We propose an optimization algorithm to compute the optimal sensor locations in experimental design in the formulation of Bayesian inverse problems, where the parameter-to-observable mapping is described through an integral equation and its discretization results in a continuously indexed matrix whose size depends on the mesh size n. By approximating the gradient and Hessian of the objective design criterion from Chebyshev interpolation, we solve a sequence of quadratic programs and achieve the complexity $mathcal{O}(nlog^2(n))$. An error analysis guarantees the integrality gap shrinks to zero as $ntoinfty$, and we apply the algorithm on a two-dimensional advection-diffusion equation, to determine the LIDARs optimal sensing directions for data collection.
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