Physics 6730 Final Exam

Rules

• The test is open book and notes. You may use the internet to consult references. However, you may not seek or receive assistance from anyone other than the instructor or TA.
  
  
• Answer all questions (5 pts each) in a single file answers.txt. Please submit them just as you have been submitting homework. Use

      submit p6730 final answers.txt
  

  Please be sure you submit your final version. Replacing the file is permitted until the end of the test.
  
  
• For questions requiring an algebraic answer, please use either LaTeX or Maple notation. For questions requiring an explanation, please answer in one or two sentences.

Questions

1. A data set has 51 (x,y,sigma) values. A least chisquare fit to the data is used to determine the best values of four model parameters. The minimum chisquare value is 53.5. What is the confidence level of the fit?
   
   
2. What does the Levenberg-Marquardt scheme do if, in attempting to update the fit parameters, it finds that the value of chisquare has increased?
   
   
3. What is the main advantage of the simulated annealing optimization method?
   
   
4. You would like to solve the equation A x = b for x where A is an n by n matrix, x is an unknown n-component vector, and b is an n-component vector.

   You do a singular value decomposition of A and find that one of the singular values is zero.

   Does the equation Au = 0 have a nontrivial solution for an n-component vector u?
   
   

5. Continuing with the above question, does the equation A x = b always have at least one solution?
   
   
6. Find the mean, sample (population) standard deviation, and the standard deviation of the mean of the following values:

       4.6
       3.5
       4.7
       6.6
       3.2
   

   
   
   
7. Continuing with the previous question, find the single elimination jackknife averages of the above values.
   
   
8. When we attempted to reverse the smoothing of an image by starting from the smoothed ppm file, we found that noise was troublesome. What was the largest source of noise?
   
   
9. What partial differential equation does our Monte Carlo random walk simulate in the limit of very small coordinate spacing and time step? Give the name and write the one-dimensional version of the equation using Maple notation as in diff (f(x,y), x) to denote the partial derivative of f(x,y) with respect to x.
   
   
10. The Monte Carlo simulation of a one-dimensional walk solves a finite difference equation exactly in the limit of an infinite number of walks. Write the equation. Use the notation w_t,x to specify the solution at discrete time t and discrete spatial point x on a space-time grid with grid spacing equal to 1. Let p be the probability of taking a step left or right in one time inteval.
    
    
11. How does the error in a Monte Carlo integration depend on the number of sample points N for a sufficiently large sample?
    
    
12. Suppose you are solving Laplace's equation in two dimensions using the basic three-point formula for the second derivative. When you apply the Gauss-Seidel SOR method to the approximate solution w_x,y at spatial coordinate x,y you first calculate the average of the approximate solution at the four nearest neighbor points. Call this average avg = w_x,y,avg. Call the old approximate solution old = w_x,y,old. Give the formula for the new value new = w_x,y,new if the relaxation parameter is omega. (Write your answer in terms of new, old, avg, and omega.)
    
    
13. We used cfourier -1 to do the spatial Fourier transform of Schroedinger wave packets in order to get the correct quantum mechanics convention for momentum. This code gives results with wavenumbers scaled to the interval [0,1]. Suppose a wavepacket gives a single strong peak at approximately 0.9. If we make a movie of the evolution of the wavepacket, what direction does the wavepacket go (toward positive x or negative x)? If the momentum is h (Planck's constant) times this scaled wavenumber, what is the magnitude of the momentum? (Express your answer in terms of h.)
    
    
14. Which of the following numerical methods lead to solving a large, sparse linear system? (In your answer, list the numbers.)
    1. Levenberg-Marquardt optimization method
    2. Simulated annealing optimization method
    3. Monte Carlo integration
    4. Finite difference method for PDEs
    5. Crank-Nicholson integration method
    6. Finite element method for PDEs
    7. Monte Carlo method for Ising model.
    
    
    
15. Write an awk command to print the first two numbers of each line of the file spectrum, skipping the 17th line.
    
    
16. Name two reasons for preferring the Crank-Nicholson method for integrating the time-dependent Schroedinger equation.
    
    
17. The conjugate gradient method generates a sequence of search vectors p_k for solving a linear system A x = b with positive definite A. In what sense are two such vectors in this sequence conjugate to each other?
    
    
18. Each node in a parallel process computes its single precision floating point value called my_x. You would like to sum these values across all of the machine and arrange so each node gets the same sum as a single precision floating point value called sum_x. Write an MPI function call that does this sum.
    
    
19. Write the declaration for a double precision complex vector called psi consisting of 50 components. Use the Numerical Recipes class NRVec for the vector and the standard library complex class for the components. Precede your definition with #include statements for the required header files.
    
    
20. You have a file called ising2d.cc on the machine icebox and want to copy it to your account on a physics machine in your subdirectory a11. Suppose you are logged in to icebox. Write the Unix command that does the copy. Assume you have the same username on both machines.