Homework 1 due Feb. 10 
 
  Do marked problems and any 4 other problems.  2 more are considered as extra credit.

1. Discuss the atmospheric optics problem. Show that the reconstruction problem is ill-posed and discuss the parameters that determine ill-posedness of the problem. Show (analytically or numerically) how ill-posedness is changing with changing these parameters.

2*. Discretize the integral equation by using midpoint quadrature. Calculate the data vector generated by the test function given in problem 1.14 (p.12).
    Compute and plot singular values and several singular vectors.  Show how they behave for different discretizations.

3. Problems   1.2 - 1.3  from  Vogel,  page 11 .

4*. Compute the inverse solution using SVD decomposition and using data with different levels of error.
    Compare these solutions with solutions constructed using TSVD and Tikhonov regularization.

5. Plot the regularization filter functions and discuss how regularization influences the solution.

6. Show that Tikhonov(-Phillips) representation of the regularized solution given in (1.15) is equivalent to Tikhonov solution in (1.14).

7. Problem   1.10  or  1.11  from  Vogel,  page 12 .

8. Derive the regularized solution error in (1.16) - (1.18) , page 6, using singular value decomposition of the regularization operator.

Homework 2  due Mar. 3

  Do any 6 problems.  1 more is an extra credit.

1. Discuss the discrepancy principle and use it to find the regularization parameter. Apply the method to the atmospheric optics problem.

2. Compute the regularized solution using Landweber iteration method. Show a regularizing role of the iteration count parameter.

3. Problem   1.16  from  Vogel,  page 12 .

4. Describe the steepest descent method and show how to find optimal stepsize. Compute solution of the 1D atmospheric optics problem using steepest descent method.
 
5. Calculate the gradient of the Tikhonov regularized functional for the case when K is integral operator on the unit interval and both f and g belong to the set of square-integrable functions.

6. Describe and prove the Newton's method. Use it to compute solution of the 1D atmospheric optics problem.  

7. Describe a quasi-Newton's method and use it to compute solution of the same problem.  


Homework 3  due Apr. 5

  Do any 6 problems.  3 other problems are extra credit problems.

1. Discuss the deconvolution problem using Fourier method in application to the atmospheric optics problem (1D or 2D).

2. Use Fourier transform to analytically derive the Tikhonov regularized solution for 1D problem (in continuous setting).

3. Expand the previous derivation to 2D .

4. Use Fourier transform to derive the regularized solution for H1 regularization functional.

5. Problem   5.1  from  Vogel,  page 83 .

6. Numerically calculate the regularized solution for the atmospheric optics problem (2D) .

7. Describe the adjoint method in the 1D inverse problem of finding diffusion coefficient in the  steady-state diffusion problem. Write down the algorithm of computation of the gradient of the least squares functional .

8. Calculate solution of the forward problem with some known diffusion coefficent.

9. Use the simulated in (8) solution as data for the inverse problem and compute  an estimate for the diffusion coefficient.