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Simulating Normalized Constants: From Importance Sampling to Bridge Sampling to Path Sampling
 Statistical Science, 13, 163–185. COMPARISON OF METHODS FOR COMPUTING BAYES FACTORS 435
, 1998
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Cited by 146 (4 self)
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Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
The Art of Data Augmentation
, 2001
"... The term data augmentation refers to methods for constructing iterative optimization or sampling algorithms via the introduction of unobserved data or latent variables. For deterministic algorithms,the method was popularizedin the general statistical community by the seminal article by Dempster, Lai ..."
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Cited by 22 (3 self)
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The term data augmentation refers to methods for constructing iterative optimization or sampling algorithms via the introduction of unobserved data or latent variables. For deterministic algorithms,the method was popularizedin the general statistical community by the seminal article by Dempster, Laird, and Rubin on the EM algorithm for maximizing a likelihood function or, more generally, a posterior density. For stochastic algorithms, the method was popularized in the statistical literature by Tanner and Wong’s Data Augmentation algorithm for posteriorsampling and in the physics literatureby Swendsen and Wang’s algorithm for sampling from the Ising and Potts models and their generalizations; in the physics literature,the method of data augmentationis referred to as the method of auxiliary variables. Data augmentationschemes were used by Tanner and Wong to make simulation feasible and simple, while auxiliary variables were adopted by Swendsen and Wang to improve the speed of iterative simulation. In general,however, constructing data augmentation schemes that result in both simple and fast algorithms is a matter of art in that successful strategiesvary greatlywith the (observeddata) models being considered.After an overview of data augmentation/auxiliary variables and some recent developments in methods for constructing such
Random Number Generators for Parallel Applications
 in Monte Carlo Methods in Chemical Physics
, 1998
"... this article is devoted, because these com1 putations require the highest quality of random numbers. The ability to do a multidimensional integral relies on properties of uniformity of ntuples of random numbers and/or the equivalent property that random numbers be uncorrelated. The quality aspect i ..."
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Cited by 17 (7 self)
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this article is devoted, because these com1 putations require the highest quality of random numbers. The ability to do a multidimensional integral relies on properties of uniformity of ntuples of random numbers and/or the equivalent property that random numbers be uncorrelated. The quality aspect in the other uses is normally less important simply because the models are usually not all that precisely specified. The largest uncertainties are typically due more to approximations arising in the formulation of the model than those caused by lack of randomness in the random number generator. In contrast, the first class of applications can require very precise solutions. Increasingly, computers are being used to solve very welldefined but hard mathematical problems. For example, as Dirac [1] observed in 1929, the physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are completely known and it is only necessary to find precise methods for solving the equations for complex systems. In the intervening years fast computers and new computational methods have come into existence. In quantum chemistry, physical properties must be calculated to "chemical accuracy" (say 0.001 Rydbergs) to be relevant to physical properties. This often requires a relative accuracy of 10
The complexity of stoquastic local hamiltonian problems. Arxiv: quantph/0606140
, 2006
"... We study the complexity of the Local Hamiltonian Problem (denoted as LHMIN) in the special case when a Hamiltonian obeys conditions of the PerronFrobenius theorem: all offdiagonal matrix elements in the standard basis are real and nonpositive. We will call such Hamiltonians, which are common in ..."
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Cited by 6 (1 self)
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We study the complexity of the Local Hamiltonian Problem (denoted as LHMIN) in the special case when a Hamiltonian obeys conditions of the PerronFrobenius theorem: all offdiagonal matrix elements in the standard basis are real and nonpositive. We will call such Hamiltonians, which are common in the natural world, stoquastic. An equivalent characterization of stoquastic Hamiltonians is that they have an entrywise nonnegative Gibbs density matrix for any temperature. We prove that LHMIN for stoquastic Hamiltonians belongs to the complexity class AM — a probabilistic version of NP with two rounds of communication between the prover and the verifier. We also show that 2local stoquastic LHMIN is hard for the class MA. With the additional promise of having a polynomial spectral gap, we show that stoquastic LHMIN belongs to the class PostBPP=BPPpath — a generalization of BPP in which a postselective readout is allowed. This last result also shows that any problem solved by adiabatic quantum computation using stoquastic Hamiltonians is in PostBPP ∩ AM. 1
Large deviations for many Brownian bridges with symmetrised initialterminal condition
, 2006
"... Consider a large system of N Brownian motions in R d with some nondegenerate initial measure on some fixed time interval [0, β] with symmetrised initialterminal condition. That is, for any i, the terminal location of the ith motion is affixed to the initial point of the σ(i)th motion, where σ is ..."
