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Introduction: Big data and partial differential equations

  • YVES VAN GENNIP (a1) and CAROLA-BIBIANE SCHÖNLIEB (a2)
Abstract

Partial differential equations (PDEs) are expressions involving an unknown function in many independent variables and their partial derivatives up to a certain order. Since PDEs express continuous change, they have long been used to formulate a myriad of dynamical physical and biological phenomena: heat flow, optics, electrostatics and -dynamics, elasticity, fluid flow and many more. Many of these PDEs can be derived in a variational way, i.e. via minimization of an ‘energy’ functional. In this globalised and technologically advanced age, PDEs are also extensively used for modelling social situations (e.g. models for opinion formation, mathematical finance, crowd motion) and tasks in engineering (such as models for semiconductors, networks, and signal and image processing tasks). In particular, in recent years, there has been increasing interest from applied analysts in applying the models and techniques from variational methods and PDEs to tackle problems in data science. This issue of the European Journal of Applied Mathematics highlights some recent developments in this young and growing area. It gives a taste of endeavours in this realm in two exemplary contributions on PDEs on graphs [1, 2] and one on probabilistic domain decomposition for numerically solving large-scale PDEs [3].

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Footnotes
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CBS acknowledges support from the Leverhulme Trust project Breaking the non-convexity barrier, the EPSRC grant EP/M00483X/1, EPSRC centre EP/N014588/1, the Cantab Capital Institute for the Mathematics of Information, the CHiPS (Horizon 2020 RISE project grant), the Global Alliance project ‘Statistical and Mathematical Theory of Imaging’ and the Alan Turing Institute.

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References
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European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
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