These notes are available online at
A slightly revised version appeared as
Mathematical Model Building,
Chapter 3 in: Modeling Languages in Mathematical Optimization
(J. Kallrath, ed.),
Applied Optimization, Vol. 88,
Kluwer, Boston 2004.
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
- 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
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
- Modeling, classifying and reconstructing skulls
- Reconstruction of objects from preserved fragments
- Classifying ancient artifices
- Computer vision
- Image interpretation
- Speech recognition
- Optical character recognition
- Reasoning under uncertainty
- Computer animation (Jurassic Park)
- Detection of planetary systems
- Correcting the Hubble telescope
- Origin of the universe
- Evolution of stars
- Protein folding
- Humane genome project
- Population dynamics
- Evolutionary pedigrees
- Spreading of infectuous diseases (AIDS)
- Animal and plant breeding (genetic variability)
- Chemical equilibrium
- Planning of production units
- Chemical reaction dynamics
- Molecular modeling
- Electronic structure calculations
- Image processing
- Realistic computer graphics (ray tracing)
- Finger print recognition
- Face recognition
- Stability of electric curcuits
- Microchip analysis
- Power supply network optimization
- Risk analysis
- Value estimation of options
- Prediction of oil or ore deposits
- Map production
- Earth quake prediction
- Web search
- Optimal routing
- Microchip production
- Semiconductor modeling
- Stability of structures (high rise buildings, bridges, air planes)
- Structural optimization
- Crash simulation
- Radiation therapy planning
- Computer-aided tomography
- Blood circulation models
- Weather prediction
- Climate prediction (global warming, what caused the ozone hole?)
- Analysis and synthesis of sounds
- Neural networks
- Signal transmission in nerves
- Docking of molecules to proteins
- Screening of new compounds
- Elementary particle tracking
- Quantum field theory predictions (baryon spectrum)
- Laser dynamics
- Formalizing diaries of therapy sessions
- Trajectory planning
- Flight simulation
- Shuttle reentry
- 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
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
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
- Function evaluation
- Automatic and numerical differentiation
- 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)
- Time series analysis (signal processing, filtering, time correlations,
- 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)
- Symbolic methods (computer algebra)
- 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
- Priorities/Quality assignments
- of Application
- of Mathematics
- Literature search
N. Numerical Methods
- Software libraries
- Free software from WWW
- Background information
- Flow diagrams
- User interface
Using the modeling diagram
- The modeling diagram breaks the modeling task into 16=6+10 different
- 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
- 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
- Collect/ask for background information needed to understand the problem.
- Start with simple models; add details as they become known and useful
- 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
- 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
- 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
- Be not discouraged. Failures inform you about important missing details
in your understanding of the problem (or the customer/boss) - utilize
- 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
- 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.''
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
- 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
Jesus: ''Ask, and you will receive. Search, and you will find.
Knock, and the door will be opened for you.''
For more information about mathematics, software, and applications,
see, e.g., my home page, at