@article{RoyBorziHabbal2017,
  author    = {Roy, S. and Borz{\`i}, A. and Habbal, A.},
  title     = {Pedestrian motion modelled by Fokker-Planck Nash games},
  series = {Royal Society Open Science},
  volume    = {4},
  journal   = {Royal Society Open Science},
  number    = {9},
  doi       = {10.1098/rsos.170648},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-170395},
  pages     = {170648},
  year      = {2017},
  abstract  = {A new approach to modelling pedestrians' avoidance dynamics based on a Fokker-Planck (FP) Nash game framework is presented. In this framework, two interacting pedestrians are considered, whose motion variability is modelled through the corresponding probability density functions (PDFs) governed by FP equations. Based on these equations, a Nash differential game is formulated where the game strategies represent controls aiming at avoidance by minimizing appropriate collision cost functionals. The existence of Nash equilibria solutions is proved and characterized as a solution to an optimal control problem that is solved numerically. Results of numerical experiments are presented that successfully compare the computed Nash equilibria to the output of real experiments (conducted with humans) for four test cases.},
  language  = {en}
}
@article{GathunguBorzi2017,
  author    = {Gathungu, Duncan Kioi and Borz{\`i}, Alfio},
  title     = {Multigrid Solution of an Elliptic Fredholm Partial Integro-Differential Equation with a Hilbert-Schmidt Integral Operator},
  series = {Applied Mathematics},
  volume    = {8},
  journal   = {Applied Mathematics},
  number    = {7},
  doi       = {10.4236/am.2017.87076},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-158525},
  pages     = {967-986},
  year      = {2017},
  abstract  = {An efficient multigrid finite-differences scheme for solving elliptic Fredholm partial integro-differential equations (PIDE) is discussed. This scheme combines a second-order accurate finite difference discretization of the PIDE problem with a multigrid scheme that includes a fast multilevel integration of the Fredholm operator allowing the fast solution of the PIDE problem. Theoretical estimates of second-order accuracy and results of local Fourier analysis of convergence of the proposed multigrid scheme are presented. Results of numerical experiments validate these estimates and demonstrate optimal computational complexity of the proposed framework.},
  language  = {en}
}