Julia and the numerical homogenization of PDEs

Clemens Heitzinger, Gerhard Tulzer

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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.

Original languageEnglish (US)
Title of host publicationProceedings of HPTCDL 2014: 1st Workshop for High Performance Technical Computing in Dynamic Languages - Held in Conjunction with SC 2014: The International Conference for High Performance Computing, Networking, Storage and Analysis
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages36-40
Number of pages5
ISBN (Print)9781479970209
DOIs
StatePublished - Mar 26 2015
Event1st Workshop for High Performance Technical Computing in Dynamic Languages, HPTCDL 2014 - Held in Conjunction with the International Conference for High Performance Computing, Networking, Storage and Analysis, SC 2014 - New Orleans, United States
Duration: Nov 17 2014 → …

Other

Other1st Workshop for High Performance Technical Computing in Dynamic Languages, HPTCDL 2014 - Held in Conjunction with the International Conference for High Performance Computing, Networking, Storage and Analysis, SC 2014
CountryUnited States
CityNew Orleans
Period11/17/14 → …

Keywords

  • high-performance computing
  • Julia
  • numerical homogenization
  • PDEs

ASJC Scopus subject areas

  • Computational Theory and Mathematics
  • Software
  • Modeling and Simulation
  • Computational Mathematics

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  • Cite this

    Heitzinger, C., & Tulzer, G. (2015). Julia and the numerical homogenization of PDEs. In Proceedings of HPTCDL 2014: 1st Workshop for High Performance Technical Computing in Dynamic Languages - Held in Conjunction with SC 2014: The International Conference for High Performance Computing, Networking, Storage and Analysis (pp. 36-40). [7069902] Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/HPTCDL.2014.8