APSC608/Phys690: (Spring 2000)
Math and Computational Methods II

Course calendar:

There is a brief description of the content for each lecture, together with links to handouts (HW, etc). The references (in red) point to parts in the text and reference books that may be helpful in complementing your lecture notes. The names of the main references are: 
              T&G: H. Gould and J. Tobochnik
              Gi:  Text, by N.J.Giordano
              K&W: M.H. Kalos and P.A. Whitlock
              NR:  Numerical Recipes
W 1/19
  • General information.

  • Why is a course focusing on computations useful? 
    
  
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  M
  1/24
  

  • What is Monte Carlo? Example of pi. 

        -  T&G, Sec. 11.2, 11.3; K&W, p1-6; Gi, P157-169. 
    

  • Random numbers and their testing.

        -  T&G, Sec. 12.6; K&W, Appendix; Gi, P157-169.  
    

  • Intro. to the central limit theorem.

        -  T&G, Sec. 11.4, Prob 12.9; K&W, p25-28. 
    
  
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  W
  2/26
  
--
No class today --- W&M closed due to storm.

  

    

    
    
     
    
    

    
  M
  1/31
  

  • MC error analysis, the central limit theorem.

        -  T&G, Sec. 11.4, Prob 12.9; Gi, Sec. 7.3.
 
    

  • General reading assignment on probability:

        -  K&W, p2-15.

    

  • Random walks (diffusion), Monte Carlo sampling of discrete probabilities.

        -  T&G, Sec.'s 7.3, 12.1; Gi, Sec. 7.4. 
    

-- Homework Assignment 1 out.
  
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    W
  2/2
  

  • Monte Carlo sampling, concept; probability.

        -  T&G, Sec. 11.5; K&W, p39-50; Gi, P160-162.  
    

  • The accumulant and inversion technique.

        -  T&G, Sec. 11.5; K&W, p39-50.
 
    

  • Gaussian distributions --- the Box-Muller algorithm & testing of results
        -  T&G, Sec. 11.5; K&W, p39-50, p86-87.
    

    
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  M
  2/7
  

  • Particle transport simulations.

        -  T&G, Sec. 11.6; K&W, Chap. 6.
  
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    W
  2/9
  

  • Monte Carlo integration.

        -  T&G Sec. 11.2; K&W, p89-92.  

    

  • How does the computing time scale with dimensionality? Comparison of
MC and quadrature integration.

        -  K&W, Chap. 6.

    

-- Homework Assignment 2 out.
  
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  M
  2/14
  

  • Monte Carlo integration, cont'd.

        -  T&G, Sec. 11.6; K&W, Chap. 6.
    

  • Introduction to the concept of importance sampling.

        -  K&W, p92-103; T&G Sec. 11.7. 

    
  
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    W
  2/16
  

  •  Importance sampling

        -  T&G Sec. 11.2; K&W, p89-92.  

    

  •  Introduction to the Metropolis algorithm --- the pi game
again, grown-up's version.

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  M
  2/21
  

  •  The scrambling game --- detailed balance and the concept of
rejecting a move.

    

  •  The grown-up's version of the pi game, piles at edges and
corners.

    

  •  Brief review of statistical mechanics, the double-well problem.
  
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    W
  2/23
  

  •  The double-well problem by the Metropolis algorithm.

    

  •  The generalized Metropolis algorithm.

        -  T&G Sec. 11.8 
    

-- Homework Assignment 3 out.

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  M
  2/28
  

  • The generalized Metropolis algorithm, understanding T(x -> x').

        -  T&G, Chap. 16, Chap. 17; K&W, p73-86, p117-126. 
    
  
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    W
  3/1
  

  • The Metropolis algorithm and applications.

        -  T&G, Chap. 16, Chap. 17; K&W, p73-86, p117-126. 
    

  • Hard disks and spheres.

    

  •  -- Materials on Metropolis and applications:

        -  TXT, Sec's 8.3, 8.4, 8.5;
       T&G, Chap. 16, Chap. 17; K&W, p73-86, p117-126. 

    

    
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  M
  3/6
  
 Have a good Spring Break! 

  

    

    
  W
  3/8
  
  

    

    
    
     
    
    

    
  M
  3/13
  

    

  •  Classical liquids.

    

  •  Ising model and phase transitions.

    

  •  -- Materials related to past few lectures:

        -  T&G, Chap. 16, Chap. 17; Gi, Sec's 8.3, 8.4, 8.5;
       K&W, p73-86, p117-126. 

    

    
  
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    W
  3/15
  

    

  •  Ising model and phase transitions, cont'd.

    

  •  Simulated annealing.

    

  •  Travelling salesman problem.

