Solving Optimal Experimental Design with Sequential Quadratic Programming and Chebyshev Interpolation

TL;DR

This paper introduces an efficient optimization algorithm combining sequential quadratic programming and Chebyshev interpolation to determine optimal sensor placements in Bayesian inverse problems, achieving low computational complexity and high accuracy.

Contribution

The authors develop a novel algorithm that approximates gradients and Hessians via Chebyshev interpolation, enabling scalable optimal experimental design for large discretizations.

Findings

Achieves $ ext{O}(n ext{log}^2(n))$ complexity in sensor placement optimization.

Error analysis shows the integrality gap diminishes as mesh size increases.

Successfully applied to a 2D advection-diffusion problem for LIDAR sensor optimization.

Abstract

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 $\mathcal{O}(n\log^2(n))$. An error analysis guarantees the integrality gap shrinks to zero as $n\to\infty$, and we apply the algorithm on a two-dimensional advection-diffusion equation, to determine the LIDAR's optimal sensing directions for data collection.