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(a) Four configurations of vortices that would generate identical measurements for the set of pressure sensors (brown squares). (b) Two distinct vortex states that differ only in the vortex labeling but generate identical flow fields.
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We study the instantaneous inference of an unbounded planar flow from sparse noisy pressure measurements. The true flow field comprises configurations of one or more regularized point vortices of various strength and size. We interpret the true flow's measurements with a vortex estimator, also consisting of regularized point vortices, and attempt t...
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... there are many situations in which multiple solutions arise due to symmetries in the vortex-sensor arrangement. This is easy to see from a simple thought experiment, depicted in Figure 1(a): Suppose that we wish to estimate a single vortex from pressure sensors arranged in a straight line. A vortex on either side of this line of sensors will induce the same pressure on the sensors, and a vortex of either sign of strength will, as well. ...
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... symmetry arises when there is more than one vortex to estimate, as in Figure 1(b), because in such a case, there is no unique ordering of the vortices in the state vector. With each of the vortices assigned a fixed set of parameters, any of the í µí±! permutations of the ordering leads to the same pressure measurements. ...
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... this basic configuration, the vortices are widely separated so that the estimator's challenge is similar to that of two isolated single vortices, each estimated with four sensors. However, unique challenges arise as the true vortices become closer, as Figure 10 shows. Here, we keep the strength and vertical position of each true vortex the same as in the basic case, but vary both vortices' horizontal position-the left one is moved rightward and the right one is moved leftward-in such a manner that their average is invariant, (í µí±¥ 1 + í µí±¥ 2 )/2 = −0.125. ...
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... different numbers of sensors are used, í µí± = 6, 7, 8, all uniformly distributed between [−1, 1]. In Figure 10(a), it is clear that using six sensors, though ostensibly sufficient to estimate the six states, is actually insufficient in a few isolated cases in which the maximum uncertainty becomes infinite. These cases are examples of rank deficiency in the vortex estimator. ...
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... this rank deficiency disappears when more than six sensors are used. An example of the estimator's behavior in one of these rank-deficient configurations is depicted in Figure 10(b,c). When six sensors are used (panel (b)), the MCMC samples are distributed more widely, along a manifold in the vicinity of the true state, with the eigenvector of the most-uncertain eigenvalue tangent to this manifold. ...
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... we can avoid rank deficiency by using more sensors than states. As a demonstration, we show in the left panels of Figure 11 the expected vorticity field that results from estimating four different true vortex configurations with eight sensors. In each case, the locations of the vortices are accurately estimated with relatively little uncertainty, even as the vortices become closer to each other than they are to the array of sensors. ...
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... each case, the locations of the vortices are accurately estimated with relatively little uncertainty, even as the vortices become closer to each other than they are to the array of sensors. However, with closer vortices there is considerable uncertainty in estimating the strengths of the individual vortices, as exhibited in the right panels of Figure 11, each corresponding to the vortex configuration on the left. As the true vortices become even closer than in the examples in Figure 11, multiple solutions emerge. ...
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... with closer vortices there is considerable uncertainty in estimating the strengths of the individual vortices, as exhibited in the right panels of Figure 11, each corresponding to the vortex configuration on the left. As the true vortices become even closer than in the examples in Figure 11, multiple solutions emerge. This is illustrated in Figure 12, depicting the extreme case of one vortex just above the other. ...
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... the true vortices become even closer than in the examples in Figure 11, multiple solutions emerge. This is illustrated in Figure 12, depicting the extreme case of one vortex just above the other. The MCMC identifies three modes of the posterior, each representing a different candidate solution for the estimator. ...
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... MCMC identifies three modes of the posterior, each representing a different candidate solution for the estimator. One mode consists of vortices of opposite sign to either side of the true set, shown in the top row of Figure 12. The second mode, in the middle row, comprises vortices very near the true set, though the strengths of the vortices are quite uncertain, as evidenced by the long ridge of samples in the strength plot in Figure 12(d). ...
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... mode consists of vortices of opposite sign to either side of the true set, shown in the top row of Figure 12. The second mode, in the middle row, comprises vortices very near the true set, though the strengths of the vortices are quite uncertain, as evidenced by the long ridge of samples in the strength plot in Figure 12(d). Finally, the bottom row shows a mode that has positive vortices further apart than in the other two modes. ...
