2,792 research outputs found
Almost uniform sampling via quantum walks
Many classical randomized algorithms (e.g., approximation algorithms for
#P-complete problems) utilize the following random walk algorithm for {\em
almost uniform sampling} from a state space of cardinality : run a
symmetric ergodic Markov chain on for long enough to obtain a random
state from within total variation distance of the uniform
distribution over . The running time of this algorithm, the so-called {\em
mixing time} of , is , where
is the spectral gap of .
We present a natural quantum version of this algorithm based on repeated
measurements of the {\em quantum walk} . We show that it
samples almost uniformly from with logarithmic dependence on
just as the classical walk does; previously, no such
quantum walk algorithm was known. We then outline a framework for analyzing its
running time and formulate two plausible conjectures which together would imply
that it runs in time when is
the standard transition matrix of a constant-degree graph. We prove each
conjecture for a subclass of Cayley graphs.Comment: 13 pages; v2 added NSF grant info; v3 incorporated feedbac
On the Use of Surrogate Functions for Mixed Variable Optimization of Simulated Systems
This research considers the efficient numerical solution of linearly constrained mixed variable programming (MVP) problems, in which the objective function is a black-box stochastic simulation, function evaluations may be computationally expensive, and derivative information is typically not available. MVP problems are those with a mixture of continuous, integer, and categorical variables, the latter of which may take on values only from a predefined list and may even be non-numeric. Mixed Variable Generalized Pattern Search with Ranking and Selection (MGPS-RS) is the only existing, provably convergent algorithm that can be applied to this class of problems. Present in this algorithm is an optional framework for constructing and managing less expensive surrogate functions as a means to reduce the number of true function evaluations that are required to find approximate solutions. In this research, the NOMADm software package, an implementation of pattern search for deterministic MVP problems, is modified to incorporate a sequential selection with memory (SSM) ranking and selection procedure for handling stochastic problems. In doing so, the underlying algorithm is modified to make the application of surrogates more efficient. A second class of surrogates based on the Nadaraya-Watson kernel regression estimator is also added to the software. Preliminary computational testing of the modified software is performed to characterize the relative efficiency of selected surrogate functions for mixed variable optimization in simulated systems
Consistency properties of a simulation-based estimator for dynamic processes
This paper considers a simulation-based estimator for a general class of
Markovian processes and explores some strong consistency properties of the
estimator. The estimation problem is defined over a continuum of invariant
distributions indexed by a vector of parameters. A key step in the method of
proof is to show the uniform convergence (a.s.) of a family of sample
distributions over the domain of parameters. This uniform convergence holds
under mild continuity and monotonicity conditions on the dynamic process. The
estimator is applied to an asset pricing model with technology adoption. A
challenge for this model is to generate the observed high volatility of stock
markets along with the much lower volatility of other real economic aggregates.Comment: Published in at http://dx.doi.org/10.1214/09-AAP608 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Binary sequences with prescribed autocorrelations
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