Differential Equations Solutions 78

Differential Equations Solutions 78 - there are only a...

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88 Chapter 14. Solutions: Case Study: Blind Deconvolution: Errors method is more robust to errors in assumptions about the model, such as underes- timating the variance or not accounting for errors in the matrix.) An alternate heuristic is to look for a value of ˜ n that makes the residual look most like white noise, but since our error is not normally distributed, this heuristic doesn’t have much meaning for our problem. An excellent way to approach this problem is to generate your own test data, for which you know the true solution, and use it to gain insight into the choice of ˜ n . Model 2: TLS. Sample solutions are shown in Figure 14.3. The discrepancy principle doesn’t give much insight for TLS, so we use more heuristic methods, giving us even less con±dence in our choice. For example, from Figure 14.4 we see that the L-curve corner occurs when 15 singular values are retained, giving a solution that looks very much like the L-curve least squares solution. Because the number of peaks is reasonable, and because
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Unformatted text preview: there are only a small number of negative values in the solution, and these have small magnitude, we might accept this solution. Now we need to extract the energies and estimated counts for the 4 types of particles. I have normalized so that the count for the lowest energy peak is 1. The Computed Estimate to Energy Levels and Counts bin centers 2.55 3.25 3.55 3.85 relative counts 1.00 1.39 1.91 0.90 A spectroscopist would actually estimate the counts by taking the integral under each of the 4 peaks, and estimate the energy by the centroid of the peak, but this is difficult since three of the peaks are not well separated. The Truth. The program used to generate the data is posted. The variance of the error is 10 − 4 . The True Energy Levels and Counts energy 2.54 3.25 3.53 3.85 relative counts 1 1.5 2 1 So, despite all of the errors, our computed solution estimates the energy levels to 2 digits and the relative counts to within 10%....
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