All sources of **correlations** studied **in** this paper — namely, **the** autocorrelation, nuclear **correlations**, and **the** twin **correlations** — involve scales much shorter than **the** nuclear radius. **The** eccentricity and size fluctuations **in** **the** **Glauber** **model** appear then to be created by uncor- related, small-scale fluctuations **in** **the** transverse plane. Subnucleonic fluctuations, which are not incorporated **in** **the** **Glauber** **model**, are also intrinsically small-scale phe- nomena. They typically increase **the** magnitude of **the** lo- cal fluctuations, but do not give rise to any large-distance correlation. To a good approximation, **the** **Monte**-**Carlo** **Glauber** **model** provides a picture of energy deposition for **the** RHIC at LHC energies **in** terms of independent sources, that seems to capture **the** main features of these fluctuations and their **correlations**.

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V. CONCLUSIONS
We have used **Monte** **Carlo** ray tracing simulations with a variety of sample kernels to accurately **model** **the** total scattering response observed **in** a single crystal of uranium nitride on **the** SEQUOIA and ARCS time-of-flight chopper spectrometers. **The** simulations have verified that multiple scattering creates an essentially Q-independent background at **the** oscillator mode positions. **The** simulations have also shown that **the** measured scattering for uranium nitride can be reproduced extremely well by including QHO single- scattering events, acoustic phonon scattering events (both single and multiphonon), and multiple-scattering events that create any combination of oscillator excitations or acoustic phonons. Finally, **the** temperature dependence of **the** oscillator modes has been investigated on SEQUOIA from T = 8 to 300 K. **Monte** **Carlo** ray tracing simulations, incorporating intrinsic broadening of **the** oscillator modes according to **the** binary solid **model**, agree extremely well with **the** experimental data. This work shows that **Monte** **Carlo** ray tracing simulations can be an extremely effective tool for **the** accurate modeling of complex neutron scattering spectra.

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On the other hand, the duality of the prior distribution and the fixed regional shape parameter allows the proposed estimator to be at least as efficient as the index flood model.. Thus [r]

patients. **In** consequence, we can find many dedicated small field of view scanners for SPECT (Single Photon Emission Tomography) [2, 3] and PET (Positron Emission Tomography) [4] which have been designed **in** **the** last decade for these purposes. SPECT images have a very poor quality because generally used models do not incorporate all physical interactions such as scattering (30% of photons **in** SPECT images are scattered). **Monte** **Carlo** Simulations (MCS) have greatly contributed to these developments thanks to their accuracy and utility **in** **the** field of nuclear medicine. **The** use of **the** MCS has limitations **in** computing time. Different strategies have been suggested to reduce **the** computing time such as **the** acceleration techniques [5]. Another solution to speed up MCS is to combine **Monte** **Carlo** and non- **Monte** **Carlo** modelling [5]. A third option that has recently been explored is **the** deployment of computing grids, also known as **the** parallelization of **the** MCS [6]. **The** parallelization consists **in** sub- dividing a long simulation into short ones. Each MCS uses a Random Number Stream (RNS) to produce **the** physical interactions **in** question. **The** distribution of MCS on multiple computing resources requires that **the** generated streams are independent. **In** this work, we report **the** validation of **the** Biospace dedicated to small field of view SPECT scanner using **the** GATE MC simulation toolkit on **the** CIMENT Grid, a grid deployed on **the** university campuses **in** Grenoble.

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1 Introduction
Many critical applications (e.g. medical diagnosis or autonomous driving) require accurate fore- casts while detecting unreliable predictions, that may arise from anomalies, missing information, or unknown situations. While neural networks excel at predictive tasks, they often solely out- put a single-point estimate, lacking uncertainty measures to assess their confidence about their predictions. To overcome this limitation, an open research question is **the** design of neural gen- erative models able to output a predictive distribution instead of single point-estimates. First, such distributions would naturally provide **the** desired uncertainty measures over **the** **model** pre- dictions. Secondly, learning algorithms can build upon such uncertainty measurements to improve their predictive performance such as active learning [Campbell et al., 2000] or exploration **in** re- inforcement learning [Geist and Pietquin, 2010, Fortunato et al., 2018]. Thirdly, they may better **model** incoming sources of variability, such as observation noise, missing information, or **model** misspecification.

