What Is Mathematical Modelling?
What is a model?Mathematical modellingExamplesConclusionsWhat is Mathematical Modelling?John StockieCFD Research Group Meetinghttp://www.math.sfu.ca/ stockie/research/cfdgroup.htmlMay 21, 2014May 21, 2014What is a Model?1/33
What is a model?Mathematical modellingExamplesConclusionsOutline1What is a model?Disciplinary differences2Mathematical modellingDefinitionQuotesModelling process3ExamplesExample 1: Atmospheric dispersion modellingExample 2: Maple sap exudation4ConclusionsMay 21, 2014What is a Model?2/33
What is a model?Mathematical modellingExamplesConclusionsOutline1What is a model?Disciplinary differences2Mathematical modellingDefinitionQuotesModelling process3ExamplesExample 1: Atmospheric dispersion modellingExample 2: Maple sap exudation4ConclusionsMay 21, 2014What is a Model?3/33
What is a model?Mathematical modellingExamplesConclusionsWhat is a “model”?The word model has many different meanings:Even when restricted to the scientific endeavour (#3) thenumber of shades of meaning is huge, which can lead toconfusion!In the literature, the use of “model” varies greatly dependingon discipline . . .May 21, 2014What is a Model?4/33
What is a model?Mathematical modellingExamplesConclusionsModel for an experimentalist (biologist, chemist, . . . ) conceptual orbiological orchemical model hypothesis or picture May 21, 2014(to explain experimentalobservations) What is a Model? 5/33
What is a model?Mathematical modellingExamplesConclusionsModel for (some) engineers experimental orengineering modelor system May 21, 2014 device that mimicsa more complex one What is a Model? 6/33
What is a model?Mathematical modellingExamplesConclusionsModel for a software engineer# software model system description viamodelling language,diagrams or graphs(model theory)"May 21, 2014What is a Model?!7/33
What is a model?Mathematical modellingExamplesConclusionsModel for a computational scientist/engineer numerical orcomputationalmodel May 21, 2014 computer code thatsimulates a realsystem What is a Model? 8/33
What is a model?Mathematical modellingExamplesConclusionsNumerical or computational models:Similar to mathematical models in that they are based onmathematical equations implemented in code.Focus is much more on expert use of code, understandingassumptions and limitations, and fiddling with parameters.Example (CFD modelling):Running a commercial CFDcode involves defining problemgeometry (CAD) and selectingphysical and numerical parameters. The underlying equationsare well-known.May 21, 2014What is a Model?9/33
What is a model?Mathematical modellingExamplesConclusionsModel for a computer graphics or animation expert 3D model orgraphical model May 21, 2014 mathematicalrepresentation of a3D surface or object What is a Model? 10/33
What is a model?Mathematical modellingExamplesConclusionsModel for a statistician statistical model fitting to a curve ordistribution, parameterestimation May 21, 2014What is a Model? 11/33
What is a model?Mathematical modellingExamplesConclusionsOther uses of the word modelTheoretical model hypothesisData model data structure (in computer science)Climate model massive climate code (in geosciences)Other (non-scientific) uses listed in Wikipedia:Mental modelMetaphysical modelLogical modelEpistemlological modeletc.May 21, 2014What is a Model?12/33
What is a model?Mathematical modellingExamplesConclusionsPainfully overloaded use of “model”In a paper by McCray, Nieber, Poeter (2008), the word model isused in so many different ways it’s confusing and even painful!“An analytical model is a closed form mathematical solution”(approximate solution to PDE)“Conceptual model for the analytical solution” (a figure)“Numerical models discretize the flow domain into subdomains ofspace and time” (discretized PDE)“The errors associated with theoretical considerations of ananalytical model described previously, however, are probably lesssevere than errors associated with using incorrect values forhydraulic conductivity in the analytical model” (yikes) tool solution model ? May 21, 2014What is a Model?13/33
What is a model?Mathematical modellingExamplesConclusionsOutline1What is a model?Disciplinary differences2Mathematical modellingDefinitionQuotesModelling process3ExamplesExample 1: Atmospheric dispersion modellingExample 2: Maple sap exudation4ConclusionsMay 21, 2014What is a Model?14/33
What is a model?Mathematical modellingExamplesConclusionsWhat is a mathematical model?There are many, many definitions of mathematical model:Wikipedia: “A description of a system using mathematicalconcepts and language.” (too vague)Vahid Dabbaghian (2013): “A principled activity that hasboth principles behind it and methods that can be successfullyapplied.” (Huh?)Rutherford Aris (1978): “Any complete and consistent set ofmathematical equations which is thought to correspond tosome other entity, its prototype.” (fantastic book!)Jeremy Gunawardena (2014): “A logical machine forconverting assumptions into conclusions.”Gerda de Vries (2001, paraphrased): “The use of mathematicsto describe and explain real-world phenomena, investigateimportant questions about the observed world, test ideas, andmake predictions.”May 21, 2014What is a Model?15/33
