Fast Multipole Method for Two-Dimensional Pressure Reconstruction from Noisy Gradient Information*

Accurately obtaining fields from noisy gradient information is crucial in many computational physics applications. Examples include deriving velocity potentials from sparse velocity measurements or reconstructing pressure fields from particle image velocimetry (PIV) data in high Reynolds number scenarios. The computational burden often arises from the need to average different integration paths and solve for unknown boundary conditions, which can be formulated as a boundary integral equation using the Green's function of the Laplace operator. In this paper, we apply the Fast Multipole Method (FMM) to reduce the computational complexity comparable to the number of measurements for two-dimensional problems. We present the formulation of the integral equation and describe the process of solving the boundary pressure using the Boundary Element Method (BEM) in conjunction with the FMM. We demonstrate the effectiveness of our approach using a test case from the Johns Hopkins turbulent database, reconstructing pressure fields from material acceleration data. Our results are compared with those obtained using existing methods, highlighting the advantages of our proposed approach in terms of accuracy and computational efficiency.