Mathematical Modeling - A tutorialMathematical Modeling A tutorial ... Tamás Biró Mathematical...
Transcript of Mathematical Modeling - A tutorialMathematical Modeling A tutorial ... Tamás Biró Mathematical...
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Mathematical ModelingA tutorial
Tamás Biró
Eötvös Loránd University
Network Theory and Computer Modeling in the Study of ReligionAugust 31, 2016
Tamás Biró Mathematical Modeling (tutorial) 1
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
A note on myself
June 1997: two exams on the same day
János Kertész: Computer simulations (physics major)
József Schweitzer: Jewish liturgy (Hebrew major)
Anything common in these two topics?
HOPEFULLY. . .
Tamás Biró Mathematical Modeling (tutorial) 2
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
A note on myself
June 1997: two exams on the same day
János Kertész: Computer simulations (physics major)
József Schweitzer: Jewish liturgy (Hebrew major)
Anything common in these two topics?
HOPEFULLY. . .
Tamás Biró Mathematical Modeling (tutorial) 2
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
A note on myself
Theses on
Analysis of DNA sequences using text analysis methods(physics major, supervisor: Tamás Vicsek)
Modeling Mathematically the Statistical Properties of Written Texts(theoretical linguistics major)
From physics to linguistics: was it a big step?
NO!
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
A note on myself
Theses on
Analysis of DNA sequences using text analysis methods(physics major, supervisor: Tamás Vicsek)
Modeling Mathematically the Statistical Properties of Written Texts(theoretical linguistics major)
From physics to linguistics: was it a big step?
NO!
Tamás Biró Mathematical Modeling (tutorial) 3
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Overview
1 Computation 6= computers
2 Mathematics and computer simulations as methodologies
3 On differential equations
4 Differential equations for dynamic systems
5 Conclusions
Tamás Biró Mathematical Modeling (tutorial) 4
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Overview
1 Computation 6= computers
2 Mathematics and computer simulations as methodologies
3 On differential equations
4 Differential equations for dynamic systems
5 Conclusions
Tamás Biró Mathematical Modeling (tutorial) 5
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Let us create a calculating machine
The machine has to be able to sum up (two) numbers.
Input: Tamás BiróOutput: István CzacheszProgrammer: Luther MartinProcessing units: everybody else
Only rule type allowed for each processing unit:
if you hear X 1 [and X 2 [and X 3 . . . ] ],then say Y 1 to Z 1 [and say Y 2 to Z 2 [and Y 3 to Z 3 [. . . ] ]
20 minutes for the project!
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Let us create a calculating machine
The machine has to be able to sum up (two) numbers.
Input: Tamás BiróOutput: István CzacheszProgrammer: Luther MartinProcessing units: everybody else
Only rule type allowed for each processing unit:
if you hear X 1 [and X 2 [and X 3 . . . ] ],then say Y 1 to Z 1 [and say Y 2 to Z 2 [and Y 3 to Z 3 [. . . ] ]
20 minutes for the project!
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Let us create a calculating machine
The machine has to be able to sum up (two) numbers.
Input: Tamás BiróOutput: István CzacheszProgrammer: Luther MartinProcessing units: everybody else
Only rule type allowed for each processing unit:
if you hear X 1 [and X 2 [and X 3 . . . ] ],then say Y 1 to Z 1 [and say Y 2 to Z 2 [and Y 3 to Z 3 [. . . ] ]
20 minutes for the project!
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
What have we learned?
Computation 6= computers!Seemingly intelligent processes can be automated.Computational resources: memory (# of processing units) and time.Human resources: the time to create the program.Need to precisely define everything. Bugs and debugging.Continuous time vs. discrete time ticks.. . .
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
What have we learned?
Computation 6= computers!Seemingly intelligent processes can be automated.Computational resources: memory (# of processing units) and time.Human resources: the time to create the program.Need to precisely define everything. Bugs and debugging.Continuous time vs. discrete time ticks.. . .
Tamás Biró Mathematical Modeling (tutorial) 7
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
David Marr: Three levels of analysis
1. Computational level: What does the system do? What is thefunction (i.e., mapping input onto output) performed by the system?E.g., summation; face recognition; ritual performance.
2. Algorithmic/representational level: How is it performed?Representations, and manipulations of those representation.E.g., summation digit-by-digit.
3. Implementational/physical level: How is this algorithmphysically realized?E.g., in silico; wetware; workshop participants.