What is a model?Mathematical modellingExamplesConclusionsWhat is a mathematical model?There are many, many definitions of mathematical model:Wikipedia: “A description of a system using mathematicalconcepts and language.” (too vague)Vahid Dabbaghian (2013): “A principled activity that hasboth principles behind it and methods that can be successfullyapplied.” (Huh?)Rutherford Aris (1978): “Any complete and consistent set ofmathematical equations which is thought to correspond tosome other entity, its prototype.” (fantastic book!)Jeremy Gunawardena (2014): “A logical machine forconverting assumptions into conclusions.”Gerda de Vries (2001, paraphrased): “The use of mathematicsto describe and explain real-world phenomena, investigateimportant questions about the observed world, test ideas, andmake predictions.”May 21, 2014What is a Model?15/33
What is a model?Mathematical modellingExamplesConclusionsWhat is a mathematical model?There are many, many definitions of mathematical model:Wikipedia: “A description of a system using mathematicalconcepts and language.” (too vague)Vahid Dabbaghian (2013): “A principled activity that hasboth principles behind it and methods that can be successfullyapplied.” (Huh?)Rutherford Aris (1978): “Any complete and consistent set ofmathematical equations which is thought to correspond tosome other entity, its prototype.” (fantastic book!)Jeremy Gunawardena (2014): “A logical machine forconverting assumptions into conclusions.”Gerda de Vries (2001, paraphrased): “The use of mathematicsto describe and explain real-world phenomena, investigateimportant questions about the observed world, test ideas, andmake predictions.”May 21, 2014What is a Model?15/33
What is a model?Mathematical modellingExamplesConclusionsWhat is a mathematical model?There are many, many definitions of mathematical model:Wikipedia: “A description of a system using mathematicalconcepts and language.” (too vague)Vahid Dabbaghian (2013): “A principled activity that hasboth principles behind it and methods that can be successfullyapplied.” (Huh?)Rutherford Aris (1978): “Any complete and consistent set ofmathematical equations which is thought to correspond tosome other entity, its prototype.” (fantastic book!)Jeremy Gunawardena (2014): “A logical machine forconverting assumptions into conclusions.”Gerda de Vries (2001, paraphrased): “The use of mathematicsto describe and explain real-world phenomena, investigateimportant questions about the observed world, test ideas, andmake predictions.”May 21, 2014What is a Model?15/33
What is a model?Mathematical modellingExamplesConclusionsWhat is a mathematical model?There are many, many definitions of mathematical model:Wikipedia: “A description of a system using mathematicalconcepts and language.” (too vague)Vahid Dabbaghian (2013): “A principled activity that hasboth principles behind it and methods that can be successfullyapplied.” (Huh?)Rutherford Aris (1978): “Any complete and consistent set ofmathematical equations which is thought to correspond tosome other entity, its prototype.” (fantastic book!)Jeremy Gunawardena (2014): “A logical machine forconverting assumptions into conclusions.”Gerda de Vries (2001, paraphrased): “The use of mathematicsto describe and explain real-world phenomena, investigateimportant questions about the observed world, test ideas, andmake predictions.”May 21, 2014What is a Model?15/33
What is a model?Mathematical modellingExamplesConclusionsWhat is a mathematical model?There is no perfect definition of “mathematical model” but . . .My choice:“A well-posed system of equations that describes a real system.”May 21, 2014What is a Model?16/33
What is a model?Mathematical modellingExamplesConclusionsQuotesGeorge Box (1987): “The fact that [the model] is an approximationdoes not necessarily detract from its usefulness because models areapproximations. All models are wrong, but some are useful.”J.R. Philip (1966): “A model is a useful (and often indispensible)framework on which to organize our knowledge about aphenomenon . . . It must not be overlooked that the quantitativeconsequences of any model can be no more reliable than the a prioriagreement between the assumptions of the model and the knownfacts about the real phenomenon. When the model is known todiverge significantly from the facts, it is self-deceiving to claimquantitative usefulness for it by appeal to agreement between aprediction of the model and observation.”David Levermore (PIMS-CSC seminar, 2014): “A model can beviewed as a form of data compression, and simulation is a methodfor decompressing or unpacking the data.”May 21, 2014What is a Model?17/33