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Summary. Importance sampling methods can be iterated like MCMC algorithms, while being
more robust against dependence and starting values, as shown **in** this paper. **The** population **Monte** **Carlo** principle we describe here consists of iterated generations of importance samples, with importance functions depending on **the** previously generated importance samples. **The** advantage over MCMC algorithms is that **the** scheme is unbiased at any iteration and can thus be stopped at any time, while iterations improve **the** performances of **the** importance function, thus leading to an adaptive importance sampling. We first illustrate this method on a toy mixture example with multiscale importance functions. A second example reanalyses **the** ion channel **model** of Hodgson (1999), using an importance sampling scheme based on a hidden Markov representation, and compares population **Monte** **Carlo** with a corresponding MCMC algorithm. Keywords: Adaptive algorithm, degeneracy, hidden semi-Markov **model**, importance sampling, ion channel **model**, MCMC algorithms, mixture **model**, multiple scales, particle system, random walk, unbiasedness.

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clusters tend to increase **in** size and get localized. If AV= - 0.6E 0 < BV= - 0.4E 0 , then voids drives A spatially with A at **the** borders of **the** vacancy clusters.
Cycling of temperature between 300K and 600K do not have a marked effect on **the** distribution of **the** vacancy clusters. But when local temperature rise are taken into account **in** **the** algorithm to mimic **the** trapping of heat by **the** vacancy clusters, then a 1D spreading effect of **the** clusters is obtained which may be interpreted as crack propagation. **The** results obtained show that **the** ABV **model** can be used **in** correlation to FEM methods to study **the** thermo-mechanical evolution of interconnect alloy or material during **the** operation of a power mechatronic device and reduce by numerical simulation **the** uncertainties on **the** coupling between thermal and mechanical parameters of materials assembled **in** such devices. Then RBDO method can be applied for reliability issues.

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Figure 5: **Monte** **Carlo** simulation algorithm
from 10 5 to 10 8 steps for ratios of locked sites ranging from 0 (every site of **the** lattice that is not occupied by **the** polymer is accessible) to 0.6 (none of **the** lattice sites is accessible). Figure 6 shows **the** ratio of visited lattice sites with respect to **the** ratio of locked sites. For every tested duration, when **the** amount of locked sites reaches around 38%, **the** molecular motion is close to null. Above this value, increasing **the** simulation duration will not increase **the** polymer mobility, which is **in** a frozen-like state. **The** 38% value appears as a threshold below which **the** molecular mobility is activated and increases quickly. Remarkably, as **the** simulation duration increases, **the** change of mobility from null to full occurs **in** a very narrow span of ratio of locked sites. Another way to present this result, is to look at **the** duration required to visit a large part of **the** free sites of **the** lattice according to **the** ratio of locked sites. Figure 7 shows **the** duration needed to visit 95% of **the** unoccupied sites of **the** lattice according to **the** ratio of locked sites for **the** reference system. As one can read, such a duration grows exponentially after passing 35% of locked sites.

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All experiments use Morpion Solitaire disjoint **model**. All **the** time results are a mean over multiple runs of each algorithm, except for results **in** parenthesis which were run only once. **The** standard deviation is given between paren- thesis after **the** time results. **The** algorithms were tested on playing only **the** first move of a game, and on playing an entire game. All experiments consist **in** testing **the** al- gorithms at level 3 and 4 of nesting. Each rollout needs a time that is slightly different from others since random games inside each rollout can have different lengths. Times taken by two rollouts can be different. Standard deviations show these times variations.

