in our industrialized world several important applications require reliable and accurate wind speed forecasts . these include , but are not limited to agriculture , aviation or wind energy production .in particular , high wind speeds can cause severe damages to infrastructure and their predictions are important parts of weather warnings .wind speed forecasts are standard outputs of numerical weather prediction ( nwp ) models .nwp has traditionally been viewed as a deterministic problem , but over the last decades , a change towards probabilistic forecasts , i.e. , forecasts in the form of a full predictive distribution , can be observed .probabilistic forecasts are important in the context of weather forecasting as they allow for a quantification of the associated uncertainty of the prediction , and further allow for optimal point forecasting by using certain functionals of the predictive distribution ( see , e.g. , * ? ? ?* ) .  nowadays, weather services typically produce ensemble forecasts which consist of multiple runs of nwp models that differ in initial conditions and/or the numerical representation of the atmosphere . while the transition to ensemble forecasts is an important step towards probabilistic forecasting , ensembles are finite and do not provide full predictive densities .further , ensemble forecasts are typically underdispersive and subject to systematic bias , they thus require some form of statistical post - processing .state of the art techniques for statistical post - processing of ensemble forecasts include bayesian model averaging ( bma ) developed by , and ensemble model output statistics ( emos ) or non - homogeneous regression by .the bma approach uses weighted mixtures of parametric probability density functions ( pdfs ) which depend on the ensemble forecasts , with the mixture weights being determined based on the performance of the ensemble members in the training period .possible component choices for wind speed are given by pdfs of gamma distributions or truncated normal ( tn ) distributions .by contrast , the predictive distribution of the emos approach is given by a single parametric distribution with parameters depending on the ensemble forecasts . propose the use of truncated normal distributions for emos models of wind speed .alternative choices are given by generalized extreme value distributions , log - normal ( ln ) distributions , and combinations thereof .the article at hand builds on the emos framework and proposes models based on weighted mixtures of tn and ln distributions .this new approach allows for combining the advantages of lighter and heavier - tailed distributions , but avoids problems encountered in using previously proposed combination models .apart from the flexibility , the mixture models further exhibit desirable properties from a theoretical perspective and provide well calibrated and skillful probabilistic forecasts .the novel emos mixture approach is applied to forecasts of maximal wind speed of the 50 member ensemble of the european centre for medium - range weather forecasts ( ecmwf ; * ? ? ? * ) and the eight - member university of washington mesoscale ensemble ( uwme ; * ? ? ?* ) , and to instantaneous wind - speed forecasts of the 11 member limited area model ensemble prediction system of the hungarian meteorological service ( hms ) called aire limite adaptation dynamique dveloppement international - hungary ensemble prediction system ( aladin - huneps ; * ? ? ?* ; * ? ? ?the three ensemble prediction system differ in the generation of their members , which is accounted for in the model formulation .the tn model of , and the ln and tn - ln combination models of serve as benchmark models for the three case studies .the remainder of this article is organized as follows .section [ sec : sec2 ] contains a description of the ensembles and observational data . in section [ sec : sec3 ] the emos technique is reviewed and the novel tn - ln mixture models are introduced .section [ sec : sec4 ] summarizes the results of the three case studies .the article concludes with a discussion in section [ sec : sec5 ] .we consider three distinct data sets of ensemble forecasts and corresponding observations which differ both in the stochastic properties of the ensemble as well as the observed wind quantities .the data sets coincide with those used in .we thus limit our discussion here to a succinct summary of the data and refer to for a more detailed description .the ecmwf ensemble consists of 50 exchangeable ensemble members of one day ahead forecasts of 10 m daily maximum wind speed ( given in ms ) along with corresponding validating observations from 228 synoptic observation stations over germany .the observations are daily maxima of hourly observations of 10-minute average wind speed measured over the 10 minutes before the hour , where the maxima are taken over the 24 hours corresponding to the time frame of the ensemble forecast .the results presented in section [ subs : subs4.1 ] are based on a verification period from 1 may 2010 to 30 april 2011 , consisting of 83220 individual forecast cases .  to 11 cm  figure [ fig :fig1]a shows the verification rank histogram of the raw ensemble , that is the histogram of ranks of validating observations with respect to the corresponding ensemble forecasts computed from the ranks at all locations and dates considered ( see , e.g. , * ? ? ?* section 7.7.2 ) .the strongly u - shaped histogram indicates a highly underdispersive character of the ecmwf ensemble .the range of the ecmwf ensemble contains the validating observation only in of all cases ( the nominal value of this coverage is , that is ) , verifying the need of statistical post - processing .the aladin - huneps system of the hms consists of 11 members , 10 exchangeable forecasts initialized from perturbed initial conditions and one control member from the unperturbed analysis .the data base contains ensembles of 42-h forecasts for 10 m wind speed ( given in ms ) for 10 major cities in hungary , together with the corresponding validating observations for the one - year period between 1 april 2012 and 31 march 2013 .the validating wind speed measurements are considered as instantaneous values ( valid at a given time ) , however , they are in fact mean values over the preceding 10 minutes .the model wind speed values are also considered as instantaneous , but they are representatives for a given model time step , which is 5 min in our case .similar to the ecmwf ensemble , the verification rank histogram of the raw aladin - huneps ensemble is far from the desired uniform distribution ( see figure [ fig : fig1]b ) , however , it shows a much less underdispersive character .the better fit of the ensemble can also be observed on its coverage value of which should be compared with the nominal coverage of ( ) .the uwme covering the pacific northwest region of western north america has eight members that are obtained from different runs of the fifth generation pennsylvania state university national center for atmospheric research mesoscale model ( psu - ncar mm5 ) .our data base contains ensembles of 48 h forecasts and corresponding validating observations of 10 m maximal wind speed ( maximum of the hourly instantaneous wind speeds over the previous twelve hours , given in ms , see e.g. ) for 152 stations in the automated surface observing network .the ensemble members are not exchangeable as they are generated with initial conditions from different sources .  in the present studywe investigate forecasts for calendar year 2008 with additional data from the last month of 2007 used for parameter estimation . after removing days and locations with missing data 101 stations remain where the number of days for which forecasts and validating observations are available varies between 160 and 291 .figure [ fig : fig1]c shows the verification rank histogram of the raw ensemble where , similar to the previous cases , one can again observe a strongly underdispersive character .the ensemble coverage equals , whereas the nominal coverage for eight ensemble members equals , that is .as mentioned in the introduction , the emos predictive distribution of a future weather quantity is a single parametric distribution , where the parameters depend on the ensemble .for example , a normal distribution provides a fairly good fit for temperature and pressure , whereas wind speed requires a distribution with non - negative support .in what follows , we consider three different types of emos models : standard emos models based on a single parametric family , combination models that select one of multiple parametric distributions based on the values of suitable covariates , and new mixture models .models based on single parametric families for wind speed which employ truncated normal , log - normal or generalized extreme value distributions , as well as combination models selecting one of these distribution based on covariates have been explored in previous works ( see e.g. * ? ? ?* ; * ? ? ?* ; * ? ? ?* ) and are reviewed in section [ subs : subs3.1 ] . as an alternative choice , we propose new mixture models based on a weighted mixture of truncated normal and log - normal distributions in section [ subs : subs3.2 ] .the basic emos models from previous studies are used as benchmark models in order to assess the predictive performance of the novel mixture models in section [ sec : sec4 ] .    the emos model introduced by based on a truncated normal ( tn ) distribution , i.e. , the predictive distribution is where  denote the ensemble of distinguishable forecasts of wind speed for a given location and time ,  stands for the ensemble mean , and  denotes the tn distribution with location  , scale  , and cut - off at zero having probability density function ( pdf ) where  and  are the pdf and the cumulative distribution function ( cdf ) of the standard normal distribution , respectively .  