Mathematical Modeling

Arnold Neumaier

These notes are available online at

A slightly revised version appeared as
A. Neumaier, Mathematical Model Building, Chapter 3 in: Modeling Languages in Mathematical Optimization (J. Kallrath, ed.), Applied Optimization, Vol. 88, Kluwer, Boston 2004.

Abstract. Some notes on mathematical modeling, listing motivations, applications, a numerical toolkit, general modeling rules, modeling conflicts, useful attitudes, and structuring the modeling work into 16 related activities by means of a novel modeling diagram.

1  Why mathematical modeling?

Mathematical modeling is the art of translating problems from an application area into tractable mathematical formulations whose theoretical and numerical analysis provides insight, answers, and guidance useful for the originating application.

Mathematical modeling

Learning about mathematical modeling is an important step from a theoretical mathematical training to an application-oriented mathematical expertise, and makes the student fit for mastering the challenges of our modern technological culture.

2  A list of applications

In the following, I give a list of applications whose modeling I understand, at least in some detail. All areas mentioned have numerous mathematical challenges.

This list is based on my own experience; therefore it is very incomplete as a list of applications of mathematics in general. There are an almost endless number of other areas with interesting mathematical problems.

Indeed, mathematics is simply the language for posing problems precisely and unambiguously (so that even a stupid, pedantic computer can understand it).




Artificial intelligence




Chemical engineering


Computer science

Criminalistic science


Electrical engineering


Fluid mechanics




Materials Science

Mechanical engineering







Political Sciences


Space Sciences

Transport Science

3  Basic numerical tasks

The following is a list of categories containing the basic algorithmic toolkit needed for extracting numerical information from mathematical models.

Due to the breadth of the subject, this cannot be covered in a single course. For a thorough education one needs to attend courses (or read books) at least on numerical analysis (which usually covers some numerical linear algebra, too), optimization, and numerical methods for partial differential equations.

Unfortunately, there appear to be few good courses and books on (higher-dimensional) numerical data analysis.

Numerical linear algebra

Numerical analysis

Numerical data analysis (= numerical statistics)

Numerical functional analysis

Non-numerical algorithms

4  The modeling diagram

The nodes of the following diagram represent information to be collected, sorted, evaluated, and organized.

Information flow diagram

The edges of the diagram represent activities of two-way communication (flow of relevant information) between the nodes and the corresponding sources of information.

S. Problem Statement

M. Mathematical Model

T. Theory

N. Numerical Methods

P. Programs

R. Report

Using the modeling diagram

5  General rules

Lao Tse: ''People often fail on the verge of success; take care at the end as at the beginning, so that you may avoid failure.''

6  Conflicts

Most modeling situations involve a number of tensions between conflicting requirements that cannot be reconciled easily.

Einstein: ''A good theory'' (or model) ''should be as simple as possible, but not simpler.''

The conflicts described are creative and constructive, if one does not give in too easily. As a good material can handle more physical stress, so a good scientist can handle more stress created by conflict.

''We shall overcome'' - a successful motto of the black liberation movement, created by a strong trust in God. This generalizes to other situations where one has to face difficulties, too.

Among other qualities it has, university education is not least a long term stress test - if you got your degree, this is a proof that you could overcome significant barriers. The job market pays for the ability to persist.

7  Attitudes

Jesus: ''Ask, and you will receive. Search, and you will find. Knock, and the door will be opened for you.''

8  References

For more information about mathematics, software, and applications, see, e.g., my home page, at