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What makes it so interesting **in** contrast to classic search algorithms is that it does not rely on **the** knowledge of **the** problem beforehand. Prior to MCS, algorithms designed to decide what is **the** best possible move to execute were mostly relying on an objective function. An objective function is basically a function designed by an expert **in** **the** field that decides which move is **the** best. Obviously **the** main problem with this approach is that such a function seldom covers every situation encountered, and **the** quality of **the** decision depends on **the** quality of **the** expert. Moreover, it is very difficult to tackle new problems or problems where there is little information available. MCS algorithms do not suffer from either of these drawbacks. All it requires is a **model** of **the** problem at hand upon which they can execute (a lot of) simulations. Here lies **the** strength of these algorithms. It is not **in** **the** ability to abstract like **the** human brain, but **in** **the** raw computational power that computers excel. Computers can do a lot of simulations very quickly and if needed simulations are suitable for massive parallelization. More importantly, from an optimization point of view most MCS algorithms can theoretically converge to **the** optimal decision given enough time. Before **the** description of a well-known MCS algorithms, Section 1.2 first introduces **the** underlying framework needed to define **the** algorithms.

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simulation. Such DES-based simulators require at least a platform specification and an application description. **The** available cloud DESs can be divided **in** two categories. **In** **the** first category are **the** simulators dedicated to study **the** clouds from **the** provider point-of-view, whose purpose is to help evaluating **the** design decisions of **the** datacenter. Examples of such simulators are MDCSim [6], which offers specific and precise models for low-level components including network (e.g InfiniBand or Gigabit ethernet), operating system kernel and disks. It also offers a **model** for energy consumption. However, **the** cloud client activity that can be modeled is restricted to web-servers, application-servers, or data-base applications. GreenCloud [7] follows **the** same purpose with a strong focus on energy consumption of cloud’s network apparatus using a packet-level simulation for network communications (NS2). **In** **the** second category (which we focus on) are **the** simulators targeting **the** whole cloud ecosystem including client activity. **In** this category, CloudSim [8] is **the** most broadly used simulator **in** academic research. It offers simplified models regarding network communications, CPU, or disks. However, it is easily extensible and serves as **the** underlying simulation engine **in** a number of projects. Simgrid [3] is **the** other long-standing project, which when used **in** conjunction with **the** SchIaaS cloud interface provides similar functionnalities as CloudSim. Among **the** other related projects is iCanCloud [9] proposed to address scalability issues encountered with CloudSim (written **in** Java) for **the** simulation of large use-cases. Most recently, PICS [10] has been proposed to evaluate specifically **the** simulation of public clouds. **The** configura- tion of **the** simulator uses only parameters that can be measured by **the** cloud client, namely inbound and outbound network bandwidths, average CPU power, VM boot times, and scale-**in**/scale-out policies. **The** data center is therefore seen as a black box, for which no detailed description of **the** hardware setting is re- quired. **The** validation study of PICS under a variety of use-cases has nonetheless shown accurate predictions.

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magnitude and/or present strong **correlations**.
Some families of AIS methods use geometric information about **the** target for **the** adaptation of **the** location parame- ters, yielding to optimization-based adaptation schemes. For example, **the** GAPIS algorithm [18] is an AIS method that exploits **the** gradient and **the** Hessian of **the** logarithm of **the** target, and also introduces an artificial repulsion among proposals to promote **the** diversity (without any resampling step). Other methods such as [19], [20] adapt **the** location parameters by performing at each sample several steps of **the** unadjusted Langevin algorithm (ULA) [21], which can also be seen as an instance of a stochastic gradient descent method. **The** covariance is also adapted **in** those methods by either com- puting **the** sample autocorrelation [19] or using second-order information [18], [20]. A covariance adaptation has been also explored via robust moment-matching mechanisms **in** [22], [23]. We refer **the** interested reader to **the** survey [6]. **The** use of optimization techniques within PMC framework remains however unexplored. It is worth mentioning that optimization inspired schemes have also shown to be an efficient strategy to improve practical convergence rate **in** MCMC algorithms (see **the** survey paper [24] and references therein). **In** particular, **the** works [25], [26], [27], [28], [29] fall **in** **the** framework of **the** so-called Metropolis adjusted Langevin algorithms (MALA), where **the** ULA scheme is combined with a Metropolis- Hastings step. **The** Langevin-based strategy yields proposed samples that are more likely drawn from a highly probable re- gion, with **the** consequence of a larger acceptance probability. MALA can be further improved by rescaling **the** drift term by a preconditioning matrix encoding local curvature information about **the** target density, through **the** Fisher metric [30], **the** Hessian matrix [31], [32], [28] or a tractable approximation of it [33], [34], [35], [36]. Optimization-based methods for accelerating MCMC sampling of non-differentiable targets have also been considered, for instance **in** [26], [37].

