@article{MenzLatorreSchuetteetal.2012,
  author    = {Menz, Stephan and Latorre, Juan C. and Sch{\"u}tte, Christof and Huisinga, Wilhelm},
  title     = {Hybrid stochastic-deterministic solution of the chemical master equation},
  series = {Multiscale modeling \& simulation : a SIAM interdisciplinary journal},
  volume    = {10},
  journal   = {Multiscale modeling \& simulation : a SIAM interdisciplinary journal},
  number    = {4},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {1540-3459},
  doi       = {10.1137/110825716},
  pages     = {1232 -- 1262},
  year      = {2012},
  abstract  = {The chemical master equation (CME) is the fundamental evolution equation of the stochastic description of biochemical reaction kinetics. In most applications it is impossible to solve the CME directly due to its high dimensionality. Instead, indirect approaches based on realizations of the underlying Markov jump process are used, such as the stochastic simulation algorithm (SSA). In the SSA, however, every reaction event has to be resolved explicitly such that it becomes numerically inefficient when the system's dynamics include fast reaction processes or species with high population levels. In many hybrid approaches, such fast reactions are approximated as continuous processes or replaced by quasi-stationary distributions in either a stochastic or a deterministic context. Current hybrid approaches, however, almost exclusively rely on the computation of ensembles of stochastic realizations. We present a novel hybrid stochastic-deterministic approach to solve the CME directly. Our starting point is a partitioning of the molecular species into discrete and continuous species that induces a partitioning of the reactions into discrete-stochastic and continuous-deterministic processes. The approach is based on a WKB (Wentzel-Kramers-Brillouin) ansatz for the conditional probability distribution function (PDF) of the continuous species (given a discrete state) in combination with Laplace's method of integral approximation. The resulting hybrid stochastic-deterministic evolution equations comprise a CME with averaged propensities for the PDF of the discrete species that is coupled to an evolution equation of the related expected levels of the continuous species for each discrete state. In contrast to indirect hybrid methods, the impact of the evolution of discrete species on the dynamics of the continuous species has to be taken into account explicitly. The proposed approach is efficient whenever the number of discrete molecular species is small. We illustrate the performance of the new hybrid stochastic-deterministic approach in an application to model systems of biological interest.},
  language  = {en}
}
@article{ZaourarHamoudiMandeaetal.2013,
  author    = {Zaourar, N. and Hamoudi, M. and Mandea, M. and Balasis, G. and Holschneider, Matthias},
  title     = {Wavelet-based multiscale analysis of geomagnetic disturbance},
  series = {EARTH PLANETS AND SPACE},
  volume    = {65},
  journal   = {EARTH PLANETS AND SPACE},
  number    = {12},
  publisher = {TERRA SCIENTIFIC PUBL CO},
  address   = {TOKYO},
  issn      = {1343-8832},
  doi       = {10.5047/eps.2013.05.001},
  pages     = {1525 -- 1540},
  year      = {2013},
  abstract  = {The dynamics of external contributions to the geomagnetic field is investigated by applying time-frequency methods to magnetic observatory data. Fractal models and multiscale analysis enable obtaining maximum quantitative information related to the short-term dynamics of the geomagnetic field activity. The stochastic properties of the horizontal component of the transient external field are determined by searching for scaling laws in the power spectra. The spectrum fits a power law with a scaling exponent beta, a typical characteristic of self-affine time-series. Local variations in the power-law exponent are investigated by applying wavelet analysis to the same time-series. These analyses highlight the self-affine properties of geomagnetic perturbations and their persistence. Moreover, they show that the main phases of sudden storm disturbances are uniquely characterized by a scaling exponent varying between 1 and 3, possibly related to the energy contained in the external field. These new findings suggest the existence of a long-range dependence, the scaling exponent being an efficient indicator of geomagnetic activity and singularity detection. These results show that by using magnetogram regularity to reflect the magnetosphere activity, a theoretical analysis of the external geomagnetic field based on local power-law exponents is possible.},
  language  = {en}
}
@misc{ZaourarHamoudiMandeaetal.2013,
  author    = {Zaourar, Naima and Hamoudi, Mohamed and Mandea, Mioara and Balasis, Georgios and Holschneider, Matthias},
  title     = {Wavelet-based multiscale analysis of geomagnetic disturbance},
  series = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  volume    = {65},
  journal   = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  number    = {12},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-43691},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-436912},
  pages     = {1525 -- 1540},
  year      = {2013},
  abstract  = {The dynamics of external contributions to the geomagnetic field is investigated by applying time-frequency methods to magnetic observatory data. Fractal models and multiscale analysis enable obtaining maximum quantitative information related to the short-term dynamics of the geomagnetic field activity. The stochastic properties of the horizontal component of the transient external field are determined by searching for scaling laws in the power spectra. The spectrum fits a power law with a scaling exponent β, a typical characteristic of self-affine time-series. Local variations in the power-law exponent are investigated by applying wavelet analysis to the same time-series. These analyses highlight the self-affine properties of geomagnetic perturbations and their persistence. Moreover, they show that the main phases of sudden storm disturbances are uniquely characterized by a scaling exponent varying between 1 and 3, possibly related to the energy contained in the external field. These new findings suggest the existence of a long-range dependence, the scaling exponent being an efficient indicator of geomagnetic activity and singularity detection. These results show that by using magnetogram regularity to reflect the magnetosphere activity, a theoretical analysis of the external geomagnetic field based on local power-law exponents is possible.},
  language  = {en}
}