122,855 research outputs found
Splitting for Rare Event Simulation: A Large Deviation Approach to Design and Analysis
Particle splitting methods are considered for the estimation of rare events.
The probability of interest is that a Markov process first enters a set
before another set , and it is assumed that this probability satisfies a
large deviation scaling. A notion of subsolution is defined for the related
calculus of variations problem, and two main results are proved under mild
conditions. The first is that the number of particles generated by the
algorithm grows subexponentially if and only if a certain scalar multiple of
the importance function is a subsolution. The second is that, under the same
condition, the variance of the algorithm is characterized (asymptotically) in
terms of the subsolution. The design of asymptotically optimal schemes is
discussed, and numerical examples are presented.Comment: Submitted to Stochastic Processes and their Application
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
Splitting algorithm for DMT optimal cooperative MAC protocols in wireless mesh networks
A cooperative protocol for wireless mesh networks is proposed in this paper. The protocol implements both on-demand relaying and a selection of the best relay terminal so only one terminal is relaying the source message when cooperation is needed. Two additional features are also proposed. The best relay is selected with a splitting algorithm. This approach allows fast relay selection within less than three time-slots, on average. Moreover, a pre-selection of relay candidates is performed prior to the splitting algorithm. Only terminals that are able to improve the direct path are pre-selected. So efficient cooperation is now guaranteed. We prove that this approach is optimal in terms of diversity-multiplexing trade-off. The protocol has been designed in the context of Nakagami-mfading channels. Simulation results show that the performance of the splitting algorithm does not depend on channel statistics
RAM-Efficient External Memory Sorting
In recent years a large number of problems have been considered in external
memory models of computation, where the complexity measure is the number of
blocks of data that are moved between slow external memory and fast internal
memory (also called I/Os). In practice, however, internal memory time often
dominates the total running time once I/O-efficiency has been obtained. In this
paper we study algorithms for fundamental problems that are simultaneously
I/O-efficient and internal memory efficient in the RAM model of computation.Comment: To appear in Proceedings of ISAAC 2013, getting the Best Paper Awar
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