Ill-Posed Rectangular System of Equations and Misfit Functions

OBJECTIVE: Learn that an overdetermined system of equations L s = t can be characterized by narrow valleys in misfit function.

PROCEDURE:

1. Define the 3x2 matrix L=[1 2; 2 4; 3 7] by typing

   
    L=[1 2; 2 4; 3 7] 
   
   
   

   Define the known RHS data vector by typing

   
   
   t= [3 ; 6 ; 9]
   
   
   

   There are non-unique solutions to Ls=t if any column vector in L is a linear combination of the other column vectors, otherwise column vectors are linearly independent. Are the columns in L linearly independent? A numerical check for a poorly conditioned system of equations is to determine condition number (max eigenvalue/min eigenvalue) by typing

   cond(L'*L)
   

   Large values of condition number indicate that columns are nearly linearly independent.

2. The least squares solution is found by solving the normal equations [L^T L] s = L ^T t , which can be done in MATLAB by typing

   
   
   s = inv(L'*L)*L'*t
   
   
   

3. Plot misfit function for a range of (s(1),s(2)) values.

   L=[1 2; 2 4; 3 7] ;t= [3 ; 6 ; 9];
   xstart=-30;xend=30;hold off;
   for i=xstart:xend;
   ii=i-xstart+1;
   for j=xstart:xend
   jj=j-xstart+1;xx=[i j];eps(ii,jj)=(L*xx'-t)'*(L*xx'-t);
   end;end
   imagesc([xstart:xend],[xstart:xend],eps');colorbar;hold on;
   x=[xstart:xend];
   for m=1:3;plot(x,(-L(m,1)*x+t(m))/L(m,2),'-w');end;
   title('Sum of Squared Error Misfit Function')
   xlabel('x');ylabel('y')
   

   Are the equations consistent or inconsistent? Why? Are the lines nearly parallel? Why should nearly parallel lines lead to an ill-posed system of equations?

4. Repeat above but now add damping to system of equations so that e=||Ls-t||+a||s-s0|| where s0 is the apriori solution you want least squares solution to be near.

5. Load in MATLAB file misfit.m and enjoy the movie where misfit function is plotted for different sets of lines. Is inconsistency related to ill-conditioning? Explain.