the case of researching a complex system, large teams of developers are involved, with different stages performed by different teams. Therefore, the results obtained at each stage must be transferable to subsequent stages, that is, have a unified presentation form and content that is understandable to other interested specialists.

Thirdly, the result of each stage must be a valuable product in its own right. For example, conceptual model may not be used for further conversion into a mathematical model, but rather be a description that stores information about the system, which can be used as an archive, as a teaching tool, etc.