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Overview
1 Computation 6= computers
2 Mathematics and computer simulations as methodologies
3 On differential equations
4 Differential equations for dynamic systems
5 Conclusions
Tamás Biró Mathematical Modeling (tutorial) 9
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Methodologies in physics
A. Experimental physics: data collection(exploratory research vs. hypothesis testing)
B. Theoretical physics: mathematics for modeling the word/nature
+ [Thought experiments]
+ Computer simulations (e.g., Kertész and Vicsek)
Analogy in other disciplines?
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
“Complicatedness” of theories
0. Thought experiments: handled mentally.
1. Mathematical models: handled analytically.
2. Computer simulations: can be more complex thanmathematically tractable models, but simpler than real life.
Are we happy with
Level of abstraction?
Simplifications?
3. Experiments: complexities of real life controlled.
4. Observations: complexities of real life at their best.
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Numerical solution vs. analytic solution
1 + 2 + 3 + . . . + 98 + 99 + 100 =?
Numerical solution: go and compute it with sheer force.For more complex problems: often an approximate solution, only.
Analytic solution: clever math provides a closed formula.Exact solution with pencil and paper
→ on the condition that an analytic solution exists!
1 + 2 + . . . + 99 + 100 =100× (100 + 1)
2= 50× 101 = 5050
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Numerical solution vs. analytic solution
1 + 2 + 3 + . . . + 98 + 99 + 100 =?
Numerical solution: go and compute it with sheer force.For more complex problems: often an approximate solution, only.
Analytic solution: clever math provides a closed formula.Exact solution with pencil and paper
→ on the condition that an analytic solution exists!
1 + 2 + . . . + 99 + 100 =100× (100 + 1)
2= 50× 101 = 5050
Tamás Biró Mathematical Modeling (tutorial) 12
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Issues with computer simulations
Helps you better understand your theory/hypothesis.
Forces you to formulate details of theory/hypothesis precisely.
Faster. Can also be applied to past/remote/unreal conditions. Etc.
Level of optimal abstractions:
– If too simple: no connection to reality? What do the results tell us?
– If too complex, too many parameters: easy to tweak the model.What do the results tell us?
→ Possible answer:
Understand the behavior of the model as a function of its parameters.
Seek results that are not too dependent on parameter setting.
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Overview
1 Computation 6= computers
2 Mathematics and computer simulations as methodologies
3 On differential equations
4 Differential equations for dynamic systems
5 Conclusions
Tamás Biró Mathematical Modeling (tutorial) 14
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Derivatives
https://en.wikipedia.org/wiki/Derivative#/media/File:
Tangent_function_animation.gif
x
y = f (x)
x
y = f ′(x)
f (x) = x sin(x2) + 1 ⇒ f ′(x) = sin(x2) + 2x2 cos(x2)
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Derivatives
f (x) f ′(x)
c 0x 1x2 2xx3 3x2
c · f (x) c · f ′(x)
f (x) + g(x) f ′(x) + g′(x)
ex ex
sin(x) cos(x)
cos(x) − sin(x)
. . . . . .
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Differential equations
What is f (x), iff ′(x) = 2x
Solution:
f (x) = x2
f (x) = x2 + c
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Differential equations
What is f (x), iff ′(x) = 2x
Solution:
f (x) = x2
f (x) = x2 + c
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Differential equations
What is f (x), iff ′(x) = f (x)
Solution:
f (x) = ex
f (x) = c · ex
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Differential equations
What is f (x), iff ′(x) = f (x)
Solution:
f (x) = ex
f (x) = c · ex
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Differential equations
What is f (x), iff ′′(x) = −f (x)
Solution:
f (x) = sin(x)
f (x) = cos(x)
f (x) = c1 · sin(x) + c2 · cos(x)
Thus far: analytic solutions
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Differential equations
What is f (x), iff ′′(x) = −f (x)
Solution:
f (x) = sin(x)
f (x) = cos(x)
f (x) = c1 · sin(x) + c2 · cos(x)
Thus far: analytic solutions
Tamás Biró Mathematical Modeling (tutorial) 19
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Differential equations
What is f (x), iff ′′(x) = −f (x)
Solution:
f (x) = sin(x)
f (x) = cos(x)
f (x) = c1 · sin(x) + c2 · cos(x)
Thus far: analytic solutions
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Numerical solutions for differential equations
What is f (x), if
f ′′(x) + x · f ′(x)− 2 · x2 · f (x) + cos(x3)− 15 = 0
Solution:
Use computers to solve this problem.Numerical solutions: e.g., using step-by-step approximations.
