Grazing Incidence Optics for Wide-field X-ray Survey Imaging: A   Comparison of Optimization Techniques
  Utilizing a ray-tracing program, we have modeled the angular resolution of a short focal length (~2m), large field-of-view (3.1 square degrees), grazing incidence mirror shell. It has been previously shown in the literature that the application of a polynomial to the surface of grazing incidence mirror shells enhances the global performance of the mirror over the entire field-of-view. The objective of this project was to efficiently locate the optimal polynomial coefficients that would provide a 15 arcsec response over the entire field-of-view. We have investigated various techniques for identifying the optimal coefficients in a large multi-dimensional polynomial space. The techniques investigated include the downhill simplex method, fractional factorial, response surface (including Box-Behnken and central composite) designs, artificial neural networks (such as back-propagation, general regression, and group method of data handling neural networks), and the Metropolis-Coupled Markov-Chain Monte-Carlo (MC-MCMC) method. We find of the methods examined, the MC-MCMC approach performs the best. This project demonstrates that the MC-MCMC technique is a powerful tool for designing irreducible algorithms that optimize arbitrary, bounded functions and that it is an efficient way of probing a multi-dimensional space and uncovering the global minimum in a function that may have multiple minimums. 
Peter W. A. Roming, John C. Liechty, Jared R. Shoemaker, David H.   Sohn, William B. Roush, David N. Burrows, Gordon P. Garmire