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RKKY-type interactions. We then consider **the** 2D and 3D Heisenberg models with oscillatory Ruderman- Kittel-Kasuya-Yosida (RKKY) interactions J ij = J 0 (cos(2k F r ij )/r d ij ) exp(−r ij /λ), where d is **the** spatial dimension, k F is **the** Fermi vector (k F ≈4.91 for **the** spin- glass system CuMn), and λ is **the** characteristic length **in** **the** damping term [37, 39–43]. Due to their approximate description of real materials, rich behaviors, and impor- tant roles **in** bridging **the** experimental study of glassy materials and **the** spin-glass theory of short-range inter- actions [37, 39], these systems are under extensive stud- ies. For simplicity, we set J 0 = 1 and k F = π, and take λ = 3 for 3D and λ = ∞ for 2D, so that **the** system is **in** **the** class of strict (3D) and marginal (2D) extensivities. **The** simulations are at β(2D) = 1 and β(3D) = 0.693, close to **the** critical temperature β c = 0.693 003(2) for **the** 3D pure Heisenberg **model** [44]. Box sizes are set to 1 and **the** achieved acceleration is again A∼ O(N) for **the** strict extensivity, and A ∼ O(N/ ln N) for **the** marginal extensivity, as illustrated **in** Fig. 4.

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Online verification of dose delivery during radiotherapy treatments has become essential **in** order to ensure that **the** dose planned by **the** treatment planning system (TPS) is delivered as accurately as possible and to detect possible deviations.
One strategy using EPIDs for dosimetric verification consists **in** comparing predicted dose images with acquired portal images before and/or during treatment [1].
Our goal is twofold:

Abstract
This paper provides an analysis of **the** generation-to-generation **correlations** as observed when solving full core eigenvalue problems on PWR systems. Many studies have **in** **the** past looked at **the** impact of these **correlations** on reported variance and this paper extends **the** analysis to **the** observed convergence rate on **the** tallies, **the** effect of tally size and **the** effect of generation size. Since performing meaningful analysis on such a large problem is inherently difficult, a simple homogeneous reflective cube problem with analytical solution was developed that exhibits similar behavior to **the** full core PWR benchmark. **The** data **in** this problem was selected to match **the** dimensionality of **the** reactor problem and preserve **the** migration length traveled by neutrons. Results demonstrate that **the** variance will deviate signifi- cantly from **the** 1/N (N being **the** number of simulated particles) convergence rate associated with truly independent generations, but will eventually asymptote to 1/N after 1000’s of generations regardless of **the** numbers of neutrons per generation. This indicates that optimal run strategies should emphasize lower number of active generations with greater number of neutrons per generation to produce **the** most accurate tally results. This paper also describes and compares three techniques to evaluate suitable confi- dence intervals **in** **the** presence of **correlations**, one based on using history statistics, one using generation statistics and one batching generations to reduce batch-to-batch correlation.