NB: various sources of errors.
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Numerical solutions for differential equations
What is f (x), if
f ′′(x) + x · f ′(x)− 2 · x2 · f (x) + cos(x3)− 15 = 0
Solution:
Use computers to solve this problem.Numerical solutions: e.g., using step-by-step approximations.
NB: various sources of errors.
Tamás Biró Mathematical Modeling (tutorial) 20
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Overview
1 Computation 6= computers
2 Mathematics and computer simulations as methodologies
3 On differential equations
4 Differential equations for dynamic systems
5 Conclusions
Tamás Biró Mathematical Modeling (tutorial) 21
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Population dynamics
y(t): size of the population at time t .
∆y(t) = y(t + 1)− y(t): population growth at time t .
Suppose that population growth is equal to population size:
y(t + 1)− y(t) = y(t)y(t + 1) = 2y(t)
Then: y(1) = 2y(0), y(2) = 2y(1) = 4y(0),y(3) = 2y(2) = 8y(0),. . . , y(t) = 2ty(0).
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Population dynamics
y(t): size of the population at time t .
∆y(t) = y(t + 1)− y(t): population growth at time t .
Suppose that population growth is equal to population size:
y(t + 1)− y(t) = y(t)∆y(t) = y(t)
dydt
= y ′(t) = y(t)
And so: y(t) = et .
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Dynamic system
1,2, . . .n: the components of the dynamics system.
y1(t), y2(t), . . . yn(t):“value” of each component in the dynamics system at time t .
The equations defining the dynamic system (discrete time!):
y1(t + 1) = . . . y1(t) + . . . y2(t) + . . . yn(t) + . . . t + . . .
y2(t + 1) = . . . y1(t) + . . . y2(t) + . . . yn(t) + . . . t + . . .
. . .
yn(t + 1) = . . . y1(t) + . . . y2(t) + . . . yn(t) + . . . t + . . .
So what functions are y1(t), y2(t), . . . yn(t)?Solve those differential equations either numerically, or analytically.
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Dynamic system
1,2, . . .n: the components of the dynamics system.
y1(t), y2(t), . . . yn(t):“value” of each component in the dynamics system at time t .
The equations defining the dynamic system (continuous time!):
y1′(t) = . . . y1(t) + . . . y2(t) + . . . yn(t) + . . . t + . . .
y2′(t) = . . . y1(t) + . . . y2(t) + . . . yn(t) + . . . t + . . .
. . .
yn′(t) = . . . y1(t) + . . . y2(t) + . . . yn(t) + . . . t + . . .
So what functions are y1(t), y2(t), . . . yn(t)?Solve those differential equations either numerically, or analytically.
Tamás Biró Mathematical Modeling (tutorial) 25
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Overview
1 Computation 6= computers
2 Mathematics and computer simulations as methodologies
3 On differential equations
4 Differential equations for dynamic systems
5 Conclusions
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Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Conclusions
And now: I am expected to provide smart conclusions!
But what if
you gave them
or postpone them to the end of the day/end of the week?
Anyway. . .
Tamás Biró Mathematical Modeling (tutorial) 27
![Page 39: Mathematical Modeling - A tutorialMathematical Modeling A tutorial ... Tamás Biró Mathematical Modeling (tutorial) 1. Computation6= computersMath. & comp. sim.Diff. equationsDiff.](https://reader030.fdocuments.us/reader030/viewer/2022040105/5e849d88f12a9a37a81e7ded/html5/thumbnails/39.jpg)
Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Conclusions
And now: I am expected to provide smart conclusions!
But what if
you gave them
or postpone them to the end of the day/end of the week?
Anyway. . .
Tamás Biró Mathematical Modeling (tutorial) 27
![Page 40: Mathematical Modeling - A tutorialMathematical Modeling A tutorial ... Tamás Biró Mathematical Modeling (tutorial) 1. Computation6= computersMath. & comp. sim.Diff. equationsDiff.](https://reader030.fdocuments.us/reader030/viewer/2022040105/5e849d88f12a9a37a81e7ded/html5/thumbnails/40.jpg)
Computation6=computers Math. & comp. sim. Diff. equations Diff. eq. dynamics Conclusions
Thank you for your attention!
Tamás Biró:[email protected]
http://birot.web.elte.hu/, http://www.birot.hu/
Tools for Optimality Theoryhttp://www.birot.hu/OTKit/
Work supported by:
Tamás Biró Mathematical Modeling (tutorial) 28