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Mathematical/Numerical optimization

cliff avatar
cliff
4 diapositives · 185 Views · Jul 17, 2014

This study investigates the implications of including correlated observation errors in mathematical optimization and how these errors impact the minimization process. Focus is placed on the Hessian matrix's conditioning, eigenvalue spectrum, and eigenvectors related to highly accurate observations. It also explores the sensitivity of scales between B and R matrices, particularly in cases of sustained correlation, such as diurnal cycles. The balance between accuracy and computational efficiency is discussed, alongside potential methods for regularizing covariance and preconditioning techniques.

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Mathematical/Numerical optimization
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  1. Mathematical/Numerical optimization

  2. What are the effects of including correlated observation errors on the minimization? • How does it affect the hessian conditionning • It affects the eigen spectrum. We are making the observation very accurate. • It would be interesting to identify the eigenvectors corresponding to these very accurate obs. • How does the eigenvalues and vectors of the hessian change when we account for correlated obs • Sensitivity of scales in B respect to those in R • Special case when the correlation does not decrease in space/or time (e.g. diurnal cycle), how does it affect the above? • How does it affect the statistics in ensemble methods?

  3. How should we regularize R to improve the numerical behaviour of the problem? • Isn’t it dangerous to fiddle with the statistical just to improve the numerical aspects? • It is a matter of balance between accuracy and computing time • Should we be that confident to the diagnose R anyway? • It is probably not too harmful to bump-up the stddev, if they are quite small. • There is already a literature on covariance regularization (e. g. in finance), maybe we should look into it. • Should we use raw estimates or try to fit a correlation function?

  4. What preconditioning techniques should we use? • Do we need to completely rethink the whole preconditioning? • The second level preconditioning should still apply • Probably very different answers depending on the correlation structures in R