12,927 research outputs found
Optimization of mesh hierarchies in Multilevel Monte Carlo samplers
We perform a general optimization of the parameters in the Multilevel Monte
Carlo (MLMC) discretization hierarchy based on uniform discretization methods
with general approximation orders and computational costs. We optimize
hierarchies with geometric and non-geometric sequences of mesh sizes and show
that geometric hierarchies, when optimized, are nearly optimal and have the
same asymptotic computational complexity as non-geometric optimal hierarchies.
We discuss how enforcing constraints on parameters of MLMC hierarchies affects
the optimality of these hierarchies. These constraints include an upper and a
lower bound on the mesh size or enforcing that the number of samples and the
number of discretization elements are integers. We also discuss the optimal
tolerance splitting between the bias and the statistical error contributions
and its asymptotic behavior. To provide numerical grounds for our theoretical
results, we apply these optimized hierarchies together with the Continuation
MLMC Algorithm. The first example considers a three-dimensional elliptic
partial differential equation with random inputs. Its space discretization is
based on continuous piecewise trilinear finite elements and the corresponding
linear system is solved by either a direct or an iterative solver. The second
example considers a one-dimensional It\^o stochastic differential equation
discretized by a Milstein scheme
Capacity-achieving ensembles for the binary erasure channel with bounded complexity
We present two sequences of ensembles of non-systematic irregular
repeat-accumulate codes which asymptotically (as their block length tends to
infinity) achieve capacity on the binary erasure channel (BEC) with bounded
complexity per information bit. This is in contrast to all previous
constructions of capacity-achieving sequences of ensembles whose complexity
grows at least like the log of the inverse of the gap (in rate) to capacity.
The new bounded complexity result is achieved by puncturing bits, and allowing
in this way a sufficient number of state nodes in the Tanner graph representing
the codes. We also derive an information-theoretic lower bound on the decoding
complexity of randomly punctured codes on graphs. The bound holds for every
memoryless binary-input output-symmetric channel and is refined for the BEC.Comment: 47 pages, 9 figures. Submitted to IEEE Transactions on Information
Theor
Massive MIMO Multicasting in Noncooperative Cellular Networks
We study the massive multiple-input multiple-output (MIMO) multicast
transmission in cellular networks where each base station (BS) is equipped with
a large-scale antenna array and transmits a common message using a single
beamformer to multiple mobile users. We first show that when each BS knows the
perfect channel state information (CSI) of its own served users, the
asymptotically optimal beamformer at each BS is a linear combination of the
channel vectors of its multicast users. Moreover, the optimal combining
coefficients are obtained in closed form. Then we consider the imperfect CSI
scenario where the CSI is obtained through uplink channel estimation in
timedivision duplex systems. We propose a new pilot scheme that estimates the
composite channel which is a linear combination of the individual channels of
multicast users in each cell. This scheme is able to completely eliminate pilot
contamination. The pilot power control for optimizing the multicast beamformer
at each BS is also derived. Numerical results show that the asymptotic
performance of the proposed scheme is close to the ideal case with perfect CSI.
Simulation also verifies the effectiveness of the proposed scheme with finite
number of antennas at each BS.Comment: to appear in IEEE JSAC Special Issue on 5G Wireless Communication
System
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