In this work, we exploit the ill-posedness of linear inverseproblems to design algoithms to release differentially private data ormeasurements of the physical system. We discuss the spectralrequirements on a matrix such that only a small amount of noise isneeded to achieve privacy and contrast this with the poor conditioningof the system. We then instantiate our framework with severaldiffusion operators and explore recovery via l1 constrainedminimisation. Our work indicates that it is possible to producelocally private sensor measurements that both keep the exact locationsof initial heat sources private and permit recovery of the “generalgeographic vicinity” of the sources.Joint work with Audra McMillan