TY  - JOUR
A1  - Redmann, Martin
A1  - Freitag, Melina A.
T1  - Optimization based model order reduction for stochastic systems
JF  - Applied mathematics and computation
N2  - In this paper, we bring together the worlds of model order reduction for stochastic linear systems and H-2-optimal model order reduction for deterministic systems. In particular, we supplement and complete the theory of error bounds for model order reduction of stochastic differential equations. With these error bounds, we establish a link between the output error for stochastic systems (with additive and multiplicative noise) and modified versions of the H-2-norm for both linear and bilinear deterministic systems. When deriving the respective optimality conditions for minimizing the error bounds, we see that model order reduction techniques related to iterative rational Krylov algorithms (IRKA) are very natural and effective methods for reducing the dimension of large-scale stochastic systems with additive and/or multiplicative noise. We apply modified versions of (linear and bilinear) IRKA to stochastic linear systems and show their efficiency in numerical experiments.
KW  - Model order reduction
KW  - Stochastic systems
KW  - Optimality conditions
KW  - Sylvester equations
KW  - Levy process
Y1  - 2021
U6  - https://doi.org/10.1016/j.amc.2020.125783
SN  - 0096-3003
SN  - 1873-5649
VL  - 398
PB  - Elsevier
CY  - New York
ER  - 
TY  - INPR
A1  - Murr, Rüdiger
T1  - Reciprocal classes of Markov processes : an approach with duality formulae
N2  - In this work we are concerned with the characterization of certain classes of stochastic processes via duality formulae. First, we introduce a new formulation of a characterization of processes with independent increments, which is based on an integration by parts formula satisfied by infinitely divisible random vectors. Then we focus on the study of the reciprocal classes of Markov processes. These classes contain all stochastic processes having the same bridges, and thus similar dynamics, as a reference Markov process. We start with a resume of some existing results concerning the reciprocal classes of Brownian diffusions as solutions of duality formulae. As a new contribution, we show that the duality formula satisfied by elements of the reciprocal class of a Brownian diffusion has a physical interpretation as a stochastic Newton equation of motion. In the context of pure jump processes we derive the following new results. We will analyze the reciprocal classes of Markov counting processes and characterize them as a group of stochastic processes satisfying a duality formula. This result is applied to time-reversal of counting processes. We are able to extend some of these results to pure jump processes with different jump-sizes, in particular we are able to compare the reciprocal classes of Markov pure jump processes through a functional equation between the jump-intensities.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)26 
KW  - Duality formula
KW  - reciprocal class
KW  - Levy process
KW  - infinite divisibility
KW  - counting process
KW  - Malliavin calculus
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-63018
ER  -