Figure 3

Flowchart for the multiple ellipsoid-based sampling (MEBS) procedure. Given V _MC , the set of viable parameter points found by OEAMC, and an initial viable parameter point, the method finds viable parameter points near the boundary of the viable region. Then, it calculates the minimum volume enclosing ellipsoid (MVEE) that encloses those viable parameter points and samples inside an ellipsoid with the same orientation but smaller axes. In the figure, the ellipsoids inside of which sampling is carried out are represented by solid curves; dark blue and black points correspond to viable and nonviable points found in the last sampling, respectively; the MVEE ellipsoids are represented by dashed curves. After the sampling step just desribed, the method again calculates the MVEE of the viable points found so far (light blue points in the figure), and samples inside a scaled ellipsoid with the same orientation but larger axes (7). If the scaling factor tends to one, or a fixed number of iterations is reached, the initial exploration finishes. If this does not happen the method calculates the MVEE of the viable parameter points found and performs a new uniform sampling inside a new scaled ellipsoid. At the end of every new ellipsoid expansion, the algorithm checks if MEBS must stop, which occurs if the algorithm does not find any new viable points in viable nonexplored regions (grey ellipsoids). If MEBS does not stop, it carries out another ellipsoid expansion starting from a different viable parameter point. The result of the MEBS is the set of the viable parameter points found during all the ellipsoid expansions.