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... mode with a significantly larger maximum (i.e., is significantly closer to zero, since (2.12) is negative semidefinite) is a superior candidate solution. For the two modes shown in Figure 12, the maximum log-posteriors are −0.20, −0.11, and −14.83, respectively, suggesting that the mode in the middle row is mildly superior to that of the top row and clearly superior to that of the bottom row. ...
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... Figure 13 we carry out the same procedure of bringing two vortices closer together as in Figure 11, but now we do so for one vortex of positive strength (1.2) and another of negative strength (−1.0). We get similar results as before, successfully estimating the vortex locations and strengths. ...
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... Figure 13 we carry out the same procedure of bringing two vortices closer together as in Figure 11, but now we do so for one vortex of positive strength (1.2) and another of negative strength (−1.0). We get similar results as before, successfully estimating the vortex locations and strengths. ...
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... no spurious solutions arise as the vortices become very close together, as they did in the previous example. In fact, when the two opposite-sign vortices are vertically aligned, as in Figure 14, the estimator has no difficulty in identifying the individual vortices and their strengths. The figure ostensibly depicts two modes identified by the estimator, but in fact these modes are identical aside from the sign of their strengths. ...
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... 1.10), x 2 = (0.25, 0.51, −1.25), x 3 = (0.68, 0.75, 1.36). This is shown in Figure 15. Both the expected vorticity field and the pressure field are captured very well by the estimator. ...
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... a counterexample, when two of the three vortices form a compact pair that is well-separated from the third, the estimator tends to be less able to prefer one choice of sign for the vortex pair over the other. An example is shown in Figure 16, in which the rightmost pair of vortices has opposite sign in each mode. The corresponding pressure fields shown on the right are nearly identical because the coupling of the pair with the leftmost vortex Right panels depict contours of the estimated pressure field for that mode. ...
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... it is useful to restrict the estimator to search a lower-dimensional space, and the easiest way to achieve this is by using fewer vortices in the estimator. In Figure 17, we illustrate the behavior a two-vortex estimator on the three-vortex configuration in Figure 15, in variations in which the signs of the right two true vortices are changed. The range of strengths in the prior is expanded in this problem to (−4, 4). ...
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... it is useful to restrict the estimator to search a lower-dimensional space, and the easiest way to achieve this is by using fewer vortices in the estimator. In Figure 17, we illustrate the behavior a two-vortex estimator on the three-vortex configuration in Figure 15, in variations in which the signs of the right two true vortices are changed. The range of strengths in the prior is expanded in this problem to (−4, 4). ...
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... third case is the most interesting. Here, the true vortex configuration consists of positive, negative, and positive vortices from left to right, so there is no pairing of like-sign vortices as in the previous two cases. The estimator identifies a solution consisting of x 1 = (0.40, 1.03, 3.50 and x 2 = (0.90, 0.97, −1.2). Neither of these vortices bears an obvious connection with one of the true vortices, so no aggregation is possible. The estimator has done the best in can in the lower-dimensional space available to it, aliasing the true flow onto a dissimilar flow state. The maximum log-posterior is −88.6, significantly ...
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... µí±(í µí²) Figure 18. Triadic interaction between two vorticity-laden elements and the pressure at some point. ...
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... form reveals an essentially triadic relationship between vorticity and pressure, illustrated in Figure 18: the pressure at í µí² comprises a double sum of elementary interactions between vorticity at í µí² ′ and í µí² ′′ . Interestingly, a consequence of this relationship is that the pressure is invariant to a change of sign of the entire vorticity field. ...
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... a consequence of this relationship is that the pressure is invariant to a change of sign of the entire vorticity field. Figure 19. Vortex interaction kernel for a pair of unit-strength point vortices at í µí² 1 = (−1/2, 0) and í µí² 2 = (1/2, 0). ...
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... kernel, Π(í µí² − í µí² J , í µí² − í µí² K ), is dependent only on the relative positions of the observation point í µí² from each of the two vortex positions, í µí² J and í µí² K . It is symmetric with respect to the members of the pair, J and K, as is apparent from Figure 19, which shows the kernel and its two additive parts. ...
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... vortex interaction kernel is centered midway between the pair at í µí² JK = (í µí² K + í µí² J )/2 and has directivity as indicated in the left panel of Figure 19. It is apparent that the interaction kernel has much less influence along the pair's axis (the í µí¼ direction); its primary influence is perpendicular to this line, in the í µí± direction. ...
