This four-year Project Grant from the National Science Foundation's Division of Information and Intelligent Systems and Computer and Information Science and Engineering program will provide $1,199,743 to develop grid-free Monte Carlo methods for digital geometry processing problems. Carnegie Mellon University will receive funding to expand the set of partial differential equations that can be solved using scalable, reliable Monte Carlo techniques without traditional discretization. The university aims to provide these tools via free and open source software that is easily usable by non-experts to address problems in structural analysis, engineering design, robotics path planning, and other applications. Key outcomes will include algorithms for closest point queries on complex geometric representations and a domain-specific language to automatically translate partial differential equation specifications into unbiased walk on spheres estimators. The methods developed under this award are intended to eliminate meshing requirements, enable trivial parallelism, and allow evaluation of solutions at any point without solving a global system of equations.