1,305 research outputs found
Evaluating functions of positive-definite matrices using colored noise thermostats
Many applications in computational science require computing the elements of
a function of a large matrix. A commonly used approach is based on the the
evaluation of the eigenvalue decomposition, a task that, in general, involves a
computing time that scales with the cube of the size of the matrix. We present
here a method that can be used to evaluate the elements of a function of a
positive-definite matrix with a scaling that is linear for sparse matrices and
quadratic in the general case. This methodology is based on the properties of
the dynamics of a multidimensional harmonic potential coupled with colored
noise generalized Langevin equation (GLE) thermostats. This "thermostat"
(FTH) approach allows us to calculate directly elements of functions of a
positive-definite matrix by carefully tailoring the properties of the
stochastic dynamics. We demonstrate the scaling and the accuracy of this
approach for both dense and sparse problems and compare the results with other
established methodologies.Comment: 8 pages, 4 figure
Optimized auxiliary oscillators for the simulation of general open quantum systems
A method for the systematic construction of few-body damped harmonic
oscillator networks accurately reproducing the effect of general bosonic
environments in open quantum systems is presented. Under the sole assumptions
of a Gaussian environment and regardless of the system coupled to it, an
algorithm to determine the parameters of an equivalent set of interacting
damped oscillators obeying a Markovian quantum master equation is introduced.
By choosing a suitable coupling to the system and minimizing an appropriate
distance between the two-time correlation function of this effective bath and
that of the target environment, the error induced in the reduced dynamics of
the system is brought under rigorous control. The interactions among the
effective modes provide remarkable flexibility in replicating non-Markovian
effects on the system even with a small number of oscillators, and the
resulting Lindblad equation may therefore be integrated at a very reasonable
computational cost using standard methods for Markovian problems, even in
strongly non-perturbative coupling regimes and at arbitrary temperatures
including zero. We apply the method to an exactly solvable problem in order to
demonstrate its accuracy, and present a study based on current research in the
context of coherent transport in biological aggregates as a more realistic
example of its use; performance and versatility are highlighted, and
theoretical and numerical advantages over existing methods, as well as possible
future improvements, are discussed.Comment: 23 + 9 pages, 11 + 2 figures. No changes from previous version except
publication info and updated author affiliation
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