8,411 research outputs found
Global integration of the Schr\"odinger equation: a short iterative scheme within the wave operator formalism using discrete Fourier transforms
A global solution of the Schr\"odinger equation for explicitly time-dependent
Hamiltonians is derived by integrating the non-linear differential equation
associated with the time-dependent wave operator. A fast iterative solution
method is proposed in which, however, numerous integrals over time have to be
evaluated. This internal work is done using a numerical integrator based on
Fast Fourier Transforms (FFT). The case of a transition between two potential
wells of a model molecule driven by intense laser pulses is used as an
illustrative example. This application reveals some interesting features of the
integration technique. Each iteration provides a global approximate solution on
grid points regularly distributed over the full time propagation interval.
Inside the convergence radius, the complete integration is competitive with
standard algorithms, especially when high accuracy is required.Comment: 25 pages, 14 figure
On Approximate Nonlinear Gaussian Message Passing On Factor Graphs
Factor graphs have recently gained increasing attention as a unified
framework for representing and constructing algorithms for signal processing,
estimation, and control. One capability that does not seem to be well explored
within the factor graph tool kit is the ability to handle deterministic
nonlinear transformations, such as those occurring in nonlinear filtering and
smoothing problems, using tabulated message passing rules. In this
contribution, we provide general forward (filtering) and backward (smoothing)
approximate Gaussian message passing rules for deterministic nonlinear
transformation nodes in arbitrary factor graphs fulfilling a Markov property,
based on numerical quadrature procedures for the forward pass and a
Rauch-Tung-Striebel-type approximation of the backward pass. These message
passing rules can be employed for deriving many algorithms for solving
nonlinear problems using factor graphs, as is illustrated by the proposition of
a nonlinear modified Bryson-Frazier (MBF) smoother based on the presented
message passing rules
Adapting Predictive Feedback Chaos Control for Optimal Convergence Speed
Stabilizing unstable periodic orbits in a chaotic invariant set not only
reveals information about its structure but also leads to various interesting
applications. For the successful application of a chaos control scheme,
convergence speed is of crucial importance. Here we present a predictive
feedback chaos control method that adapts a control parameter online to yield
optimal asymptotic convergence speed. We study the adaptive control map both
analytically and numerically and prove that it converges at least linearly to a
value determined by the spectral radius of the control map at the periodic
orbit to be stabilized. The method is easy to implement algorithmically and may
find applications for adaptive online control of biological and engineering
systems.Comment: 21 pages, 6 figure
How single neuron properties shape chaotic dynamics and signal transmission in random neural networks
While most models of randomly connected networks assume nodes with simple
dynamics, nodes in realistic highly connected networks, such as neurons in the
brain, exhibit intrinsic dynamics over multiple timescales. We analyze how the
dynamical properties of nodes (such as single neurons) and recurrent
connections interact to shape the effective dynamics in large randomly
connected networks. A novel dynamical mean-field theory for strongly connected
networks of multi-dimensional rate units shows that the power spectrum of the
network activity in the chaotic phase emerges from a nonlinear sharpening of
the frequency response function of single units. For the case of
two-dimensional rate units with strong adaptation, we find that the network
exhibits a state of "resonant chaos", characterized by robust, narrow-band
stochastic oscillations. The coherence of stochastic oscillations is maximal at
the onset of chaos and their correlation time scales with the adaptation
timescale of single units. Surprisingly, the resonance frequency can be
predicted from the properties of isolated units, even in the presence of
heterogeneity in the adaptation parameters. In the presence of these
internally-generated chaotic fluctuations, the transmission of weak,
low-frequency signals is strongly enhanced by adaptation, whereas signal
transmission is not influenced by adaptation in the non-chaotic regime. Our
theoretical framework can be applied to other mechanisms at the level of single
nodes, such as synaptic filtering, refractoriness or spike synchronization.
These results advance our understanding of the interaction between the dynamics
of single units and recurrent connectivity, which is a fundamental step toward
the description of biologically realistic network models in the brain, or, more
generally, networks of other physical or man-made complex dynamical units
- …