Inverse Problem Crimes

§ The representation is equivalent to reconstruction algorithm fallacy

Claim: Equation (x.x) gives a representation from which it trivially follows we can reconstruct f.

Example: Compute f(t) on [0,1] from g(x) on [100,∞) where g(x) = ∫_o¹ e^-xt f(t) dt

§ The perfect data method

The algorithm used for reconstruction requires a solution(s) of the direct problem. The direct problem is solved by the identical method used to get simulated data and the implementation in each case is with identical parameters.

Example: g(x) = Kf = ∫₀¹ e^-xt f(t) dt x in [0,1].
Take g(x) = (1 - e^-x)/x so that the solution is f = 1.

We will solve the integral equation by the Nyström method with the trapezoidal rule as quadrature and using 64 points. This will lead a 64 × 64 system of the form [g] := g_vec = K[f]. Of course, all computations will be in double precision. We recognise the need to regularise and will do so by spectral cut-off taking a value of 10^-12.

To get data we use one of two methods. First, we compute the value of g(x) at the 64 equally spaced points using IEEE arithmetic to obtain g_vec and call this [g₁]. Second we will use the direct solver to get [g₂] from [g₂] = K[1].

The results are shown below, that on left is from data set [g₁], that on the right from data [g₂].

If you are looking for good reconstructions, Inverse crimes certainly pay.

§ The best of all worlds routine

If the data is generated numerically and has "random noise added", the reconstructions obtained will vary slightly. Which do you take?

The two reconstructions shown below were made from the automated routine applied to sound soft obstacle Scattering: Quasi-Newton Method using spectral cut-off.

Obtain numerically-generated data.
Add random noise to this data (at a prescribed level) and pass to the inversion routine.
Inversion routine accepts data and value of a single parameter controlling the basis set of ∂D, (in this case N the maximum number of frequencies). Regularisation is by SVD cut-off.
Output reconstruction.

The actual bean-shaped function can be reconstructed to almost within naked-eye similarity by a fourier mode approximation using only 5 Fourier modes.

Does this not seem a bit strange? With N=5 and only 1% error the reconstruction looks very poor, while that for N=6 with the larger error hardly seems believable.

The explanation is quite simple: The above algorithm was run 1,000 times in each of the two cases (N=5, 1% error and N=6, 2% error); and the worst looking picture was selected for the reconstruction shown on the left (of course no one would ever show this). Naturally, you have to know in advance what you are looking for to do this. The picture on the right for N=6 was obtained by selecting the best looking reconstruction.