Need Math Model

8/3/2019 Need Math Model

1/23

Why do we Need a

Mathematical Model?SAMSI/CRSC

Undergraduate Workshop 2006

Moustapha Pemy


2/23

1. The Reality, the experiment andthe model.

Let us denote by ytruethe true displacement of thebeam, ydata the data collected in the lab, and ythe solution of the beam model.


3/23

1. The Reality, the experiment andthe model.

Let us denote by ytrue the true displacement of thebeam, ydata the data collected in the lab, and ythe solution of the beam model.

Question:

Is it always possible to find a mathematical

formula to express ytrue as function of time?


4/23

1.The Reality, the experiment and

the model.

1. Let us denote by ytrue the true displacement ofthe beam, ydata the data collected in the lab,and y the solution of the beam model.

Question: Is it always possible to find a mathematical

formula to express ytrue as function of time?

How can we compare ytrue and ydata ? How can we compare ytrue and y?

How can we compare ydata and y?


5/23

The Reality, the experiment andthe model.

1. Answer: It is not always possible to find a mathematical

expression of the reality.

Due to measurement errors the data collected alwaysdiffer from the reality.


6/23

The Reality, the experiment andthe model.

1. Answer: It is not always possible to find a mathematical

expression of the reality.

Due to measurement errors the data collected alwaysdiffer from the reality.

In a mathematical model it is impossible to take intoaccount all parameters of the experiment, for example inthe beam model we do not take into account

temperature of lab and any other gravitational forces thatexist in the lab. This is why ytrue differs from y.

As we have seen throughout this week, ydata differs fromy. This leads to the errors analysis in the model.


7/23

2. The data and the black box

Assume we do not have a model. Can we just with datacollected in the CRSC lab derive a good understandingof the system?


8/23

2.The data and the black box

Assume we do not have a model. Can we justwith data collected in the CRSC lab derive agood understanding of the system?

Yes, just with data collected in CRSC lab we caninterpolate the data using the least squareapproach or any other interpolation technique toderive functional relationship between the

displacements of the beam and the times.


9/23


Assume we do not have a model, and that we haveinterpolated the data from the Beam in the CRSC laband obtained a functional relationship between the

displacements and the times.


10/23


Assume we do not have a model, and that we haveinterpolated the data from the Beam in the CRSC laband obtained a functional relationship between thedisplacements and the times.

Can we use this functional relationship to study a largerbeam?


11/23


Assume we do not have a model, and that we haveinterpolate the data from the Beam in the CRSC lab andobtain a functional relationship between thedisplacements and the times.

Can we use this functional relationship to study a largerbeam?

No, without a model we cannot use the functionalrelationship of a smaller beam to study a larger beam.


12/23

3. Interdependence of the modeland the data

Is it possible, just with experiments to obtain allnecessary or desirable data of a physical system?


13/23


Is it possible, just with experiments to obtain allnecessary or desirable data of a physical system?

No, there are certain data that are not observable, wecannot obtain them just with experiments. In fact, weneed experiments and models in order to filter outunobservable data.


14/23


Can we fit the model without experiments?


15/23


Can we fit the model without experiments?

No, in order to calibrate and validate themodel we need experiments


16/23


Why do scientists use both models andobserved data?


17/23


Why do scientists use both models andobserved data?

Control and design

Navigation (Space shuttles, Satellites,Rockets)

Predictions and Forecasting etc


18/23

4. Models without data

Give examples from science and industry where it isextremely difficult to collect data, and scientists mainlyrely on mathematical models?


19/23


Give examples from science and industry where it isextremely difficult to collect data, and scientists mainlyrely on mathematical models.

Astrophysics: due to the large scale.


20/23




Nanotechnology, quantum physics (very small scale).


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The design and the testing of nuclear weapons.


22/23






In Biomedical sciences, certain measurements in vivo

can be destructive


23/23






In Biomedical sciences, certain measurements in vivo

can be destructive In the design and the development of jet engines and big

airplanes like the Airbus A380 etc

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Need Math Model