  We propose a novel method for solving optimal sensor placement problem for
high-dimensional system using an annealing machine. The sensor points are
calculated as a maximum clique problem of the graph, the edge weight of which
is determined by the proper orthogonal decomposition (POD) mode obtained from
data based on the fact that a high-dimensional system usually has a
low-dimensional representation. Since the maximum clique problem is equivalent
to the independent set problem of the complement graph, the independent set
problem is solved using Fujitsu Digital Annealer. As a demonstration of the
proposed method, the pressure distribution induced by the K\'arm\'an vortex
street behind a square cylinder is reconstructed based on the pressure data at
the calculated sensor points. The pressure distribution is measured by
pressure-sensitive paint (PSP) technique, which is an optical flow diagnose
method. The root mean square errors (RMSEs) between the pressure measured by
pressure transducer and the reconstructed pressures (calculated from the
proposed method and an existing greedy method) at the same place are compared.
As the result, the similar RMSE is achieved by the proposed method using
approximately 1/5 number of sensor points obtained by the existing method. This
method is of great importance as a novel approach for optimal sensor placement
problem and a new engineering application of an annealing machine.