What is a model?Mathematical modellingExamplesConclusionsMore quotesGerda de Vries (2001): “A modeller undertakes experiments onmathematical representations of the real world.”Howard Emmons: The challenge in mathematical modelling is “notto produce the most comprehensive descriptive model but toproduce the simplest possible model that incorporates the majorfeatures of the phenomenon of interest.” (see Gunawardena)Reinhard Illner et al. (2005): “Mathematical modelling is a subjectwithout boundaries in every conceivable sense.”Andrew Fowler (1997): “Mathematical modelling is a subject that isdifficult to teach . . . one learns it by practice: There are no setrules, and an understanding of the ‘right’ way to model can only bereached by familiarity with a wealth of examples.”May 21, 2014What is a Model?18/33
What is a model?Mathematical modellingExamplesConclusionsFowler’s modelling process1Problem identification2Model formulation3Analysis4Computation5Model validationMay 21, 2014What is a Model?19/33
What is a model?Mathematical modellingExamplesConclusionsSource: de Vries (2001)May 21, 2014What is a Model?20/33
What is a model?Mathematical modellingExamplesConclusionsA better process diagramMay 21, 2014What is a Model?21/33
What is a model?Mathematical modellingExamplesConclusionsA more philosophical view of modellingA mathematical model is the mathematical structure that tiesthe specific situation back into a more general theoryIts validity as an “explanation” of what is going on in thegiven situation rests on the tripod of:123the adequacy of its representation of the situationits internal correctnessthe acceptability of the general theory which is involvedSource: Aris and Penn, “The mere notion of a model”Mathematical Modelling, 1:1-12, 1980.May 21, 2014What is a Model?22/33
What is a model?Mathematical modellingExamplesConclusionsOutline1What is a model?Disciplinary differences2Mathematical modellingDefinitionQuotesModelling process3ExamplesExample 1: Atmospheric dispersion modellingExample 2: Maple sap exudation4ConclusionsMay 21, 2014What is a Model?23/33
What is a model?Mathematical modellingExamplesConclusionsExample 1: Atmospheric dispersion modellingTransport of a contaminant by advection and diffusion.Governing equations are very well-understood: u · C · (K C ) Q δ(x) δ(y ) δ(z H)Gaussian plume solution: " #y2(z H)2(z H)2Qexp 2exp exp C (x, y , z) 2πUσy σz2σy2σz22σz2windspeedz Hzplume centerlineeffectiveheightxyUz 0May 21, 2014What is a Model?24/33
What is a model?Mathematical modellingExamplesConclusionsExample 1: Atmospheric dispersion modellingAim: Given measurements of C , find Q.We used the Gaussian plume solution to develop a veryefficient inverse solver.Most effort was expended initially in fitting parameters.Our results were in excellent agreement with current emissionestimates.Currently developing a much more accurate and robustinversion approach (Bamdad).May 21, 2014What is a Model?25/33
What is a model?Mathematical modellingExamplesConclusionsExample 2: Maple sap exudationMystery: Maple trees exude sap inwinter with no leaves or root uptake.Leading hypothesis due toMilburn-O’Malley assumes acombination of:multiphase flow of liquid and gasfreezing and thawing sap (Stefanproblem)dissolving gas (Henry’s law)porous flow through cell walls(Darcy’s Law)osmotic pressure acrossselectively permeable membranesMay 21, 2014What is a Model?26/33
What is a model?Mathematical modellingExamplesConclusionsExample 2: Maple sap exudationWriting down a consistent set of equations for the freezingprocess alone took Maurizio and I almost a full year!Tyree’s challenge (1983): “There is insufficient quantitativeinformation to set up a system of physical equations.” (wrong)Indeed, we had to introduce several corrections/additions tothe Milburn-O’Malley hypothesis to get things to work this is the real value of a mathematical model!May 21, 2014What is a Model?27/33
What is a model?Mathematical modellingExamplesConclusionsOutline1What is a model?Disciplinary differences2Mathematical modellingDefinitionQuotesModelling process3ExamplesExample 1: Atmospheric dispersion modellingExample 2: Maple sap exudation4ConclusionsMay 21, 2014What is a Model?28/33
What is a model?Mathematical modellingExamplesConclusionsConclusionsBe careful how you use the word model, paying attention toyour audience.If you want to succeed at modelling, then you should:12absorb as much mathematics as you can;step outside your comfort zone and become an expert onnon-math subject(s), or at least collaborate with such anexpert.Being a mathematical generalist affords you valuableopportunities to study an enormous variety of physical andbiological problems.Remember that no model is perfect, so always be critical ofyour assumptions and carefully validate them.May 21, 2014What is a Model?29/33
