TY  - JOUR
A1  - Gathungu, Duncan Kioi
A1  - Borzì, Alfio
T1  - Multigrid Solution of an Elliptic Fredholm Partial Integro-Differential Equation with a Hilbert-Schmidt Integral Operator
JF  - Applied Mathematics
N2  - 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.
KW  - elliptic problems
KW  - finite differences
KW  - fredholm operator
KW  - multigrid schemes
KW  - numerical analysis
Y1  - 2017
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-158525
VL  - 8
IS  - 7
ER  - 
TY  - JOUR
A1  - Gaviraghi, Beatrice
A1  - Schindele, Andreas
A1  - Annunziato, Mario
A1  - Borzì, Alfio
T1  - On Optimal Sparse-Control Problems Governed by Jump-Diffusion Processes
JF  - Applied Mathematics
N2  - 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.
KW  - jump-diffusion processes
KW  - partial integro-differential Fokker-Planck Equation
KW  - optimal control theory
KW  - nonsmooth optimization
KW  - proximal methods
Y1  - 2016
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-147819
VL  - 7
IS  - 16
SP  - 1978
EP  - 2004
ER  -