151,372 research outputs found
Krylov subspaces associated with higher-order linear dynamical systems
A standard approach to model reduction of large-scale higher-order linear
dynamical systems is to rewrite the system as an equivalent first-order system
and then employ Krylov-subspace techniques for model reduction of first-order
systems. This paper presents some results about the structure of the
block-Krylov subspaces induced by the matrices of such equivalent first-order
formulations of higher-order systems. Two general classes of matrices, which
exhibit the key structures of the matrices of first-order formulations of
higher-order systems, are introduced. It is proved that for both classes, the
block-Krylov subspaces induced by the matrices in these classes can be viewed
as multiple copies of certain subspaces of the state space of the original
higher-order system
A phenomenological cluster-based model of Ca2+ waves and oscillations for Inositol 1,4,5-trisphosphate receptor (IP3R) channels
Clusters of IP3 receptor channels in the membranes of the endoplasmic
reticulum (ER) of many non-excitable cells release calcium ions in a
cooperative manner giving rise to dynamical patterns such as Ca2+ puffs, waves,
and oscillations that occur on multiple spatial and temporal scales. We
introduce a minimal yet descriptive reaction-diffusion model of IP3 receptors
for a saturating concentration of IP3 using a principled reduction of a
detailed Markov chain description of individual channels. A dynamical systems
analysis reveals the possibility of excitable, bistable and oscillatory
dynamics of this model that correspond to three types of observed patterns of
calcium release -- puffs, waves, and oscillations respectively. We explain the
emergence of these patterns via a bifurcation analysis of a coupled two-cluster
model, compute the phase diagram and quantify the speed of the waves and period
of oscillations in terms of system parameters. We connect the termination of
large-scale Ca2+ release events to IP3 unbinding or stochasticity.Comment: 18 pages, 10 figure
Reduced-order modeling of large-scale network systems
Large-scale network systems describe a wide class of complex dynamical
systems composed of many interacting subsystems. A large number of subsystems
and their high-dimensional dynamics often result in highly complex topology and
dynamics, which pose challenges to network management and operation. This
chapter provides an overview of reduced-order modeling techniques that are
developed recently for simplifying complex dynamical networks. In the first
part, clustering-based approaches are reviewed, which aim to reduce the network
scale, i.e., find a simplified network with a fewer number of nodes. The second
part presents structure-preserving methods based on generalized balanced
truncation, which can reduce the dynamics of each subsystem.Comment: Chapter 11 in the book Model Order Reduction: Volume 3 Application
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