as an alternative to the tn distribution propose the use of a log - normal ( ln ) distribution where the mean  and variance  are affine functions of the ensemble members and ensemble variance , respectively ,i.e. , usually the pdf of the ln distribution  is expressed using location  and shape  parameters and has the form however , it can be easily expressed in terms of mean and variance with the help of transformations propose an emos approach based on a generalized extreme value ( gev ) distribution , however , this model has the disadvantage of assigning positive probability to negative wind speed values ( see , e.g. , * ? ? ?location and scale / shape parameters of models and can be estimated from the training data consisting of ensemble members and verifying observations from the preceding  days , by optimizing an appropriate verification score ( see section [ subs : subs3.3 ] ) .emos models and are valid only in the cases when the sources of the ensemble members are clearly distinguishable , which is the case for the uwme described in section [ subs : subs2.3 ] or for the consortium for small - scale modeling germany ( cosmo - de ) ensemble of the german meteorological service .however , in most of the currently used epss some members are obtained with the help of perturbations of the initial conditions .these members are statistically indistinguishable and can be considered as exchangeable .this the case for the ecmwf and aladin - huneps ensembles described in sections [ subs : subs2.1 ] and [ subs : subs2.2 ] , respectively .suppose we have  ensemble members divided into  exchangeable groups , where the  group contains  ensemble members such that  .in such situations the ensemble members within an exchangeable group should share the same parameters resulting in a tn model and a ln model with mean and variance where  denotes the member of the group .ln and gev distributions have heavier upper tails than the tn distribution and are therefore more appropriate to model high wind speed values . to combine this advantage with the good performance of the tn model for low and medium wind speeds and also examined a regime - switching combination method , where either a tn or a heavy - tail distribution is used depending on the median value of the ensemble forecast .if the ensemble median is below a given threshold , wind speed is modeled by a tn distribution , otherwise a gev or ln distribution is employed . the optimal threshold value for a given eps is determined during a preliminary study and it is then fixed over the whole data set .the problem with this approach is that the threshold parameter is static ( rarely updated ) and can not adapt to the changes in the ensemble . also consider a more adaptive method where the threshold parameter is re - estimated as a fixed quantile of the ensemble medians in the corresponding training period for each forecast date .however , this approach is computationally more demanding without yielding a significant improvement in predictive performance .emos models based on combining two parametric families by exclusively selecting one of them at each forecast instance also suffer from the drawback that a suitable covariate has to be chosen as a selection criterion .this necessary step limits the flexibility of the combination models in practice as the adequacy of covariates might depend on the data set at hand . while the ensemble median works reasonably well in the data sets considered in this article , this observation might change for different epss .   in order to combine the advantages of lighter and heavier - tailed distributions and to avoid the aforementioned problems in the process , we introduce new emos models based on weighted mixtures of two parametric distributions .in particular , we propose to model wind speed with a weighted mixture of models and ( or and for exchangeable ensemble members ) resulting in the predictive pdf where the dependence of parameters  and  on the ensemble are given by ( or ) and ( or ) and , respectively . in case of model location and scale / shape parameters of the tn and ln models together with the weight  $ ] are estimated simultaneously by optimizing some verification score over the training data .note that instead of a ln distribution , in one can incorporate other non - negative laws with heavy right tails .a natural choice would be the generalized pareto distribution ( gpd ) used in extreme value theory ( see , e.g. , * ? ? ?* ) , however , tests for the ensemble forecasts considered here indicate a worse predictive performance of the tn - gpd model compared with the tn - ln mixture and the benchmark models .  in comparison with the basic emos models proposed in previous work , the new mixture models exhibit desirable properties from a theoretical perspective as they do not require the exclusive choice of one of multiple parametric families and are more flexible than models based on single parametric distributions .their advantages from a practical perspective such as a significantly improved calibration will be demonstrated in section [ sec : sec4 ] .the main aim of probabilistic forecasting is to access the maximal sharpness of the predictive distribution subject to calibration .calibration means a statistical consistency between the predictive distributions and the validating observations whereas sharpness refers to the concentration of the predictive distribution .a straightforward way to check the calibration of a probabilistic forecast is the use of probability integral transform ( pit ) histograms .the pit is defined as the value of the predictive cdf evaluated at the verifying observations and the closer the histogram to the uniform distribution , the better the calibration .pit histograms are continuous analogues of verification rank histograms , a comparison of these histograms can thus be used as a measure of the possible improvements due to statistical post - processing .  apart from the visual inspection of pit histograms , formal statistical test of uniformity can be used to assess calibration . as the pit values of multi - step ahead probabilistic forecast exhibit serial correlation ( see , e.g. , * ? ? ?* ) and the probabilistic forecasts can not be assumed to be independent in space and time , we employ a moment - based test of uniformity proposed by which accounts for dependence in the pit values . in particular, we use the test of that has been demonstrated to have superior size and power properties compared to alternative choices . due to the large sample size in case of the ecmwf and uwme data , the null hypothesis of uniformityis rejected for all post - processing models . however , as our focus lies on the comparative assessment of calibration , we report bootstrap estimates of the rejection rates of the test based on 10000 random samples of size 2500 each .if a model exhibits superior calibration , the null hypothesis of uniformity should be rejected in fewer cases compared to a model with inferior calibration .  another approach to assess calibrationis the investigation of the coverage of the  ,  central prediction interval , defined as the proportion of validating observations located between the lower and upper  quantiles of the predictive distribution , where  is chosen to match the nominal coverage of the raw ensemble ( ecmwf : ; aladin - huneps : ; uwme : ) .the coverage of a calibrated predictive pdf should be around  and the proposed choices of  allow direct comparisons with the raw ensembles .further , the average widths of these central prediction intervals provide information about the sharpness of the predictive distributions .calibration and sharpness can also be addressed simultaneously with the help of scoring rules which measure the predictive performance by numerical values assigned to pairs of probabilistic forecasts and observations .the most popular scoring rules are the logarithmic score ( logs ) , that is the negative logarithm of the predictive pdf  evaluated at the verifying observation , and the continuous ranked probability score ( crps ; * ? ? ?* ; * ? ? ?the crps of a cdf  and an observation  is defined as where  denotes the indicator of a set  .while both the crps and the logarithmic score are proper scoring rules , the crps can be expressed in the same unit as the observation .further , point forecasts such as emos and ensemble medians and means are evaluated with the help of mean absolute errors ( maes ) and root mean squared errors ( rmess ) .we remark that the former is optimal for the median , whereas the latter is for the mean .finally , to evaluate the goodness of fit of probabilistic forecasts to high wind speed values a useful tool to be considered is the threshold - weighted continuous ranked probability score ( twcrps ) introduced by , where  is a weight function . obviously,  yields the traditional crps defined by , while one may set   to address wind speeds above a given threshold  .similar to and , where the upper tail behaviors of regime - switching emos models are investigated , we consider threshold values approximately corresponding to the 90th , 95th and 99th percentiles of the wind speed observations .one can also quantify the improvement in twcrps with respect to some reference predictive cdf  with the help of the threshold - weighted continuous ranked probability skill score ( twcrpss ; see , e.g. , * ? ? ?* ) defined as this score is obviously positively oriented , and in this study the predictive cdf corresponding to the classical tn model is used as a reference .  in order to assess the statistical significance of observed score differences between the models we use formal statistical tests of equal predictive performance .diebold - mariano ( dm ; * ? ? ?* ) tests allow to account for dependence in the forecast errors and are widely used in the econometric literature . denote the mean values of a proper scoring rule   for two competing probabilistic forecasts   and   by   and  ,  respectively , where   denotes the forecast cases in a test set  of size  .the test statistic of the dm test is given by   where  is a suitable estimator of the asymptotic standard deviation of the sequence of score differences  .under some weak regularity assumptions ,  asymptotically follows a standard normal distribution under the null hypothesis of equal predictive performance .negative values of  indicate a better predictive performance of  ,  whereas   is preferred in case of positive values of  .the statistical significance of the observed values of the test statistic can be assessed by computing the corresponding p - values under the null hypothesis . following suggestions of and , we estimate the autocovariance of the sequence of score differences in by the sample autocovariance up to lag in case of step ahead forecasts .the results of dm tests based on the logs , the crps and the twcrps are discussed in section [ sec : sec4 ] for the individual ensembles .as mentioned in section [ sec : sec1 ] , the predictive skills of the mixture model are tested on the 50 member ecmwf ensemble , the aladin - huneps ensemble of the hms and the eight - member uwme .the three epss differ both in generation of the ensemble members and in the predicted wind speed quantity .model performances are evaluated with the help of the verification scores given in section [ subs : subs3.3 ] .the basic tn , ln and tn - ln regime - switching combination emos models proposed in previous studies are used as benchmark models to assess the predictive performance of the new mixture models . for a detailed evaluation and comparison of the different basic emos models , we refer to .we further compare the forecasts based on post - processing with the raw ensemble and with climatological forecasts where the observations of the training period are considered as an ensemble .following the ideas of and the parameters of the tn , ln and tn - ln mixture models are estimated by minimizing the mean crps of the predictive cdfs and corresponding validating observations over the training period .however , in case of model the crps can be evaluated only numerically , resulting in very long optimization procedures .numerical minimization of the right - most part of equation , i.e. ,   leads to slightly lower computation times and better verification scores compared to minimizing alternative representations of the crps integral .  