European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
08:30   Zuiderduinzaal: Keynote: Tuomo Rossi, University of Jyväskylä, Finland
45 mins
Systematisation of Systems Solving Physics Boundary Value Problems
Tuomo Rossi
Abstract: A substantial amount of contemporary science, research, and development exploits second order boundary value problems established on partial differential equations. This makes second order boundary value problems a significant field of science. Second order boundary value problems are solved with software systems. The proposed work formalizes and systematises the design of such software systems and the numerical schemes therein. The objective is a formalised approach which enables to build software systems that are not limited to finite lists of eligible problems, but instead, the users are enabled to concretise their problems from classes of physics field theories. Each class is specified with an action, and the action principle then yields the corresponding differential equations. The approach should also make it possible that the users extends the software with actions corresponding with new classes of boundary value problems. The work is established on a formalisation between theories and models; Physics field theories can be interpreted as models of abstract categories, and hierarchically, numerical schemes, such as finite element or finite difference kind of approaches can be understood as models of categories representing field theories. This makes it possible to systematise both the layout of boundary value problems and the construction of numerical schemes into the same framework. Furthermore, thanks to the Curry–Howard-Lambek correspondence, these processes can also be machine checked. The work is of multi-disciplinary nature. The main fields are mathematical physics, applied mathematics, and computer science. In more detail, the work is a combination of physics field theories, gauge theory, category theory, type theory, logic and computer science.