A Novel Sampling Technique for Probabilistic Static Coverage Problems

Abstract

Resource allocation in the presence of constraints is an important activity in many systems engineering problems such as surveillance, infrastructure planning, environmental monitoring, and cooperative task performance. The resources in many important problems are agents such as a person, machine, unmanned aerial vehicles (UAVs), infrastructures, and software. Effective execution of a given task is highly correlated with effective allocation of resources to execute the task. An important class of resource allocation problem in the presence of limited resources is static coverage problem. In static coverage problems, it is necessary to allocate resources (stationary configuration of agents) to cover an area of interest so that an event or spatial property of the area can be detected or monitored with high probability. In this paper, we outline a novel sampling algorithm for the static coverage problem in presence of probabilistic resource intensity allocation maps (RIAMs). The key intuition behind our sampling approach is to use the finite number of samples to generate an accurate representation of RIAM. The outlined sampling technique is based on an optimization framework that approximates the RIAM with piecewise linear surfaces on the Delaunay triangles and optimizes the sample placement locations to decrease the difference between the probability distribution and Delaunay triangle surface. Numerical results demonstrate that the algorithm is robust to the initial sample point locations and has superior performance in a wide range of theoretical problems and real-life applications. In a real-life application setting, we demonstrate the efficacy of the proposed algorithm to predict the position of wind stations for monitoring wind speeds across the U.S. The algorithm is also used to give recommendations on the placement of police cars in San Francisco and weather buoys in Pacific Ocean.