due to the large computational costs of minimum crps estimation, we also investigate maximum likelihood ( ml ) estimation of the parameters .ml estimation corresponds to minimizing the mean logarithmic score which has a simple and closed form . from a theoretical statistics perspective , both estimation approaches fit into a general framework of optimum score estimation and share asymptotic properties such as consistency , see for details . in applications to post - processing ensemble forecasts , the crps is often seen as the more appropriate scoring rule for parameter estimation due to the lower sensitivity to outliers and extreme events compared to the logs ( see , e.g. , * ? ? ?* ; * ? ? ?* ) . in figures and tables ,the corresponding mixture models are denoted by tn - ln mix .( crps ) and tn - ln mix .( ml ) .finally , in order to ensure the comparability with the benchmark models the same training period lengths and for the tn - ln regime - switching models the same thresholds and parameter estimation techniques as in are employed .   as the fifty members of the ecmwf ensemble are fully exchangeable , the dependencies of the parameters of the tn and ln distributions on the ensemble members are specified by and , respectively , with  and  .the preliminary study by suggested that the optimal training period length for this particular data set is 20 days , whereas the optimal value of the threshold parameter  of the tn - ln regime - switching combination model equals 8 ms , resulting in the use of an ln distribution in about of the forecast cases .as mentioned in section [ subs : subs2.1 ] , model verification is performed on 83220 forecast cases from the one year period between 1 may 2010 and 30 april 2011 .figure [ fig : fig2 ] showing the verification rank histogram of the raw ensemble and the pit histograms of the investigated emos models clearly indicates that statistical post - processing significantly improves the calibration of the raw ensemble .however , the histograms of tn and tn - ln regime - switching models are still biased , to a smaller extent the same applies for the ln model , whereas the pit values of mixture model with both parameter estimation methods suggest a better fit to the desired uniform distribution ..5 truecm  .bootstrap estimates of rejection rates of the test of uniformity based on 10000 random samples of size 2500 each at the 0.05 level for the different data sets .lower rejection rates correspond to better calibrated forecasts with the null hypothesis of uniformity being rejected on fewer occasions .[ cols="<,^,^,^,^,^",options="header " , ]   verification scores for probabilistic and point forecasts and the coverage and average width of central prediction intervals are reported in table [ tab : tab3 ] .compared with the raw ensemble and climatology post - processed forecast exhibit the same behavior as before : improved predictive skills and better calibration . in general , models based on combinations of both investigated distributions outperform the tn and ln methods , the smallest crps and mae values and the best coverage , combined with a rather narrow central prediction interval , belong to the regime - switching approach , while the mixture model with parameters optimizing the mean crps provides the lowest twcrps and rmse scores ..5 truecm  lccc & tn & ln & tn - ln r.s . +  + tn - ln mix .( crps ) & * -7.63 * & * -13.62 * & * -4.68 * + tn - ln mix . ( ml ) & * -18.44 * & * -16.45 * & * -12.24 * +  + tn - ln mix .( crps ) & * -15.73 * & * -16.70 * & -0.95 + tn - ln mix .( ml ) & * -5.62 * & * -6.52 * & * 2.24 * +  + tn - ln mix .( crps ) & * -7.68 * & * -4.63 * & 1.03 + tn - ln mix .( ml ) & * -7.04 * & * -5.11 * & 1.06 +  the results of dm tests of equal predictive performance are summarized in table [ tab : dm - uwme ] . in terms of all employed scoring rules ,the mixture models exhibit significantly better predictive performance compared to the tn and ln models . as before ,the results for the comparisons with the tn - ln combination model are mixed and depend on the scoring rule .while the mixture models show significantly better results in terms of the logs , the combination model is preferred in terms of the cprs and its threshold - weighted version .this desirable behavior of the mixture models for high wind speeds can also be observed in figure [ fig : fig9 ] where the twcrpss values of the ln , tn - ln regime - switching and mixture models with respect to the tn emos reference model are plotted as functions of the threshold .up to threshold ms the regime - switching method slightly outperforms the mixture model , whereas above it this advantage disappears .in contrast to the ecmwf and aladin - huneps ensembles , the weights of the tn component of the two versions of model plotted in figure [ fig : fig10 ] show a positive correlation of .finally , for the uwme the parameter estimates of  and  exhibit stronger correlations than the estimated location and scale parameters  and  of the tn component , the corresponding values are and , respectively .a new emos model for post - processing ensemble forecasts of wind speed is introduced , where the predictive pdf is a weighted mixture of a truncated normal and a log - normal distribution with location and scale / shape parameters depending on the ensemble .model parameters and mixture weight are estimated simultaneously by optimizing either the mean continuous ranked probabilistic score or the mean logarithmic score ( ml estimation ) of the predictive distribution over the training data .the mixture models are tested on three data sets of wind speed forecasts which differ in the generation of the ensemble members and the predicted wind quantities .the predictive skills of the new model are compared with those of the tn based emos method , the ln and the tn - ln regime - switching emos models , the raw ensemble and the climatological forecasts with the help of graphical tools , appropriate verification scores and formal statistical tests of calibration and equal predictive performance .the presented case studies clearly show that compared with the raw ensemble and climatology , statistical post - processing results in a significant improvement in calibration of probabilistic and accuracy of point forecasts .  in a comparative view of the different emos modelsit can be observed that the tn - ln regime - switching combination model and the new mixture models significantly outperform the simple emos models based on single tn and ln distributions and provide much better calibrated probabilistic forecasts as demonstrated by formal statistical tests .the novel mixture models are able to keep up with the tn - ln regime - switching combination method in terms of the various verification scores .further , the results of formal tests of uniformity indicate a superior calibration of the forecasts produced by the mixture models compared with the combination model .no substantial difference can be observed between the results corresponding to the two parameter estimation methods of the mixture model , ml estimation results in slightly worse verification scores but provides better calibrated forecasts .compared with the tn - ln regime - switching combination model , the proposed mixture models exhibit desirable properties from both a theoretical as well as an applied perspective .they are more flexible in that they do not require the exclusive choice of one of the parametric families as forecast distribution .further , it is not necessary to determine suitable covariates for the model selection , or to estimate the model selection threshold over a training period .essential part of this work was made during a visit of sndor baran at the heidelberg institute for theoretical studies .the research stay in heidelberg was funded by the daad program `` research stays for university academics and scientists , 2015 '' .sndor baran was also supported by the jnos bolyai research scholarship of the hungarian academy of sciences .sebastian lerch gratefully acknowledges support by the volkswagen foundation within the program `` mesoscale weather extremes theory , spatial modelling and prediction ( wex - mop ) '' , and by the klaus tschira foundation .the authors thank tilmann gneiting , alexander jordan and fabian krger for helpful discussions , and fabian krger for providing ` r ` code for the test .the authors further thank the university of washington muri group for providing the uwme data and mihly szcs from the hms for providing the aladin - 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Abstract: ensemble model output statistics ( emos ) is a statistical tool for post - processing forecast ensembles of weather variables obtained from multiple runs of numerical weather prediction models in order to produce calibrated predictive probability density functions ( pdfs ) . the emos predictive pdf is given by a parametric distribution with parameters depending on the ensemble forecasts . we propose an emos model for calibrating wind speed forecasts based on weighted mixtures of truncated normal ( tn ) and log - normal ( ln ) distributions where model parameters and component weights are estimated by optimizing the values of proper scoring rules over a rolling training period . the new model is tested on wind speed forecasts of the 50 member european centre for medium - range weather forecasts ensemble , the 11 member aire limite adaptation dynamique dveloppement international - hungary ensemble prediction system ensemble of the hungarian meteorological service and the eight - member university of washington mesoscale ensemble , and its predictive performance is compared to that of various benchmark emos models based on single parametric families and combinations thereof . the results indicate improved calibration of probabilistic and accuracy of point forecasts in comparison with the raw ensemble and climatological forecasts . the mixture emos model significantly outperforms the tn and ln emos methods , moreover , it provides better calibrated forecasts than the tn - ln combination model and offers an increased flexibility while avoiding covariate selection problems .   _ key words : _ continuous ranked probability score , ensemble calibration , ensemble model output statistics , truncated normal distribution , log - normal distribution .