        -  Lecture notes.  
    

-- Homework Assignment 4 out.
  
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  M
  3/20
  

  •  Simulated annealing, cont'd.

        -  General reading meterial:
       Paper by Silverman and Adler --- I'll hand out next lecture
    

    

  •  Introduction to percolation theory.

        -  T&G, Sec's 13.1 & 13.2. 

    
  
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  W
  3/22
  

  •  Forest fire simulation and relation to percolation.

        -  Reference materials are available from me.

    

  •  Fractals and their dimensions.

        -  TXT, Sec. 7.9; T&G, Sec's 14.1 and 14.2.

    

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    M
  3/27
  

  •  Introduction to the renormalization group method.

        -  T&G, Sec. 13.5.


    

  •  Diffusion in porous media.

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  W
  3/29
  

    

  •  Diffusion in porous media.

    

-- In place of the take-home mid-term, we will substitute a
course project ``proposal'', due on Wednesday 4/5.
This will constitute 10% of the course grade, with
the final project 20%.
The proposal must be complete and concise, no longer than one page
except for references, and must address:


              What is it that you plan to do?
            -- describe the problem

          Why do it?
            -- background information and significance

          How are you going to do it?
            -- method you plan to use, etc

    
  
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  M
  4/3
  

  •  Earthquakes, sandpiles, and self-orgnized criticality.

        -  T&G, Sec. 15.3.
    -  Additional reference materials available.

    

  •  Cellular automata.

        -  T&G, Sec. 15.1. 
    

  •  Game of life (an example of cellular automata simulation)

        -  The web page we visited is
       http://www.bitstorm.org/gameoflife/

    

    

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  W
  4/5
  

    

  •  Intro to finite difference -- heat conduction in a rod.

        -  Reading: T&G, Chap's 2 and 3.

    

    

  •  Finite difference -- bicycle racing and air resistance.

        -  Reading: T&G, Chap's 2 and 3;
       Gi, Sec.'s 2.2 and 2.3.

    

-- Homework Assignment 5 out.

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  M
  4/10
  

  •  Stability in finite difference methods -- advection equation.

        -  Reference materials are available from me.

    

  •  Wave propagation.

        -  T&G, Sec's 9.1, 9.3;

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  W
  4/12
  

    

  •  Finite difference --- FTCS, Lax, and Lax-Wendroff methods.

    

  •  Traffic flow.

        -  Reference materials are available from me.

    

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  M
  4/17
  

    

  •  Electric potentials -- Laplace's equation.

        -  T&G, Sec. 10.2; Gi, Sec. 5.1.
    -  Reading: T&G, Sec. Sec. 10.3 (Random walk solution ....).

    

  •  Electric potentials, with charge; convergence issues.

        -  Gi, Sec. 5.2.

    

    

-- Homework Assignment 6 out.
  
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  W
  4/19
  

    

  •  Brief intro to the finite element method.

    

  •  Introduction to molecular dynamics.

        -  T&G, Sec.'s 8.1 - 8.6; Gi, Sec. 9.1;

    

    
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  M
  4/24
  

  •  Molecular dynamics, the Verlet algorithm.

        -  T&G, Sec.'s 8.1 - 8.6; Gi, Sec. 9.1;

    

  •  The melting transition.

        -  Gi, Sec. 9.2; T&G, Chap. 8. 

    

  •  Hard disks.

        -  T&G, Sec. 8.9.

    

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  W
  4/26
  

    

  •  Molecular dynamics, cont'd.

        -  T&G, Chap. 8; Gi, Sec. 9.2.
    -  A&T, p84-98.

    

  •  Putting it all together.

        -  Lecture notes available from me.

    

  •  Outlook.

    
  
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    Tu
  5/9
  

    
-- Project presentation at 1:30pm
in Rm102 in Small (note unusual place).

        -  10 min. presentation + 2 min. for questions.

    

        -  The above time limit is STRICT.

    

        -  After talk, HAND IN HARD COPY OF TRANSPARENCY.
       You are welcome to include additional information on your project
       that you cannot cover in the talk.

    

        -  Talk should discuss:

          I.  Introduction
                what is it that you did
                what have people done related to your work
                why is it important

          II. Your work
                method
                results

          III. Summary

          ==> While part II is the meat, the other two parts are
              perhaps more difficult, particularly in a short talk
              like this. Think hard about how to do them well in a
              short time. Design your talk with one question in mind:
              what is the message that you want the audience to
              take away from the talk? Stick to that message
              and eliminate anything that does not directly
              relate to it. Do not include small details.

    


        -  Here are some hints on how to give a talk
       (by Prof. John Wilkins at OSU) that may be helpful.
       (Same as handouts.)



    

    

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