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Z
h(θ)π(θ |x) dθ
and Bayes tests also involve integration, as mentioned earlier with **the** Bayes factors—, and optimisation difficulties with **the** likelihood perspective, this clas- sification is by no way tight—as for instance when likelihoods involve unmanageable integrals—and all fields of Statistics, from design to econometrics, from genomics to psychometry and environmics, have now to rely on **Monte** **Carlo** approximations. A whole new range of statistical methodologies have entirely inte- grated **the** simulation aspects. Examples include **the** bootstrap methodology (Efron 1982), where multi- level resampling is not conceivable without a com- puter, indirect inference (Gouri´eroux et al. 1993), which construct a pseudo-likelihood from simulations, MCEM (Capp´e and Moulines 2009), where **the** E- step of **the** EM algorithm is replaced with a **Monte** **Carlo** approximation, or **the** more recent approxi- mated Bayesian computation (ABC) used **in** phylo- genics (Beaumont et al. 2002), where **the** likelihood is not manageable but **the** underlying **model** can be simulated from.

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correlation coefficients also explain **the** particular features of **the** variance convergence rates observed numerically.
Chapter 3 focuses on predicting correlation coefficients before starting **the** active generations of **the** simulation. This provides not only correction to **the** underestimated variance but also provides an estimate on when a target accuracy will be reached. A Markov Chain **Monte** **Carlo** **model** is developed to show that **the** dependence of neutron source bank between consecutive generations contributes a significant fraction of **the** correlation of tallies between generations but also misses an important part caused by **the** multiplying effect of fission. **The** method of multitype branching process (MBP) is developed to capture both **the** source bank dependence and **the** correlation due to multiplication. **The** MBP **model** is exact **in** predicting **correlations** for neutrons transported **in** a discrete phase space and provides acceptable accuracy **in** continuous problems by constructing an approximate discrete problem from tallies. Since **the** MBP **model** is a discrete approximation, it can predict **correlations** and variances on tally meshes coarser than **the** discretized **model** mesh accurately.

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Figure 4: Homogeneous cube reactor. **The** behaviour of **the** measured Shannon entropy S(g) during **Monte** **Carlo** power it- eration as a function of generations g, for different initial popu- lation sizes N and fixed L = 400 cm. **The** guess source at g = 0 consists of N neutrons located at **the** center of **the** cube. Power iteration is run for 1000 generations. Upper red curve: N = 10 5 ; central green curve: N = 10 4 ; lower blue curve: N = 10 3 . **The** dashed lines represent **the** expected entropy value S N as **in** Eq. (6) (red: N = 10 5 , green: N = 10 4 and blue: N = 10 3 , respectively), and **the** solid black line is **the** ideal expected en- tropy S∞ for an infinite number of particles per generation as **in** Eq. (5).

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Richard Fournier 3 , Mathieu Galtier 7 , Jacques Gautrais 8 , Anaïs Khuong 8 , Lionel Pelissier 9 ,
Benjamin Piaud 5 , Maxime Roger 7 , Guillaume Terrée 2 & Sebastian Weitz 1,2
**Monte** **Carlo** is famous for accepting **model** extensions and **model** refinements up to infinite dimension. However, this powerful incremental design is based on a premise which has severely limited its application so far: a state-variable can only be recursively defined as a function of underlying state- variables if this function is linear. Here we show that this premise can be alleviated by projecting nonlinearities onto a polynomial basis and increasing **the** configuration space dimension. Considering phytoplankton growth **in** light-limited environments, radiative transfer **in** planetary atmospheres, electromagnetic scattering by particles, and concentrated solar power plant production, we prove **the** real-world usability of this advance **in** four test cases which were previously regarded as impracticable using **Monte** **Carlo** approaches. We also illustrate an outstanding feature of our method when applied to acute problems with interacting particles: handling rare events is now straightforward. Overall, our extension preserves **the** features that made **the** method popular: addressing nonlinearities does not compromise on **model** refinement or system complexity, and convergence rates remain independent of dimension.

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