the _ communication complexity _ of a function is defined as the minimum amount of communication necessary among two parties , conventionally referred to as alice and bob , in order for , say , bob to acquire the value of , where , initially , alice is given and bob is given .this scenario was introduced by yao and has been widely studied ( see for a survey ) .there are a number of technical choices in the model , such as : whether the communication cost is taken as the worst - case , or the average - case with respect to some probability distribution ; whether the protocols are deterministic or probabilistic ( and , for probabilistic protocols , whether the parties have independent random sources or a shared random source ) ; and , what correctness probability is required .the communication complexity of the _ inner product modulo two ( ip ) _ function is fairly well understood in the above `` classical '' models .for worst - case inputs and deterministic errorless protocols , the communication complexity is and , for randomized protocols ( with either an independent or a shared random source ) , uniformly distributed or worst - case inputs , and with error probability required , the communication complexity is  ( see also ) .  in 1993 , yao introduced a variation of the above classical communication complexity scenarios , where the parties communicate with _ qubits _ , rather than with bits .protocols in this model are at least as powerful as probabilistic protocols with independent random sources .kremer showed that , in this model , the communication complexity of is , whenever the required correctness probability is for a constant ( kremer attributes the proof methodology to yao ) .cleve and buhrman ( see also ) introduced another variation of the classical communication complexity scenario that also involves quantum information , but in a different way . in this model ,alice and bob have an initial supply of particles in an entangled quantum state , such as einstein - podolsky - rosen ( epr ) pairs , but the communication is still in terms of classical bits .they showed that the entanglement enables the communication for a specific problem to be reduced by one bit .any protocol in yao s qubit model can be simulated by a protocol in this entanglement model with at most a factor two increase in communication : each qubit can be `` teleported '' by sending two classical bits in conjunction with an epr pair of entanglement . on the other hand, we are aware of no similar simulation of protocols in the entanglement model by protocols in the qubit model , and , thus , the entanglement model is potentially stronger .  in this paper , we consider the communication complexity of in two scenarios : with prior entanglement and qubit communication ; and with prior entanglement and classical bit communication . as far as we know, the proof methodology of the lower bound in the qubit communication model without prior entanglement does not carry over to either of these two models .nevertheless , we show lower bounds in these models .  to state our lower bounds more precisely , we introduce the following notation .let be a communication problem , and .let denote the communication complexity of in terms of _ qubits _ , where quantum entanglement is available and the requirement is that bob determines the correct answer with probability at least ( the superscript is intended to highlight the fact that prior entanglement is available ) .also , let denote the corresponding communication complexity of in the scenario where the communication is in terms of _ bits _ ( again , quantum entanglement is available and bob is required to determine the correct answer with probability at least ) .when , we refer to the protocols as _ exact _ , and , when , we refer to them as _ bounded - error _ protocols . with this notation ,our results are : note that all the lower bounds are whenever is held constant .also , these results subsume the lower bounds in , since the qubit model defined by yao differs from the bounded - error qubit model defined above only in that it does not permit a prior entanglement .our lower bound proofs employ a novel kind of `` quantum '' reduction between protocols , which reduces the problem of communicating , say , bits of information to the problem .it is noteworthy that , in classical terms , it can be shown that there is no such reduction between the two problems .the appropriate cost associated with communicating bits is then lower - bounded by the following nonstandard consequence of holevo s theorem .a slight generalization of theorem 1 is described and proven in the appendix .it should be noted that , since quantum information subsumes classical information , our results also represent new proofs of nontrivial lower bounds on the _ classical _ communication complexity of , and our methodology is fundamentally different from those previously used for classical lower bounds .finally , with respect to the question of whether quantum entanglement can _ ever _ be advantageous for protocols computing , we present a curious probabilistic scenario with where prior entanglement enables one bit of communication to be saved .in this section , we consider exact qubit protocols computing , and prove eq .( [ q ] ) .note that the upper bound follows from so - called `` superdense coding '' : by sending qubits in conjunction with epr pairs , alice can transmit her classical bits of input to bob , enabling him to evaluate . for the lower bound , we consider an arbitrary exact qubit protocol that computes , and convert it ( in two stages ) to a protocol for which theorem 1 applies .  for convenience ,we use the following notation . if an -qubit protocol consists of qubits from alice to bob and qubits from bob to alice then we refer to the protocol as an -qubit protocol .a _ clean protocol _ is a special kind of qubit protocol that follows the general spirit of the reversible programming paradigm in a quantum setting .namely , one in which all qubits incur no net change , except for one , which contains the answer .in general , the initial state of a qubit protocol is of the form where is the state of the entangled qubits shared by alice and bob , and the states can be regarded as `` ancillas '' . at each turn, a player performs some transformation ( which , without loss of generality , can be assumed to be unitary ) on all the qubits in his / her possession and then sends a subset of these qubits to the other player .note that , due to the communication , the qubits possessed by each player varies during the execution of the protocol . at the end of the protocol ,bob measures one of his qubits which is designated as his _output_.  we say that a protocol which exactly computes a function is _ clean _ if , when executed on the initial state results in the final state ( where the addition is mod 2 ) . the `` input '' , the ancilla , and initial entangled qubits will typically change states during the execution of the protocol , but they are reset to their initial values at the end of the protocol .  it is straightforward to transform an exact -qubit protocol into a clean -qubit protocol that computes the same function . to reset the bits of the input , the ancilla , and the initial entanglement, the protocol is run once , except the output is not measured , but recorded and then the protocol is run in the _ backwards _ direction to `` undo the effects of the computation '' .the output is recorded on a _ new _ qubit of bob ( with initial state ) which is control - negated with the output qubit of bob ( that is in the state ) as the control .note that , for each qubit that bob sends to alice when the protocol is run forwards , alice sends the qubit to bob when run in the backwards direction .running the protocol backwards resets all the qubits except bob s new one to their original states .the result is an -qubit protocol that maps state ( [ initial ] ) to state ( [ final ] ) .we now show how to transform a clean -qubit protocol that exactly computes for inputs of size , to an -qubit protocol that transmits bits of information from alice to bob .this is accomplished in four stages :  1 .bob initializes his qubits indicated in eq .( [ initial ] ) with and .bob performs a hadamard transformation on each of his first qubits .alice and bob execute the clean protocol for the inner product function .bob again performs a hadamard transformation on each of his first qubits .let denote the state of bob s first qubits after the stage .then where , in eq .( [ st3 ] ) , the substitution has been made ( and arithmetic over bits is taken mod 2 ) .the above transformation was inspired by the reading of ( see also ) .since the above protocol conveys bits of information ( namely , ) from alice to bob , by theorem 1 , we have .since this protocol can be constructed from an arbitrary exact -qubit protocol for , this establishes the lower bound of eq .