What is a model?Mathematical modellingExamplesConclusionsWhat next?In a few weeks, I’ll get deeper into the topic of mathematicalmodelling withChallenges and Opportunities in“Mathematics For Industry”(lecture at CAIMS meeting)May 21, 2014What is a Model?30/33
What is a model?Mathematical modellingExamplesConclusionsClosing remarksAris and Penn (1980) on modelling as craftsmanship:What goes on in the modeller’s head is not purely formalizable . . .Nor is it purely subjective . . . It has structure, it has techniquesthat can be taught and learned, but involves also a personal touch,not only in trivialities (such as the choice of notation) but indeeper considerations of skill and suitability. It also involves anelement of risk, since a wrong turn in the development of a modelmay lead to its complete stultification. Like the furniture maker,the mathematical modeller shapes the several parts of his work andfits them together. If one is marred or misshapen it must bereworked or even discarded and replaced.”Aris was a true master ofthe English language, andis a real joy to read!May 21, 2014What is a Model?31/33
What is a model?Mathematical modellingExamplesConclusionsReferences IR. Aris.Mathematical Modelling Techniques.Dover, New York, 1994.Republished from Pitman, 1978.R. Aris and M. Penn.The mere notion of a model.Mathematical Modelling, 1:1–12, 1980.M. Bakker.Are all models wrong? Absolutely not.Groundwater, 51(3):313, 2013.A. C. Fowler.Mathematical Models in the Applied Sciences.Cambridge Texts in Applied Mathematics. Cambridge University Press, 1997.J. Gunawardena.Models in biology: ‘accurate descriptions of our pathetic thinking’.BMC Biology, 12:29, 2014.May 21, 2014What is a Model?32/33
What is a model?Mathematical modellingExamplesConclusionsReferences IIR. Illner, C. S. Bohun, S. McCollum, and T. van Roode.Mathematical Modelling: A Case Studies Approach, volume 27 of StudentMathematical Library.American Mathematical Society, 2005.J. E. McCray, J. Nieber, and E. P. Poeter.Groundwater mounding in the vadose zone from on-site wastewater systems:Analytical and numerical tools.Journal of Hydrologic Engineering, 13(8):710–719, 2008.May 21, 2014What is a Model?33/33
Reinhard Illner et al. (2005): “Mathematical modelling is a subject without boundaries in every conceivable sense.” Andrew Fowler (1997): “Mathematical modelling is a subject that is difficult to teach . one learns it by practice: There are no set rules, and an understanding of the ‘right’ way to model can only be
Mathematical modelling, the socio-critical perspective and the reflexive discussions * 133 . Jonei Cerqueira Barbosa . Mathematical modelling and environmental education 145. Ademir Donizeti Caldeira . Mathematical models in
Mathematical Modelling and Mathematical Competencies: The case of Biology students. . benefits associated with engaging students in mathematical modeling. There is a ‘red thread’ . These studies include an international comparison of secondary school students’ competence pro
So, I say mathematical modeling is a way of life. Keyword: Mathematical modelling, Mathematical thinking style, Applied 1. Introduction: Applied Mathematical modeling welcomes contributions on research related to the mathematical modeling of e
Common sense (19), mathematical knowledge and experience in modelling (7), both modelling experience and common sense (3), the given information (1), idea of spread of disease (1), I have never seen such problems in mathematical context before, so I don’t
Note 3: Physical Systems Modelling XMUT315 - Note 3 - 2 Figure 1: Modelling of a physical system with scaled physical model On the other hand, to represent the physical system in addition to scaled model, we often use mathematical and numerical models. Mathematical model is described as function and variable in mathematical equation.
through MM processes not provide direct solutions. Theoretical Background Lingefjärd (2007) stated that “mathematical modelling is not a body of mathematical knowledge but rather a collection of general principles which experience has proved to be helpful in the process of applying mathematical know-how to analyze problems” (p. 476).
and simplified method to describe masonry vaults in global seismic analyses of buildings. Fig. 1 summarizes three different modelling techniques for ma sonry modelling, respectively, mi cro- , macro- and simplified micro modelling. In the case a micro modelling approach is take n, the challenge is to describe the complex behavior of the
Andreas Wagner. ERAD 2014 - THE EIGHTH EUROPEAN CONFERENCE ON RADAR IN METEOROLOGY AND HYDROLOGY ERAD 2014 Abstract ID 306 2 Using a pattern recognition scheme, single pixels or groups of pixels that show unusual signatures compared to precipitation echoes, are identified in these accumulation products. Such signatures may be straight edges, high gradients or systematic over- or .
