TY  - JOUR
A1  - Bartsch, Jan
A1  - Borzì, Alfio
A1  - Fanelli, Francesco
A1  - Roy, Souvik
T1  - A numerical investigation of Brockett’s ensemble optimal control problems
JF  - Numerische Mathematik
N2  - This paper is devoted to the numerical analysis of non-smooth ensemble optimal control problems governed by the Liouville (continuity) equation that have been originally proposed by R.W. Brockett with the purpose of determining an efficient and robust control strategy for dynamical systems. A numerical methodology for solving these problems is presented that is based on a non-smooth Lagrange optimization framework where the optimal controls are characterized as solutions to the related optimality systems. For this purpose, approximation and solution schemes are developed and analysed. Specifically, for the approximation of the Liouville model and its optimization adjoint, a combination of a Kurganov–Tadmor method, a Runge–Kutta scheme, and a Strang splitting method are discussed. The resulting optimality system is solved by a projected semi-smooth Krylov–Newton method. Results of numerical experiments are presented that successfully validate the proposed framework.
KW  - numerical analysis
KW  - Brockett
KW  - ensemble optimal control problems
Y1  - 2021
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-265352
VL  - 149
IS  - 1
ER  - 
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  -