( [ q ] ) .note that , classically , no such reduction is possible .for example , if a clean protocol for is executed in any classical context , it can never yield more than one bit of information to bob ( whereas , in this quantum context , it yields bits of information to bob ) .in this section we consider bounded - error qubit protocols for , and prove eq .( [ qe ] ) .assume that some qubit protocol computes correctly with probability at least , where .since is not exact , the constructions from the previous section do not work exactly .we analyze the extent by which they err .first , the construction of section 2.1 will not produce a protocol in clean form ; however , it will result in a protocol which _ approximates _ an exact clean protocol ( this type of construction was previously carried out in a different context by bennett _ ) .  denote the initial state as also , assume that , in protocol , bob never changes the state of his input qubits ( so the first qubits never change ) .this is always possible , since he can copy into his ancilla qubits at the beginning . after executing until just before the measurement occurs , the state of the qubits must be of the form where and . in the above ,the qubit is the designated output , denotes the inner product of and , and denotes the negation of this inner product . in general , , , , and depend on and .  now , suppose that the procedure described in section 2.1 for producing a clean protocol in the exact case is carried out for .since , in general , the answer qubit is not in the state even in a pure basis state this does not produce the final state however , let us consider the state that is produced instead .after introducing the _ new _ qubit , initialized in basis state , and applying , the state is after applying the controlled - not gate , the state is finally , after applying in reverse to this state ,the final state is where   note that the vector is the difference between what an exact protocol would produce ( state ( [ exact ] ) ) and what is obtained by using the inexact ( probabilistic ) protocol ( state ( [ finish ] ) ) .there are some useful properties of the states .first , as varies , the states are orthonormal , since is a factor in each such state ( this is where the fact that bob does not change his input qubits is used ) .also , , since only the factor in each such state depends on .call the above protocol .now , apply the four stage reduction in section 2.2 , with in place of an exact clean protocol .the _ difference _ between the state produced by using and using an exact clean protocol first occurs after the third stage and is which has magnitude bounded above by , since , for each , , and the states are orthonormal .also , the magnitude of this difference does not change when the hadamard transform in the fourth stage is applied .thus , the final state is within euclidean distance from consider the angle between this final state and ( [ correct ] ) .it satisfies , from which it follows that .therefore , if bob measures his first qubits in the standard basis , the probability of obtaining is .  now , suppose that are uniformly distributed. then fano s inequality ( see , for example , ) implies that bob s measurement causes his uncertainty about to drop from bits to less than bits , where is the binary entropy function .thus , the mutual information between the result of bob s measurement and is at least bits . by theorem 1 ,the communication from alice to bob is at least qubits , which establishes eq .( [ qe ] ) .in this section , we consider exact and bounded - error bit protocols for , and prove eqs .( [ c ] ) and ( [ ce ] ) .recall that any -qubit protocol can be simulated by a -bit protocol using teleportation ( employing epr pairs of entanglement ) .also , if the communication pattern in an -bit protocol is such that an even number of bits is always sent during each party s turn then it can be simulated by an -qubit protocol by superdense coding ( which also employs epr pairs ) .however , this latter simulation technique can not , in general , be applied directly , especially for protocols where the parties take turns sending single bits .we can nevertheless obtain a slightly weaker simulation of bit protocols by qubit protocols for that is sufficient for our purposes .the result is that , given any -bit protocol for ( that is ,  instances of size ) , one can construct an -qubit protocol for .this is accomplished by interleaving two executions of the bit protocol for to compute two independent instances of inner products of size .we make two observations .first , by taking the sum ( mod 2 ) of the two results , one obtains an inner product of size .second , due to the interleaving , an even number of bits is sent at each turn , so that the above superdense coding technique can be applied , yielding a -qubit protocol for .now , eq . ( [ q ] ) implies , which establishes the lower bound of eq .( [ c ] ) ( and the upper bound is trivial ) .if the same technique is applied to any -bit protocol computing with probability , one obtains an -qubit protocol that computes with probability . applying eq .( [ qe ] ) here , with replacing and replacing , yields . for , a better boundis obtained by simply noting that ( since qubits can always be used in place of bits ) , and applying eq .( [ qe ] ) .this establishes eq .( [ ce ] ) .here we will show that in spite of the preceding results , it is still possible that a protocol which uses prior entanglement outperforms all possible classical protocols .this improvement is done in the probabilistic sense where we look at the number of communication bits required to reach a certain reliability threshold for the function .this is done in the following setting .both alice and bob have a 2 bit vector and , for which they want to calculate the inner product modulo 2 : with a correctness - probability of at least .it will be shown that with entanglement alice and bob can reach this ratio with 2 bits of communication , whereas without entanglement 3 bits are necessary to obtain this success - ratio .initially alice and bob share a joint random coin and an epr - like pair of qubits and : with these attributes the protocol goes as follows .first alice and bob determine by a joint random coin flip who is going to be the ` sender ' and the ` receiver ' in the protocol .( we continue the description of the protocol by assuming that alice is the sender and that bob is the receiver . ) after this , alice ( the sender ) applies the rotation on her part of the entangled pair and measures this qubit in the standard basis .the result of this measurement is then sent to bob ( the receiver ) who continues the protocol .if bob has the input string ` 00 ' , he knows with certainty that the outcome of the function is zero and hence he concludes the protocol by sending the bit to alice .otherwise , bob performs the rotation on his part of the entangled pair and measure it in the standard basis yielding the value .now bob finishes the protocol by sending to alice the bit .using the rotations shown below and bearing in mind the randomization process in the beginning of the protocol with the joint coin flip , this will be a protocol that uses only 2 bits of classical communication and that gives the correct value of with a probability of at least for every possible combination of and .the unitary transformations used by the sender in the protocol are : whereas the receiver uses one of the three rotations : the matrices were found by using an optimization program that suggested certain numerical values .a closer examination of these values revealed the above analytical expressions .take the probability distribution on the input strings and , defined by : it is easily verified that for this distribution , every _ deterministic _ protocol with only two bits of communication will have a correctness ratio of at most . using theorem 3.20 of , this shows that every possible randomized protocol with the same amount of communication will have a success ratio of at most .( it can also be shown that this bound is tight but we will omit that proof here . )this implies that in order to reach the requested ration of , at least three bits of communication are required if we are not allowed to use any prior entanglement .a similar result also holds for qubit protocols without prior entanglement .this can be seen by the fact that after alice applied the rotation and measured her qubit with the result , she knows the state of bob s qubit exactly .it is therefore also possible to envision a protocol where the parties assume the measurement outcome ( this can be done without loss of generality ) , and for which alice simply sends this qubit to bob , after which bob finishes the protocol in the same way as prescribed by the ` prior entanglement'-protocol .the protocol has thus become as follows .first alice and bob decide by a random joint coin flip who is going to be the sender and the receiver in protocol .