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Cited by 5 (4 self)
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Consider a large system of N Brownian motions in R d with some nondegenerate initial measure on some fixed time interval [0, β] with symmetrised initialterminal condition. That is, for any i, the terminal location of the ith motion is affixed to the initial point of the σ(i)th motion, where σ is a uniformly distributed random permutation of 1,..., N. Such systems play an important role in quantum physics in the description of Boson systems at positive temperature 1/β. In this paper, we describe the largeN behaviour of the empirical path measure (the mean of the Dirac measures in the N paths) and of the mean of the normalised occupation measures of the N motions in terms of large deviations principles. The rate functions are given as variational formulas involving certain entropies and FenchelLegendre transforms. Consequences are drawn for asymptotic independence statements and laws of large numbers. In the special case related to quantum physics, our rate function for the occupation measures turns out to be equal to the wellknown DonskerVaradhan rate function for the occupation measures of one motion in the limit of diverging time. This enables us to prove a simple formula for the largeN asymptotic of the symmetrised trace of e −βHN, where HN is an Nparticle Hamilton operator in a trap.
Quantum Simulations of Complex ManyBody Systems: From Theory to Algorithms, Lecture Notes,
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Cited by 3 (1 self)
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Permission to make digital or hard copies of portions of this work for personal or classroom use is granted provided that the copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise requires prior specific permission by the publisher mentioned above.
Recent Advances in 2D and 2.5D Vortex statistics and dynamics
, 2005
"... This review paper offers both an overview of the recent developments in vortex statistics, and a detailed critical review of some of the advances as well as open problems in the area. A fruitful new direction in vortex dynamics concerns vortical systems with very large numbers of vortices in which s ..."
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Cited by 1 (1 self)
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This review paper offers both an overview of the recent developments in vortex statistics, and a detailed critical review of some of the advances as well as open problems in the area. A fruitful new direction in vortex dynamics concerns vortical systems with very large numbers of vortices in which statistical methods play a pivotal role. After a review of some of the older equilibrium statistical mechanics approaches, such as Onsager’s Vortex gas theory, we survey the recent burst of interest and papers on the application of statistical mechanics to a wide range of problems. They include many interesting problems such as rotating stratified flows with natural applications to atmospheric and oceanic sciences. Special models that will be discussed include point vortex as well as vortex blob models in axisymmetric flows, in the plane, and in geophysical flows involving multilayer baroclinic effects. A recent reverse application of a quantum pathintegral MonteCarlo algorithm to the LionsMajda model for nearly parallel vortex filaments will be reviewed with numerical results. A connection between these statistical theories and the classical numerical method known as the vortex method will also be made. Many of these noteworthy advances are made possible by good MonteCarlo algorithms, providing a crossdisciplinary aspect to the field. An important open problem in the field of the vortex gas is the exact partition function for any finite number of particles N and any finite temperature. 1.
Selfconsistent equation for an interacting Bose gas
, 2008
"... We consider interacting Bose gas in thermal equilibrium assuming a positive and bounded pair potential V (r) such that 0 < ∫ drV (r) = a < ∞. Expressing the partition function by the FeynmanKac functional integral yields a classicallike polymer representation of the quantum gas. With Mayer graph s ..."
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Cited by 1 (0 self)
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We consider interacting Bose gas in thermal equilibrium assuming a positive and bounded pair potential V (r) such that 0 < ∫ drV (r) = a < ∞. Expressing the partition function by the FeynmanKac functional integral yields a classicallike polymer representation of the quantum gas. With Mayer graph summation techniques, we demonstrate the existence of a selfconsistent relation ρ(µ) = F(µ − aρ(µ)) between the density ρ and the chemical potential µ, valid in the range of convergence of Mayer series. The function F is equal to the sum of all rooted multiply connected graphs. Using Kac’s scaling Vγ(r) = γ 3 V (γr) we prove that in the meanfield limit γ → 0 only tree diagrams contribute and function F reduces to the free gas density. We also investigate how to extend the validity of the selfconsistent relation beyond the convergence radius of Mayer series (vicinity of BoseEinstein condensation) and study dominant corrections to mean field. At lowest order, the form of function F is shown to depend on single polymer partition function for which we derive lower and 1 upper bounds and on the resummation of ring diagrams which can be analytically performed.