Berg, ISC 5228, Homework

Homework ISC 5228 Fall 2006

For August 31:

Prepare a five minutes talk about your science background and interest in MCMC methods.
If you have already ideas about your final project, add another five minutes to include them too.

Read the book up to chapter 1.4.1 (p.26). You will be briefed about the content.

For September 7: (HW1)

Do assignment 4 of section 1.2.1. Reproduce the plot of Fig 1.2, but insert you name as the title. This is done by adding a line title "Your Name" in hist_eps.plt. Use epstopdf Linux command to convert the eps file you get to a pdf file and mail the figure to the instructor. Up to the name the result should look like this: hist (pdf).

Read the book up to chapter 1.6.1 (p.35).

For September 14:

Complete assignment 2 of classwork 2. Read the book up to chapter 1.7.3 (p.43).

For September 19: (HW2)

Read chapter 1.8 of the book (p.47 to p.53).
Modify the program eqdf.f (used in assignment 4 of classwork2) to generate, besides the empirical peaked CDF of Gaussian random numbers, also the peaked CDFs of y=x1+x2, y=(x1+x2)/sqrt(2) and y=(x1+x2)/2, where x1 and x2 are Gaussian random numbers. Calculate in each case the empirical, peaked CDF for 10,000 events. Plot them alltogehter and compare with the previous result from Gaussian random numbers. Put your name on the combined plot and e-mail it to the instructor.
Repeat the exercise using Cauchy random numbers.

For September 26: (HW3)

Download and run the program polls.f .
The program relies on data from an early poll of the 1992 US presidential election based on 700 people. With what probability would Clinton win the elections according to the bootstrap analysis of the program? Use the crossover of Clinton's and Bush's peaked CDFs as estimate. Are there problems with this estimate? E-mail your conclusions (in percent rounded to two digits) to the instructor.
Read chapter 2.1 (p.54 to p.64) and chapter 3.1 (p.128 to p.142).

For September 28: (HW3')

Count in the previous problem explicitly how often Clinton, or there is a draw, and estimate from that the probability that Clinton wins. Compare with the previous estimate and try to explain the small difference.

For October 5: (HW4)

Do Assignment a0301_02 of the book. Then repeat the simulations for a 4 x 4 lattice, increasing the total statistics to 1,000,000 independent data points for the small lattice. Use reweighting to calculate the energy per spin at beta=0.2 for the old as well as for the new lattice size (no error bars yet). Compare with the results from ferdinand.f. Take also beta=0.02 for the big lattice.

For October 12: (HW5)

(A) In classwork 3 you found that the mean values of data sets (1, 2, 6) and data sets (3, 4, 5) were mutually consistent with one another. Plot the CDFs of each set in one figure (i.e., two figures, each showing three CDFs) and argue by visual inspection, which data sets are fully consistent with one another.

(B) Reproduce the results of assignment a0303_06. Afterwards run on the same lattice at beta=0.2. Then perform runs at the same temperatures on 4x4 lattices. In each case report the results of Gaussian difference tests with respect to the exact values from ferdinand.f. Make also a qualitative comparison with the results from the previous homework.

(C) Read chapter 3.2 up to 3.2.1 of the book (p.142 to p.148). Continue with chapter 3.3 (p.152 to p.180 for October 17).

For October 17: (HW6)

Construct your seeds for the Marsaglia random number generator as follows: iseed1 = the numbers on an US phone corresponding to the first three letters of your given name, iseed2 = the numbers corresponding to the first three letters of you last name. Repeat with this seed the run of assignment a0303_07. E-mail your seeds, the CPU time the run took, your actm mean value and your Q of the Gaussian difference test with respect to the value of Caselle et al. [Eq. (3.107) of the book].

For October 24: (HW7)

(A-80%) Go back to random sampling and reweighting on a 4x4 lattice. Initialize the random number generator with YOUR SEEDS from the previous homework. Use the fluctuation-dissipation theorem to calculate the specific heat of the 2d Ising model at beta=0.4. Then try to use the binning method to attach error bars to the result. Rely on 32x30,000 sweeps. Usefull program and routines: pottsH0.f , potts_rw0.f , potts_actl.f , potts_act2l.f .

(B-20%) Run the Kolmogoroff test (2.151) of the book between the data sets (1,2), (1,6) and (2,6) of HW5-A.
Deadline for (B) extended to October 29.

(C) Read chapters 1.8.3 (binning) and 2.3 (student distribution). Are 32 bins sufficient to get reliable error bars? Give a quantitative answer for the two sigma level.

For October 29: (HW8)

Continue HW7-A, now calculating jackknife error bars. You are supposed to calculate the energy per link (actlm) and the specific heat with jackknife error bars. Afterwards add the bias correction and keep (for simplicity) the error bars. Perform the Gaussian difference test each of the four results in compariosn with the Ferdinand-Fisher values. Read chapter 2.7 (jackknife).

For November 2: (HW9)

Vary the parameter a of the subroutine gau_metro.f (4.34) in the range 0.5 to 20. Generate for each choice 2**21 data, record the acceptance rate and estimate the integrated autocorrelation time. Plot the integrated autocorrelation time versus the acceptance rate and e-mail the figure to the instructor. Calculate also the average stepsize of the Metropolis process. Start up files: hw09.tgz . Read chapter 4.1 p.196 to 214.

For November 7: (HW10)

Reproduce the figure on p.22 of the MCMC 3 lecture. Read chapter 4.2 p.214 to 229.

For November 14: (HW11)

Assignments a0304_06 and a0304_08 of the book. Read chapter 3.2. p.148 to 151 and chapter 3.4 p.181 to 195.

For November 21: (HW12)

Specific heat of the 2D Ising model: Perform the Gausian difference test between the multicanonical and exact results for the two multicanonical data closest to the peak of the specific heat (those data we looked at in class).

For November 30: (HW13)

Modify the multicanonical analysis program, so that you get estimates of the number of groundstates by feeding in the total number of states, Z0=q**N. Run for a values of q=2 and 7 on a 20**2 lattice (N=400). Evaluate the quality of the obtained estimates.

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