Mathematical Modelling

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

is indispensable in many applications

is successful in many further applications

gives precision and direction for problem solution

enables a thorough understanding of the system modeled

prepares the way for better design or control of a system

allows the efficient use of modern computing capabilities

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).

Anthropology

Modeling, classifying and reconstructing skulls

Archeology

Reconstruction of objects from preserved fragments

Classifying ancient artifices

Architecture

Virtual reality

Artificial intelligence

Computer vision

Image interpretation

Robotics

Speech recognition

Optical character recognition

Reasoning under uncertainty

Arts

Computer animation (Jurassic Park)

Astronomy

Detection of planetary systems

Correcting the Hubble telescope

Origin of the universe

Evolution of stars

Biology

Protein folding

Humane genome project

Population dynamics

Morphogenesis

Evolutionary pedigrees

Spreading of infectuous diseases (AIDS)

Animal and plant breeding (genetic variability)

Chemical engineering

Chemical equilibrium

Planning of production units

Chemistry

Chemical reaction dynamics

Molecular modeling

Electronic structure calculations

Computer science

Image processing

Realistic computer graphics (ray tracing)

Criminalistic science

Finger print recognition

Face recognition

Economics

Labor data analysis

Electrical engineering

Stability of electric curcuits

Microchip analysis

Power supply network optimization

Finance

Risk analysis

Value estimation of options

Fluid mechanics

Wind channel

Turbulence

Geosciences

Prediction of oil or ore deposits

Map production

Earth quake prediction

Internet

Web search

Optimal routing

Linguistics

Automatic translation

Materials Science

Microchip production

Microstructures

Semiconductor modeling

Mechanical engineering

Stability of structures (high rise buildings, bridges, air planes)

Structural optimization

Crash simulation

Medicine

Radiation therapy planning

Computer-aided tomography

Blood circulation models

Meteorology

Weather prediction

Climate prediction (global warming, what caused the ozone hole?)

Music

Analysis and synthesis of sounds

Neuroscience

Neural networks

Signal transmission in nerves

Pharmacology

Docking of molecules to proteins

Screening of new compounds

Physics

Elementary particle tracking

Quantum field theory predictions (baryon spectrum)

Laser dynamics

Political Sciences

Analysis of elections

Psychology

Formalizing diaries of therapy sessions

Space Sciences

Trajectory planning

Flight simulation

Shuttle reentry

Transport Science

Air traffic scheduling

Taxi for handicapped people

Automatic pilot for cars and airplanes

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

Linear systems of equations

Eigenvalue problems

Linear programming (linear optimization)

Techniques for large, sparse problems

Numerical analysis

Function evaluation

Automatic and numerical differentiation

Interpolation

Approximation (Padé, least squares, radial basis functions)

Integration (univariate, multivariate, Fourier transform)

Special functions

Nonlinear systems of equations

Optimization = nonlinear programming

Techniques for large, sparse problems

Numerical data analysis (= numerical statistics)

Visualization (2D and 3D computational geometry)

Parameter estimation (least squares, maximum likelihood)

Prediction

Classification

Time series analysis (signal processing, filtering, time correlations, spectral analysis)

Categorical time series (hidden Markov models)

Random numbers and Monte Carlo methods

Techniques for large, sparse problems

Numerical functional analysis

Ordinary differential equations (initial value problems, boundary value problems, eigenvalue problems, stability)

Techniques for large problems

Partial differential equations (finite differences, finite elements, boundary elements, mesh generation, adaptive meshes)

Stochastic differential equations

Integral equations (and regularization)

Non-numerical algorithms

Symbolic methods (computer algebra)

Sorting

Compression

Cryptography

Error correcting codes

4  The modeling diagram

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

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

Interests of customer/boss

Often ambiguous/incomplete

Wishes are sometimes incompatible

M. Mathematical Model

Concepts/Variables

Relations

Restrictions

Goals

Priorities/Quality assignments

T. Theory

of Application

of Mathematics

Literature search

N. Numerical Methods

Software libraries

Free software from WWW

Background information

P. Programs

Flow diagrams

Implementation

User interface

Documentation

R. Report

Description

Analysis

Results

Validation

Visualization

Limitations

Recommendations

Using the modeling diagram

The modeling diagram breaks the modeling task into 16=6+10 different processes.

Each of the 6 nodes and each of the 10 edges deserve repeated attention, usually at every stage of the modeling process.

The modeling is complete only when the 'traffic' along all edges becomes insignificant.

Generally, working on an edge enriches both participating nodes.

If stuck along one edge, move to another one! Use the general rules below as a check list!

Frequently, the problem changes during modeling, in the light of the understanding gained by the modeling process. At the end, even a vague or contradictory initial problem description should have developed into a reasonably well-defined description, with an associated precisely defined (though perhaps inaccurate) mathematical model.

5  General rules

Look at how others model similar situations; adapt their models to the present situation.

Collect/ask for background information needed to understand the problem.

Start with simple models; add details as they become known and useful or necessary.

Find all relevant quantities and make them precise.

Find all relevant relationships between quantities ([differential] equations, inequalities, case distinctions).

Locate/collect/select the data needed to specify these relationships.

Find all restrictions that the quantities must obey (sign, limits, forbidden overlaps, etc.). Which restrictions are hard, which soft? How soft?

Try to incorporate qualitative constraints that rule out otherwise feasible results (usually from inadequate previous versions).

Find all goals (including conflicting ones)

Play the devil's advocate to find out and formulate the weak spots of your model.

Sort available information by the degree of impact expected/hoped for.

Create a hierarchy of models: from coarse, highly simplifying models to models with all known details. Are there useful toy models with simpler data? Are there limiting cases where the model simplifies? Are there interesting extreme cases that help discover difficulties?

First solve the coarser models (cheap but inaccurate) to get good starting points for the finer models (expensive to solve but realistic)

Try to have a simple working model (with report) after 1/3 of the total time planned for the task. Use the remaining time for improving or expanding the model based on your experience, for making the programs more versatile and speeding them up, for polishing documentation, etc.

Good communication is essential for good applied work.

The responsibility for understanding, for asking the questions that lead to it, for recognizing misunderstanding (mismatch between answers expected and answers received), and for overcoming them lies with the mathematician. You cannot usually assume your customer to understand your scientific jargon.

Be not discouraged. Failures inform you about important missing details in your understanding of the problem (or the customer/boss) - utilize this information!

There are rarely perfect solutions. Modeling is the art of finding a satisfying compromise. Start with the highest standards, and lower them as the deadline approaches. If you have results early, raise your standards again.

Finish your work in time.

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.

fast - slow

cheap - expensive

short term - long term

simplicity - complexity

low quality - high quality

approximate - accurate

superficial - in depth

sketchy - comprehensive

concise - detailed

short description - long description

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

perfecting a program - need for quick results

collecting the theory - producing a solution

doing research - writing up

quality standards - deadlines

dreams - actual results

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

Do whatever you do with love. Love (even in difficult circumstances) can be learnt; it noticeably improves the quality of your work and the satisfaction you derive from it.

Do whatever you do as a service to others. This will improve your attention, the feedback you'll get, and the impact you'll have.

Take responsibility; ask if in doubt; read to confirm your understanding. This will remove many impasses that otherwise would delay your work.

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 http://www.mat.univie.ac.at/~neum/