Optimal Sensor Placement for Linear Inverse Problems with Nonnegativity Constraints

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To optimally deploy sensors for atmospheric inverse modeling based on Gaussian plume model, closed-form designs (e.g., A or D-optimal) do not exist due to the nonnegativity constraint of emission rates. A bi-level optimization framework is proposed with a stochastic outer objective (i.e., estimation loss) and a constrained inverse model with regularization at the inner level. We solve this bi-level problem by implicit gradients considering the inner KKT system. Finally, two first-order iterative algorithms are investigated and compared using two numerical examples. The scalability of the SGD-based approach is demonstrated.