( again we assume here that alice is the sender . ) next , alice ( the sender ) sends a qubit ( according to the input string of alice and the table [ qtable ] ) to the receiver bob who continues the protocol . if bob has the input string , he concludes the protocol by sending a zero bit to alice . in the other case, bob applies the rotation to the received qubit , measures the qubit in the standard basis , and sends this measurement outcome to alice as the answer of the protocol . by doing so, the same correctness - probability of is reached for the function with two qubits of communication , whereas the classical setting requires 3 bits of communication as shown above .we would like to thank gilles brassard , harry buhrman , peter hyer , and tal mor for their comments about this research .r.c . would like to thank the laboratoire dinformatique thorique et quantique , universit de montral for their gracious hospitality while this research was initiated .thanks the office of naval research ( grant no .n00014 - 93 - 1 - 0116 ) .  99  h. araki and e.h .lieb , `` entropy inequalities '' , _ commun .phys . _ , vol . 18 , 1970 , pp . 160170.  c.h .bennett , e. bernstein , g. brassard , u. vazirani , `` strengths and weaknesses of quantum computing '' , _ siam j. on comput ._ , vol . 26 , no . 5 , 1997 , pp . 15101523 .bennett , g. brassard , c. crpeau , r. josza , a. peres , w.k .wootters , `` teleporting an unknown quantum state via dual classical and einstein - podolsky - rosen channels '' , _ phys ._ , vol . 70 , 1993 , pp .18951899 .bennett and s.j .wiesner , `` communication via one- and two - particle operators on einstein - podolsky - rosen states '' , _ phys ._ , vol .69 , no . 20 , 1992 , pp .28812884 .e. bernstein , u. vazirani , `` quantum complexity theory '' , _siam j. comput ._ , vol . 26 , no . 5 , 1997 , pp .14111473 .h. buhrman , r. cleve , and w. van dam , `` quantum entanglement and communication complexity '' , preprint available from the lanl quant - ph archive 9705033 , 1997 .b. chor and o. goldreich , `` unbiased bits from weak sources of randomness and probabilistic communication complexity '' , _ siam j. on comput ._ , vol .17 , no . 2 , pp .230261 , 1988 .r. cleve and h. buhrman , `` substituting quantum entanglement for communication '' , _ phys .a _ , vol .56 , no . 2 , pp .12011204 , 1997 .t.m . cover and j.a .thomas , _ elements of information theory _ , john wiley and sons , 1991 .a. einstein , b. podolsky , and n. rosen , `` can quantum - mechanical description of physical reality be complete ? '' , _ phys ._ , vol . 47 , 1935 , pp .777780 .holevo , `` bounds for the quantity of information transmitted by a quantum communication channel '' , _ problemy peredachi informatsii _ , vol . 9 , no . 3 , 1973 , pp .english translation _ problems of information transmission _, vol . 9 , 1973 , pp .177183 .i. kremer , `` quantum communication '' , master s thesis , the hebrew university of jerusalem , 1995 .e. kushilevitz and n. nisan , _ communication complexity _ , cambridge university press , 1997 .terhal and j.a .smolin , `` superfast quantum algorithms for coin weighing and binary search problems '' , preprint available from the lanl quant - ph archive 9705041 , 1997 .b. schumacher , m. westmoreland and w. k. wootters , `` limitation on the amount of accessible information in a quantum channel '' , _ phys ._ , vol . 76 , 1996 , pp . 34533456.  a.c .yao , `` some complexity questions related to distributed computing '' , _ proc . of the 11th ann .acm symp . on theory of computing _ , 1979 ,. 209213 .yao , `` quantum circuit complexity '' , _ proc . of the 34th ann .ieee symp . on foundations of computer science _ , 1993 ,in this appendix , we present results about the quantum resources required to transmit classical bits between two parties when two - way communication is available .these results are used in the main text in the proof of the lower bound on the communication complexity of the inner product function , and may also be of independent interest .graphically , the capacity region for the above communication problem is shown in fig .[ fig : capacity ] . note the difference with the classical result for communication with bits , where the capacity region is given by the equation ; that is , classically , communication from bob to alice does not help .( 4.5,5 ) ( 1,1)(0,1)4 ( 1,1)(1,0)4 ( 0.2,4.5) ( 4.5,0.2) ( 0.5,4) ( 0,2.5) ( 2.3,0.4) ( 1,4)(1,-1)1.5 ( 2.5,2.5)(1,0)2.5 ( 1,4)(4,0 ) ( 4.84,4)(1,0)0.16 ( 2.3,3.5)capacity region  the sufficiency of eqns .( [ app1 ] ) and ( [ app2 ] ) follows from the superdense coding technique .the nontrivial case is where .bob prepares epr pairs and sends one qubit of each pair to alice , who can use them in conjunction with sending qubits to bob to transmit bits to bob .alice uses her remaining allotment of qubits to transmit the remaining bits in the obvious way .  the proof that eqns .( [ app1 ] ) and ( [ app2 ] ) are necessary follows from an application of holevo s theorem , which we now review .suppose that a classical information source produces a random variable .depending on the value , , of , a quantum state with density operator is prepared .suppose that a measurement is made on this quantum state in an effort to determine the value of .this measurement results in an outcome .holevo s theorem states that the mutual information between and is bounded by the _ holevo bound _ where are the probabilities associated with the different values of , , and is the von neumann entropy function .the quantity on the right hand side of the holevo bound is known as the _ holevo chi quantity _ , .let be alice s bits of information , which is uniformly distributed over .without loss of generality , it can be assumed that the protocol between alice and bob is of the following form . for any value of , alice begins with a set of qubits in state and bob begins with a set of qubits in state .the protocol first consists of a sequence of steps , where at each step one of the following processes takes place .  1 .alice performs a unitary operation on the qubits in her possession .2 .  bob performs a unitary operation on the qubits in his possession .3 .  alice sends a qubit to bob .4 .  bob sends a qubit to alice .  after these steps , bob performs a measurement on the qubits in his possession , which has outcome .( note that one might imagine that the initial states could be mixed and that measurements could be performed in addition to unitary operations ; however , these processes can be simulated using standard techniques involving ancilla qubits . )let be the density operator of the set of qubits that are in bob s possession after steps have been executed . due to holevo s theorem, it suffices to upper bound the final value of is also bounded above by .we consider the evolution of and .initially , , since bob begins in a state independent of .now , consider how and change for each of the four processes above .  1 .this does not affect and hence has no effect on or .it is easy to verify that and are invariant under unitary transformations , so this does not affect and either .3 .  let denote bob s qubits after steps and denote the qubit that alice sends to bob at the step . by the subadditivity inequality and the fact that , for a single qubit , , .also , by the araki - lieb inequality , .it follows that and 4 .  in this case , is with one qubit traced out .it is known that tracing out a subsystem of any quantum system does not increase  , so .note also that for this process , by the araki - lieb inequality .now , since can only increase when alice sends a qubit to bob and by at most 2 , eq .( [ app1 ] ) follows . also , since can only increase when one party sends a qubit to the other and by at most 1 , eq .( [ app2 ] ) follows .this completes the proof of theorem 2 .

Abstract: we consider the communication complexity of the binary inner product function in a variation of the two - party scenario where the parties have an _ a priori _ supply of particles in an entangled quantum state . we prove linear lower bounds for both exact protocols , as well as for protocols that determine the answer with bounded - error probability . our proofs employ a novel kind of `` quantum '' reduction from a quantum information theory problem to the problem of computing the inner product . the communication required for the former problem can then be bounded by an application of holevo s theorem . we also give a specific example of a probabilistic scenario where entanglement reduces the communication complexity of the inner product function by one bit .


