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A linear inverse problem is given by:

$\ \mathbf{d}=\mathbf{A}\mathbf{m}+\mathbf{e}$

where d: observed data, A: theory operator, m: unknown model and e: error.

To minimize the effect of the noise; a Least Square Error (LSE) model estimate is commonly used:

$\ \mathbf{\tilde{m}}=(\mathbf{A^\top A})^{-1}\mathbf{A^\top d}$

As an example problem I am considering the model:

$\ T(x,z) = 0.5*sin(2\pi*x) - z$

My observed data is the noisy partial derivatives of T:

$\ P(x,z)=T_x+e$

$\ Q(x,z)=T_z+e$

enter image description here

By introducing matrix finite difference operators I can formulate these two pdes as linear equations:

$\ \mathbf{P=D_x*T} $

$\ \mathbf{Q=D_z*T} $

Below is the LSE solution of these equations: enter image description here

There is one big problem with this solution: in the model the line where T=0 goes between z=-0.5 and z=0.5, whereas in the estimate it goes between z=-1 and z=1. Some "scaling" of the z coordinate seems to be taking place.

Why is this happening?

Other than this the solution is quite good except for the blocky pattern. This pattern seems to be related to the sampling; the finer the sampling the finer the blocky pattern.

What is the explanation for this pattern?

In order to get a smoother solution I tested with a smoothing regularization constraint: $\ a*\mathbf{L*T}=0$

where a is a scaling parameter and L is the finite difference Laplacian matrix.

I noticed a trade-off between smoothness and correctness depending on the scaling parameter. For large scaling parameters the sine wave pattern of the model turns into a square wave pattern: enter image description here

How can I determine a suitable a?

Inverse problems are often troubled by instabilities which make it necessary to introduce a dampening regularization term. This does not seem to be the case for this problem. However since A is not square I could not find its condition number using cond() in Matlab.

Is there some way in Matlab to find if an LSE problem with a nonsquare matrix is unstable?

BTW Here is the Matlab code I used:

% number of samples in x and z direction:
n = 101;

% amount of noise added to partial derivatives:
e = 0.5;

x = linspace(-1,1,n);
z = linspace(-1,1,n);
dx = x(2) - x(1);
dz = z(2) - z(1);
[X,Z] = meshgrid(x,z);

% Hidden model:
T = 0.5*sin(2*pi*X) - Z;
title('T = 0.5*sin(2*pi*x) - z');

% Observed noisy data:
P = pi*cos(2*pi*X) + e*randn(n,n); % Tx
Q = -ones(n,n)     + e*randn(n,n); % Tz

title('P = Tx + e');
title('Q = Tz + e');

% Central difference matrices:
E = sparse(2:n, 1:n-1, 1, n, n);
D = (E' - E) / 2;
I = speye(n); 
Dx = kron(D,I)/dx;
Dz = kron(I,D)/dz;

A = []; b = [];
A = [A; Dx]; b = [b; P(:)]; 
A = [A; Dz]; b = [b; Q(:)];

% Least Squares Solution:

tlse = lsqr(A, b, 1e-3, 1000*100);
Tlse = reshape(tlse, [n n]);

% Least Squares Solution regularized by smoothing Laplace operator:
D2 = E + E' + sparse(1:n, 1:n, -2, n, n);
Dxx = kron(D2,I)/(dx^2);
Dzz = kron(I,D2)/(dz^2);
L = Dxx + Dzz;
ns = 3;
si = 1;
for si = 1 : ns;
    % regularization:
    a = (si - 1)*5e-4;
    Ar = [A; a*L]; br = [b; zeros(n^2,1)];
    tlse = lsqr(Ar, br, 1e-3, 1000*100);
    Tlse = reshape(tlse, [n n]);
    str = sprintf('Tlse a=%g',a);
    si = si + 1;

EDIT: My central difference matrix D had an edge problem. As an example for size n=5: was:

D =

         0    0.5000         0         0         0
   -0.5000         0    0.5000         0         0
         0   -0.5000         0    0.5000         0
         0         0   -0.5000         0    0.5000
         0         0         0   -0.5000         0

Rewrote this to

D =

   -1.0000    1.0000         0         0         0
   -0.5000         0    0.5000         0         0
         0   -0.5000         0    0.5000         0
         0         0   -0.5000         0    0.5000
         0         0         0   -1.0000    1.0000

With more correct difference matrices I get much nicer results: enter image description here

And this is without any regularization. Sorry for the sloppiness. I learned a lot tough :-). Thanks for your answers!

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2 Answers

up vote 5 down vote accepted

Factor of 2

Can't be sure, but it looks like your central difference matrix has an extra factor of 1/2 that shouldn't be there. That would cause the estimate to be incorrectly scaled by (1/2*1/2)^-1*(1/2)=2.

Condition number

The relevant condition number is the condition number of $A^T A$, since that is the matrix that is getting inverted.

Choice of regularization parameter

Choosing the regularization parameter is a classic problem, for which there is a vast literature and many methods. A good book on the subject is "Regularization of Inverse Problems" by Engl, Hanke, and Neubauer (though it lacks recent developments). One particularly simple but effective method is the discrepancy principle.

Discrepancy principle: Choose the regularization parameter so that the least square error residual is approximately the same size as the noise level. Eg, if the expected noise level is $\delta$, start with $k=0$ and solve the inverse problem with $\alpha=1/2^k$. Then check if $||Am-d||<\delta$. If it is, stop, if it isn't, set $k=k+1$ and repeat.

What is the source of the blocky noise

For ill-posed inverse problems, the eigenfunctions of the forward operator are generally higher and higher frequency oscillatory functions, with smaller and smaller eigenvalues decaying to zero. When you try to invert, those high frequency modes get amplified.

If you have a look at your noise, you see it looks like someone added a high frequency sinusoid on top of the true function - there was a little tiny bit of noise in that high frequency sinusoidal component, and it got amplified hugely by $(A^TA)^{-1}$. There was also noise in the other components, but it didn't get amplified as much since their eigenvalues are larger (so 1/eigenvalue is smaller).

If you want a visual demonstration of what's happening, compute eig($A^TA$), and plot the last eigenfunction.

Mesh as regularization

A coarse mesh can act as regularization, if the nature of the mesh averages out the high frequency components. Meshes can actually be chosen for the purpose of regularization - doing so adaptively is actually an area of current research!

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Wow! Thanks a lot Nick. I will need some time to take all this in. –  Andy Jul 20 '12 at 13:46
It looks like some of what I wrote isn't really applicable to this problem, since correcting the difference matrix alone fixed it (I guess the problem isn't ill-posed). On the other hand, that sort of error pattern is exactly the picture you'd expect to see if you solved an ill-posed problem with too little regularization. So.. hopefully the post can be useful to you in the future..? –  Nick Alger Jul 21 '12 at 10:59
Absolutely! Thanks Nick. –  Andy Jul 23 '12 at 21:55
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I would also consider whether your approximations to the first derivatives aren't introducing any odd-even decoupling through the central differencing you use - switch the differencing out to a one-sided approximation to see if that improves your unregularised solution.

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Thanks! Yes there was a problem with my difference scheme (se my edit above). –  Andy Jul 20 '12 at 18:01
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