CAAM 454/554 Project:
Output Least Squares for an Inverse Problem

• Please see the PDF file proj_invp.pdf for a description of the project.

• Download the instructor's codes zsolver.p, zinvprJ.p, zgetAb.p and the test code testinvp.m. You can run the test code without your codes.

• Implement two methods: the gradient and the Gauss-Newton, in a Matlab function with the interface specified below:

  function [x,iter,nf,fh,gh] = mysolver(func,x0,tol,maxit,method,varargin)
  %
  % Solver for least squares problems min f(x) = 0.5*r(x)'*r(x)
  %
  % Output:
  %   x -- solution
  %   iter -- number of iterations taken
  %   nf -- number of func evaluations
  %   fh -- function values at all iterations
  %   gh -- gradient norms at all iterations
  % Input:
  %   func -- function (handle) that returns [r, J] for given x
  %   x0 -- initial guess for x
  %   tol -- tolerance for gradient norm
  %   maxit -- maximum allowable iteration number
  %   method -- 1 for gradient, 2 for Gauss-Newton
  

  Also implement a Matlab function

  function [r,J] = myinvprJ(s,n,bvsE,cE,B)
  %   Output:
  %     r - residual at s
  %     J - Jacobian at s
  %   Input:
  %     s - parameters (3 x 1)
  %     n - number of interior nodes
  %     bvsE - boundary values used in experiments (2 x E) 
  %     cE - observed data in experiments (|B| x E)
  %     B - indices for observation locations 
  

  to evaluate r and its Jacobian J for the least squares problem where the unknowns are s(j) for j = 1,2,3 -- the coefficients of the polynomial function of x. Your solver will call this function to solve the inverse problem.

• Run the test function for 3 different noise levels (in percentage) and choose the number of experiments; specifically,

          noise level (%) = 0, 0.5, 1, (one at a time)
    	No. of experiments  = (your selection) 
  

  The purpose is to achieve a high solution accuracy at each noise level. In general, the more experiments the higher accuracy (though not always). Of course you also want to keep computing time reasonable. Compare your codes with the instructor's in terms of accuracy and effeciency.

• Write a typed Project Report on your formula derivations, implementation, results and observations. Try to provide insights into the properties of the underlying problem and the algorithms (observations without insights are less meaningful). The report should be "pledged" and no more than 4 pages.

• Attach copies of your codes and printouts from the 3 test runs (long iterations can be truncated), including graphics, to your report.

• Your project will be graded based on (i) the soundness of your approach and the correctness of your codes, (ii) the quality and efficiency of your codes, and (iii) the quality of your project report.

Rules

• Questions to the instructor are encouraged, but otherwise the project must be done independently. No late submissions are allowable unless a special permission is obtained.