generating independent uniformly distributed samples over `` simple , '' sets such as norms , has been thoroughly studied and algorithms are available which efficiently solve this problem ; e.g. , see  . however , efficient generation of independent uniformly distributed samples over general sets remains a difficult problem .this especially true in the case where the set is of low volume , hard to localize and/or nonconvex or even disconnected .although an interesting problem on its own , efficient sample generation can be used for solving complex analysis and design problems .examples of these are chance - constrained optimization , robust optimization and multi - objective controller design , see e.g. and the many references therein .  in this paper , we present some preliminary results that aim at solving the sample generation problem over semialgebraic sets with non - empty interior .more precisely , we leverage recent results on polynomial approximation of indicator functions of sets defined by polynomial inequalities  to develop algorithms that  a.  generate independent uniformly distributed samples over the given semialgebraic set ; b.  have provable bounds on sample rejection rate ; c.  have a rejection rate tends to zero as the degree of the polynomial approximation of the indicator function increases .the problem of random sample generation has been the subject of active research .techniques for univariate generation techniques are discussed e.g. in .these methods , however , are not readily extendable to the sets usually encountered in robust control .hence , specific techniques for generating uniform ( or radially symmetric ) samples in the vector norm ball are discussed in . in the papers and methods for random sample generation in real and complex spectral norm ballsare developed ; see also for generation of matrices having a toeplitz structure .the generation of causal stable dynamic operator has been the subject of different studies . in various techniques for generating random samples inside the schur stability regionare discussed , while presents an algorithm for approximately generating uniform transfer functions in the ball .we remark that the methods discussed in this paper are non - asymptotic , contrary to the markov chain monte carlo techniques discussed for instance in and references therein . in these papers ,the desired distribution is obtained by simulating a random walk on a graph . andthe independence and uniformity properties are only guaranteed asymptotically .therefore , the methods discussed in this paper can be implemented on parallel and distributed architectures , see e.g.  .the paper outline is as follows .section [ sec : statement ] provides a precise definition of the problem addressed in this paper and describes some auxiliary results needed for the development of the proposed approach . the algorithm for sample generationis described in section [ sec : unifgen ] . since this algorithm requires generation of samples from a distribution with polynomial density , details on how one can do this efficientlyare given in section [ sec : pol_sample ] .in section [ sec : examples ] , we illustrate the performance of the proposed approach with a few academic examples .finally , in section [ sec : conclusion ] , concluding remarks are presented and further research directions are delineated .consider a compact basic semialgebraic set described by polynomial inequalities where , are given real multivariate polynomials .the problem we consider is the following : +  _ given the semialgebraic set defined in ( [ kset ] ) , generate independent identically distributed ( i.i.d . ) random samples uniformly distributed in ._  note that the considered setup is very general , and encapsulates many problems of interest to the control community . in particular , the algorithm presented in this paper can be used to generate uniform samples in the solution set of linear matrix inequalities ( lmis ) .indeed , it is a well - known fact that lmi sets are ( convex ) basic semialgebraic sets . to see this , consider the lmi set where stands for positive semidefinite and the matrix has size .a vector belongs to the lmi set if and only if all the principal minors of are nonnegative .this immediately leads to a set of polynomial inequalities in .we define by the vector space of multivariate real polynomials in variables of degree less than or equal to .the uniform density over a set of nonzero volume is defined as where denotes the indicator function of the set  and is the lebesgue measure ( volume ) of ; e.g. , see for details on lebesgue measures and integration .the idea at the basis of the method we propose is to find a suitable approximation of the set , using the framework introduced in . to this end , let us consider a polynomial of degree  where the sum ranges over all integer vectors of size summing up to or less .let us now introduce the polynomial super - level set the use of super - level sets as efficient nonconvex approximations of generic semialgebraic sets has been proposed in . in particular , the following optimization problem is considered the above problem amounts at finding the super - level set that better approximates , in terms of minimum volume , the original set .since is compact by assumption , for problem ( [ optp ] ) to have a finite minimum , it was further assumed in that a compact semialgebraic set is given such that and hence   in this paper , we additionally assume that the set is the cartesian product of one - dimensional sets , i.e.  .for instance , the set can be taken as the -dimensional hyper - rectangle }:=\{x \in { \mathbb r}^n : a_i\leq x_i \leq b_i , \ : i=1,2,\ldots , n \}.\ ] ] as noted in ( * ? ? ?* remark 1 ) , an outer - bounding box of a given semialgebraic set can be found by solving relaxations of the following polynomial optimization problems which compute the minimum and maximum value of each component of the vector over the semialgebraic set .note that arbitrarily tight lower and upper bounds can be obtained by means of the the techniques discussed e.g. in based on sos / moment convex relaxations .  the algorithm presented in this paper leverages some recent results presented in , which , as a side - result , provide an optimal polynomial approximation of indicator functions .more precisely , it was shown in that paper that problem ( [ optp ] ) can be approximated by the following convex optimization problem in particular , the following result holds . see ( * ? ? ?* lemma 1 ) .[ cvg ] the minimum of problem ( [ l1 ] ) monotonically converges from above to the minimum of problem ( [ optp ] ) , i.e. for all , and .in particular , the convergence follows from the fact that the optimal polynomial solution to problem ( [ l1 ] ) converges in , or equivalently almost uniformly in , to the indicator function as its degree goes to infinity .this crucial property is the one we will exploit in section [ sec : unifgen ] to construct an efficient rejection algorithm for generating uniform samples in .from the proof of ( * ? ? ?* lemma 1 ) it can be seen that , for any , the optimal solution to problem ( [ l1 ] ) has the property of being an _ upper approximation _ of the indicator function , that is for all .moreover , this approximation becomes tight when goes to infinity .therefore , this polynomial is a `` dominating density '' of the uniform density for all , that is there exists a value such that for all .hence , the rejection method from a dominating density , discussed for instance in ( * ? ? ?* section 14.3.1 ) , can be applied leading to the random sampling procedure described in the following algorithm .algorithm 1 : uniform sample generation in semialgebraic set   1 .  fora given integer , compute the solution of 2 .generate a random sample with density proportional to over .3 .  if go to step 1 . 4 .  generate a sample uniform on  ] ) , yielding the optimal solution as it can be seen , is `` dominating '' the indicator function for all .then , uniform random samples are drawn in the hypograph of .this is done by generating uniform samples distributed according to a probability density function ( pdf ) proportional to ( step 2 ) , and then selecting its vertical coordinate uniformly in the interval  ] .2 .  compute the unique root in  ] .  in step 2 ,the numerical computation of the root can be performed , up to a given accuracy , using some standard method such as bisection or newton raphson .we also remark that more efficient methods for generating samples from polynomial densities exist , see for instance the method in  , based on finite mixtures .as a first illustration of the ideas described in this paper , we consider the outer approximations by polynomial super - level sets of the third - degree discrete - time stability region obtained by solving the convex optimization problem in step 1 of algorithm 1 .tightness of these approximations is crucial for a good performance of the sample generation method .a third - degree monic polynomial with real coefficients is discrete - time stable ( i.e. all its roots lie in the interior of the unit disk of the complex plane ) if and only if its coefficient vector belongs to the interior of the basic semialgebraic set which is nonconvex , see e.g. ( * ? ? ?* example 4.3 ) . set in included in the bounding box \times[-3,3]\times[-1,1] ] . in figure[ fig : schur4 ] we represent the graph of the degree 10 polynomial constructed by solving optimization problem ( [ l1p ] ) with the polynomial positivity constraints replaced with polynomial sos constraints . from thiswe get the outer approximation used in our sampling algorithm .  ]   to demonstrate the behavior of algorithms 1 and 2 , we revisit a numerical example originally introduced in .the considered semialgebraic set is the two - dimensional nonconvex region described as : in , the outer - bounding box was considered .figure [ fig : cerone8 ] shows a two - dimensional plot of the indicator function , and the corresponding optimal solution for . the results of algorithm 1 are reported in figure [ fig : cerone8samples ] .the red points represent the points which have been discarded . to this regard, it is important to notice that also some point falling inside has been rejected .this is fundamental to guarantee uniformity of the discarded points .  ]defined in ( [ k - es ] ) , the pink area is the superlevel set .the red dots are the discarded samples .the remaining samples ( blue ) are uniformly distributed inside .[ fig : cerone8samples ] ]in this paper , a numerically efficient procedure for generating `` truly '' random samples inside a given ( possibly nonconvex ) semialgebraic set has been proposed .the algorithm is based on an acceptance / rejection scheme constructed upon an optimal polynomial superlevel set guaranteed to contain .a key feature of the method is that , for a given set and polynomial degree , this approximation , which is undoubtedly the most time - consuming step of the sampling scheme , can be constructed _ a priori _ and once for all , and then be used for online generation .the rejection rate is shown to become asymptotically optimal when the degree of the polynomial approximation increases .future work will concentrate on the application of this to specific fixed - order controller design problems .this work was partially supported by a bilateral project cnr - cnrs .

Abstract: we propose efficient techniques for generating independent identically distributed uniform random samples inside semialgebraic sets . the proposed algorithm leverages recent results on the approximation of indicator functions by polynomials to develop acceptance / rejection based sample generation algorithms with guaranteed performance in terms of rejection rate ( the number of samples that should be generated in order to obtain an accepted sample ) . moreover , the acceptance rate is shown to be is asymptotically optimal , in the sense that it tends to one ( all samples accepted ) as the degree of the polynomial approximation increases . the performance of the proposed method is illustrated by a numerical example .


