12,156 research outputs found
A direct solver with O(N) complexity for variable coefficient elliptic PDEs discretized via a high-order composite spectral collocation method
A numerical method for solving elliptic PDEs with variable coefficients on
two-dimensional domains is presented. The method is based on high-order
composite spectral approximations and is designed for problems with smooth
solutions. The resulting system of linear equations is solved using a direct
(as opposed to iterative) solver that has optimal O(N) complexity for all
stages of the computation when applied to problems with non-oscillatory
solutions such as the Laplace and the Stokes equations. Numerical examples
demonstrate that the scheme is capable of computing solutions with relative
accuracy of or better, even for challenging problems such as highly
oscillatory Helmholtz problems and convection-dominated convection diffusion
equations. In terms of speed, it is demonstrated that a problem with a
non-oscillatory solution that was discretized using nodes was solved
in 115 minutes on a personal work-station with two quad-core 3.3GHz CPUs. Since
the solver is direct, and the "solution operator" fits in RAM, any solves
beyond the first are very fast. In the example with unknowns, solves
require only 30 seconds.Comment: arXiv admin note: text overlap with arXiv:1302.599
Unifying Projected Entangled Pair States contractions
The approximate contraction of a Projected Entangled Pair States (PEPS)
tensor network is a fundamental ingredient of any PEPS algorithm, required for
the optimization of the tensors in ground state search or time evolution, as
well as for the evaluation of expectation values. An exact contraction is in
general impossible, and the choice of the approximating procedure determines
the efficiency and accuracy of the algorithm. We analyze different previous
proposals for this approximation, and show that they can be understood via the
form of their environment, i.e. the operator that results from contracting part
of the network. This provides physical insight into the limitation of various
approaches, and allows us to introduce a new strategy, based on the idea of
clusters, that unifies previous methods. The resulting contraction algorithm
interpolates naturally between the cheapest and most imprecise and the most
costly and most precise method. We benchmark the different algorithms with
finite PEPS, and show how the cluster strategy can be used for both the tensor
optimization and the calculation of expectation values. Additionally, we
discuss its applicability to the parallelization of PEPS and to infinite
systems (iPEPS).Comment: 28 pages, 15 figures, accepted versio
Modeling of linear fading memory systems
Motivated by questions of approximate modeling and identification, we consider various classes of linear time-varying bounded-input-bounded output (BIBO) stable fading memory systems and the characterizations are proved. These include fading memory systems in general, almost periodic systems, and asymptotically periodic systems. We also show that the norm and strong convergence coincide for BIBO stable causal fading memory system
Squeezing the limit: Quantum benchmarks for the teleportation and storage of squeezed states
We derive fidelity benchmarks for the quantum storage and teleportation of
squeezed states of continuous variable systems, for input ensembles where the
degree of squeezing is fixed, no information about its orientation in phase
space is given, and the distribution of phase space displacements is a
Gaussian. In the limit where the latter becomes flat, we prove analytically
that the maximal classical achievable fidelity (which is 1/2 without squeezing,
for ) is given by , vanishing when the degree of squeezing
diverges. For mixed states, as well as for general distributions of
displacements, we reduce the determination of the benchmarks to the solution of
a finite-dimensional semidefinite program, which yields accurate, certifiable
bounds thanks to a rigorous analysis of the truncation error. This approach may
be easily adapted to more general ensembles of input states.Comment: 19 pages, 4figure
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