@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}
}
@article{GaviraghiSchindeleAnnunziatoetal.2016,
  author    = {Gaviraghi, Beatrice and Schindele, Andreas and Annunziato, Mario and Borz{\`i}, Alfio},
  title     = {On Optimal Sparse-Control Problems Governed by Jump-Diffusion Processes},
  series = {Applied Mathematics},
  volume    = {7},
  journal   = {Applied Mathematics},
  number    = {16},
  doi       = {10.4236/am.2016.716162},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-147819},
  pages     = {1978 -- 2004},
  year      = {2016},
  abstract  = {A framework for the optimal sparse-control of the probability density function of a jump-diffusion process is presented. This framework is based on the partial integro-differential Fokker-Planck (FP) equation that governs the time evolution of the probability density function of this process. In the stochastic process and, correspondingly, in the FP model the control function enters as a time-dependent coefficient. The objectives of the control are to minimize a discrete-in-time, resp. continuous-in-time, tracking functionals and its L2- and L1-costs, where the latter is considered to promote control sparsity. An efficient proximal scheme for solving these optimal control problems is considered. Results of numerical experiments are presented to validate the theoretical results and the computational effectiveness of the proposed control framework.},
  language  = {en}
}