in recent times , there has been an increased use of nuclear reaction codes in the nuclear data evaluation process for the computation of cross sections , spectra and angular distributions required for a large variety of applications  .the use of model codes in the evaluation of nuclear data offers several advantages , among them ; the preservation of energy balance and coherence of partial cross sections together with total and reaction cross sections , the prediction of data for unstable nuclei and providing data where experimental data are unavailable . where experimental data are available , they are used for fine tuning input parameters to model codes .the total monte carlo ( tmc ) method , developed around talys by the nuclear research group , petten , has the capability of incorporating microscopic nuclear physics and macroscopic nuclear reactor design into one simulation scheme  . in this way, uncertainties can be assigned to macroscopic reactor parameters together with their sensitivity to uncertainties in specific cross - sections . the probability distribution for various reactor parameters can also be observed .tmc has been tested for a variety of systems  . in this paper , we applied the total monte carlo approach in assessing the impact of pu-239 cross section uncertainties on the full core 3-d serpent monte carlo model of the european lead cooled training reactor(electra ) .electra is a 0.5 mw reactor proposed within the genius ( generation iv research in universities of sweden ) framework  .it aims at developing technology necessary for safe and economic deployment of generation iv nuclear reactors , in particular the lead cooled fast reactor . since pu-239 makes up of 30% of the total mass of the fuel , its uncertaintiesare expected to impact significantly on the core behavior of the reactor . a novel approach for the selection of benchmarks for applicationsis present in this work .the selection of benchmarks for applications is usually done with by eye judgment .this method is dependent on the experience of the user and therefore not suitable for the total monte carlo methodology which lays emphasis on automation and reproducibility  .also , an accept / reject criterion for reducing uncertainties in reactor macroscopic parameters based on limiting values is investigated using the pu-239 jezebel criticality benchmark .the input files used in this study are the serpent geometry input file developed at kth , sweden  and 740 random endf files obtained from the tendl project  .the pu-239 jezebel benchmark mcnp input file obtained from the international handbook of evaluated criticality safety benchmark experiments  was converted into a serpent file for simulations .the random files were processed into ace format using the njoy99.336 processing code at 1200k .simulations were performed with the serpent monte carlo code version 1.1.17  for the electra core at zero burnup with the absorber drums set at startup position .criticality calculations were performed for 20 inactive and 500 active cycles with 50,000 neutrons per cycle corresponding to 25 million histories resulting in a statistical uncertainty in of 21.5 pcm .endf-6 formatted files for pu-239 obtained from the latest major nuclear data libraries jeff-3.1.2 , endf / b - vii.1 and jendl-4.0 were processed in a similar way and criticality calculations performed .the results obtained are compared with mean of the distribution of the random nuclear data files .the fast tmc methology presented in  is used in this work .however , instead of using it to obtain short computation times it is used to obtain a very accurate and precise estimate of the uncertainty in reactor parameters due to nuclear data ( ) . in tmc , ( )is obtained by running n different nd files .the result is a spread in the data ( ) that is both due to statistics ( ) and due to nuclear data ( ) : since both the parameter , e.g , , and its statistical uncertainty , , is determined by most transport codes , we obtain a distribution of both and when we run with our n random files .the spread in the distribution of is then and the is equal to .[ yi ] holds assuming that there is no correlation between statistics and nuclear data ; the correlation between and the corresponding statistical error was investigated and no correlation was observed .the methodology produces an accurate estimate of if is accurately determined by the transport code . to test this we executed the serpent input file 740 times with constant nuclear data , but with different random seeds .if = then the transport code determines correctly ; in our case with serpent and electra and this was found to be true .criticality benchmarks are used mostly for validation of calculational techniques in reactor calculations and for establishing minimum subcritical margins for operations with fissile material  . in the nuclear data evaluation process , benchmarking evaluated data against differential and integral data is very important .this is mostly done by selecting large sets of criticality , dosimetry , fusion , activation benchmarks among others .the selection varies from one application group to another as a results of different expertise , purpose of the evaluation and accessibility to benchmarks  . to solve this problem , we proposed the tmc methodology for the selection of benchmarks .first , random files obtained from the tendl project are processed into usable format for reactor codes .criticality simulations are performed for 1 ) the benchmark under consideration and for 2 ) the specific reactor system using same processed random files .a correlation test is then performed by plotting the obtained for the reactor system against that of the benchmark .if a strong correlation exists between the two systems , this can be interpreted as a good representation of your system .this methodology has been tested only on the pu-239 jezebel benchmark and we plan to investigate this in more detail taking into consideration fusion and shielding benchmarks .the proposed method also opens up possibilities for assigning weights to each benchmark based on the goodness of fit or the correlation coefficient . in fig .[ fig1 ] , a correlation between the computed for electra and that of the pu-239 jezebel benchmark is presented .   determined for electra and the pu-239 jezebel benchmark for pu-239 random files . ]   an accept / reject criterion is investigated to study how limiting values for accepting random files could reduce nuclear data uncertainties in safety parameters .simulations were performed for pu-239 random files  and with pu-239 jezebel criticality benchmark and with the electra serpent monte carlo code input file .limiting chi squares were set and corresponding nuclear data uncertainty calculated.in fig .[ fig2 ] , we present a plot of limiting chi square verses pu-239 random files index .a distribution in was obtained for 740 pu-239 random endf files and the mean compared with results from other nuclear data libraries . the jeff-3.1 ( the reference library )temperature dependent cross - section libraries were maintained for all other isotopes except for pu-239 files which was changed with files from tendl , jendl-3.1.2 , jendl-4.0 and end / b - vii.1 .it was observed that the mean value of of the random files compared favorably with the other nuclear data libraries : endf / b - vii.1 was , jeff-3.1.1 was , jendl-4.0 was .this can be attributed to the contribution of the well known fission cross section of pu-239 which has a relatively high impact on the effective multiplication factor .the uncertainty due to nuclear data was determined and found to be pcm . in fig .[ fig2 ] , we present a plot of limiting values against the random file index .nuclear data uncertainties were determined for each limit and the results is illustrated in fig .[ fig3 ] .    the correlation between computed for electra and for the pu-239 jezebel benchmarkis shown in fig .it can be seen from the plot that there exists a strong correlation between the two systems implying that , the pu-239 jezebel benchmark is a good representation of electra .the pearson correlation coefficient which is a measure of the strength of the relationship between the two systems was computed and found to be r=0.84 , further underscoring the strong correlation observed . for the computation of for a large set of benchmarks, we propose that a weighted factor ( ) , which should be equal to the absolute value of the correlation coefficient ( ) be set for the benchmark .we plan to test this methodology on a large set of benchmarks and for different nuclides .the approach has however been applied am-241 in burnup evaluation of electra  .it was observed that , a 25% reduction in nuclear data uncertainty was achieved after a setting a more rigid acceptance criteria . in fig .[ fig3 ] , we present the nuclear data uncertainty against limiting . it can be observed from the figure that by setting a minimum limiting of 5.31e-06 , we were able to reduce the uncertainty in nuclear data from 748 to 443 pcm .this therefore implies that , by implementing an accept / reject criterion in the total monte carlo method for accepting random files , nuclear data uncertainty can be reduced significantly .a study has been carried out by using benchmarks within the total monte carlo methodology on nuclear data uncertainty propagation for the european lead cooled training reactor ( electra ) .it was observed that , nuclear data uncertainties could be reduced considerably in criticality calculations after introducing an accept / reject criterion based on integral benchmarks into the tmc calculation chain .a method for the selection of benchmarks with tmc was also proposed which opens up possibilities for assigning weights to benchmarks .9 iaea , final report of a coordinated research project ripl-2 , august 2006 .a. koning and d. rochman , nuclear data sheets * 113 * , 2841 ( 2012 ) .d. rochman , a. koning nuclear science and engineering , * 168 * , 1 ( 2011 ) .j. wallenius , e. suvdantsetseg and a. fokau , nuclear technology * 177 * , 303 - 313 ( 2012 ) .briggs , international handbook of evaluated criticality safety benchmark experiments nea / nsc / doc(95)03 , september 2012 .j. leppnen , serpent monte carlo users manual ( 2012 ) .d. rochman __ , presented at the int . conf . on nuclear data for sc . and techh. sjstrand _et al . _ ,presented at nd(2013 ) .

Abstract: analyses are carried out to assess the impact of nuclear data uncertainties on for the european lead cooled training reactor ( electra ) using the total monte carlo method . a large number of pu-239 random endf - formated libraries generated using the talys based system were processed into ace format with njoy99.336 code and used as input into the serpent monte carlo neutron transport code to obtain distribution in . the distribution obtained was compared with the latest major nuclear data libraries  jeff-3.1.2 , endf / b - vii.1 and jendl-4.0 . a method is proposed for the selection of benchmarks for specific applications using the total monte carlo approach . finally , an accept / reject criterion was investigated based on values obtained using the pu-239 jezebel criticality benchmark . it was observed that nuclear data uncertainties in were reduced considerably from 748 to 443 pcm by applying a more rigid acceptance criteria for accepting random files .