# Surrogate-Based Robust Design for a Non-Smooth Radiation Source Detection Problem
## 1 Introduction  
The problem of determining the location and intensity of a radiation source arises in several settings including emergency response to mitigate nuclear threats, structural and nuclear health monitoring in nuclear reactors, and environmental cleanup of biomedical and industrial nuclear waste. In this paper, we consider the development of a robust sensor network design for determining the location and intensity of a radiation source in a simulated urban environment. Specifically, we consider source localization in a simulated 250 m \(\times\) 180 m block in downtown Washington, DC. There are several difficulties that are intrinsic to this source localization problem. The first is that inverse problems of this nature are inherently ill-posed and require some form of regularization to obtain reasonable approximate solutions [1]. This difficulty is exacerbated by the fact that sensor observations are often coarsely spaced, which dictates that one cannot estimate source attributes that are more oscillatory than the grid spacing. As detailed in [2], this can yield erroneous results if ignored. The computational complexity of deterministic [3] and stochastic, Monte Carlo [4] radiation transport models poses a second challenge since it limits the number of model realizations that can be obtained for optimization, or Bayesian or frequentist inference. This has led to the development of alternative parameterizations or surrogate models. For example, in [5] the authors modeled the radiation source as a point gamma source and employed a physics-based parameterization of gamma particle transport. A fast radiation transport model is also available as a component ofSynth, a gamma-ray simulation code written by Pacific Northwest National Laboratory [6; 7]. In [8], the authors employ a Gaussian mixture to model the radiation field. In [9], we addressed challenges associated with optimization and Bayesian inference as a prelude to the robust sensor design problem considered in this paper. Specifically, we implemented a fast piecewise-continuously differentiable radiation transport model and solved the associated inverse problem using combined global [10; 11] and local [12] optimization algorithms, and Bayesian inference techniques [13; 14]. As in [9], we assume here that the threat is a point source and that the model accounts only for photons that travel directly from source to detector, with no intervening collisions. The radiation source is parameterized with three components: its 2-D location coordinates and intensity. To improve computational efficiency and permit gradient-based optimization and the implementation of a robust design algorithm [15], we implement and verify a continuously differentiable surrogate model based on radial basis functions to approximate the response for all possible detector locations. We employ this surrogate for subsequent optimization and robust design. There are three different main strategies for taking measurements for general applications. The first searches for a given number of stationary sensors, the second one relies on moving sensors, whereas the third method, entitled scanning, activates only a subset of a given number of stationary sensors at a given moment in time. The existing methods for identification of sensor locations usually employ random fields analysis [16], information theory [17] and optimum experimental design theory [15]. Moreover, sensor placement algorithms can be classified into discrete and continuous depending on the nature of the search space. In the context of nuclear source identification, [PERSON] [18] used the Gaussian process optimization [19] to solve a continuous detector placement problem and [PERSON] [20] applied Shannon entropy [21; 22; 23] to guide mobile sensors over a discrete grid of possible measurement sites. To form a basis for comparing different networks, a quantitative measure of efficiency is required.