Julia and the Numerical Homogenization of PDEs

Abstract

We discuss the advantages of using Julia for solving multiscale problems involving partial differential equations (PDEs). Multiscale problems are problems where the coefficients of a PDE oscillate rapidly on a microscopic length scale, but solutions are sought on a much larger, macroscopic domain. Solving multiscale problems requires both a theoretic result, i.e., a homogenization result yielding effective coefficients, as well as numerical solutions of the PDE at the microscopic and the macroscopic length scales. Numerical homogenization of PDEs with stochastic coefficients is especially computationally expensive. Under certain assumptions, effective coefficients can be found, but their calculation involves subtle numerical problems. The computational cost is huge due to the generally large number of stochastic dimensions. Multiscale problems arise in many applications, e.g., in uncertainty quantification, in the rational design of nanoscale sensors, and in the rational design of materials. Our code for the numerical stochastic homogenization of elliptic problems is implemented in Julia. Since multiscale problems pose new numerical problems, it is in any case necessary to develop new numerical codes. Julia is a dynamic language inspired by the Lisp family of languages, it is open-source, and it provides native-code compilation, access to highly optimized linear-algebra routines, support for parallel computing, and a powerful macro system. We describe our experience in using Julia and discuss the advantages of Julia's features in this problem domain.