Pub Date : 2020-09-01DOI: 10.1515/mcma-2020-frontmatter3
{"title":"Frontmatter","authors":"","doi":"10.1515/mcma-2020-frontmatter3","DOIUrl":"https://doi.org/10.1515/mcma-2020-frontmatter3","url":null,"abstract":"","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/mcma-2020-frontmatter3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47520213","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-08-18DOI: 10.1007/978-3-030-98319-2_4
A. Owen
{"title":"On dropping the first Sobol' point","authors":"A. Owen","doi":"10.1007/978-3-030-98319-2_4","DOIUrl":"https://doi.org/10.1007/978-3-030-98319-2_4","url":null,"abstract":"","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"109 1","pages":"71-86"},"PeriodicalIF":0.9,"publicationDate":"2020-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84944431","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract It has been shown that when using a Monte Carlo algorithm to estimate the electrostatic free energy of a biomolecule in a solution, individual random walks can become entrapped in the geometry. We examine a proposed solution, using a sharp restart during the Walk-on-Subdomains step, in more detail. We show that the point at which this solution introduces significant bias is related to properties intrinsic to the molecule being examined. We also examine two potential methods of generating a sharp restart point and show that they both cause no significant bias in the examined molecules and increase the stability of the run times of the individual walks.
{"title":"Examining sharp restart in a Monte Carlo method for the linearized Poisson–Boltzmann equation","authors":"W. Thrasher, M. Mascagni","doi":"10.1515/mcma-2020-2069","DOIUrl":"https://doi.org/10.1515/mcma-2020-2069","url":null,"abstract":"Abstract It has been shown that when using a Monte Carlo algorithm to estimate the electrostatic free energy of a biomolecule in a solution, individual random walks can become entrapped in the geometry. We examine a proposed solution, using a sharp restart during the Walk-on-Subdomains step, in more detail. We show that the point at which this solution introduces significant bias is related to properties intrinsic to the molecule being examined. We also examine two potential methods of generating a sharp restart point and show that they both cause no significant bias in the examined molecules and increase the stability of the run times of the individual walks.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"26 1","pages":"223 - 244"},"PeriodicalIF":0.9,"publicationDate":"2020-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/mcma-2020-2069","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42270569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In our paper [A. Nasroallah and K. Elkimakh, HMM with emission process resulting from a special combination of independent Markovian emissions, Monte Carlo Methods Appl. 23 2017, 4, 287–306] we have studied, in a first scenario, the three fundamental hidden Markov problems assuming that, given the hidden process, the observed one selects emissions from a combination of independent Markov chains evolving at the same time. Here, we propose to conduct the same study with a second scenario assuming that given the hidden process, the emission process selects emissions from a combination of independent Markov chain evolving according to their own clock. Three basic numerical examples are studied to show the proper functioning of the iterative algorithm adapted to the proposed model.
{"title":"Hidden Markov Model with Markovian emission","authors":"Karima Elkimakh, A. Nasroallah","doi":"10.1515/mcma-2020-2072","DOIUrl":"https://doi.org/10.1515/mcma-2020-2072","url":null,"abstract":"Abstract In our paper [A. Nasroallah and K. Elkimakh, HMM with emission process resulting from a special combination of independent Markovian emissions, Monte Carlo Methods Appl. 23 2017, 4, 287–306] we have studied, in a first scenario, the three fundamental hidden Markov problems assuming that, given the hidden process, the observed one selects emissions from a combination of independent Markov chains evolving at the same time. Here, we propose to conduct the same study with a second scenario assuming that given the hidden process, the emission process selects emissions from a combination of independent Markov chain evolving according to their own clock. Three basic numerical examples are studied to show the proper functioning of the iterative algorithm adapted to the proposed model.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"26 1","pages":"303 - 313"},"PeriodicalIF":0.9,"publicationDate":"2020-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/mcma-2020-2072","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42745651","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This study deals with a narrow escape problem, a well-know difficult problem of evaluating the probability for a diffusing particle to reach a small part of a boundary far away from the starting position of the particle. A direct simulation of the diffusion trajectories would take an enormous computer simulation time. Instead, we use a different approach which drastically improves the efficiency of the diffusion trajectory tracking algorithm by introducing an artificial drift velocity directed to the target position. The method can be efficiently applied to solve narrow escape problems for domains of long extension in one direction which is the case in many practical problems in biology and chemistry. The algorithm is meshless both in space and time, and is well applied to solve high-dimensional problems in complicated domains. We present in this paper a detailed numerical analysis of the method for the case of a rectangular parallelepiped. Both stationary and transient diffusion problems are handled.
{"title":"Monte Carlo tracking drift-diffusion trajectories algorithm for solving narrow escape problems","authors":"K. Sabelfeld, N. Popov","doi":"10.1515/mcma-2020-2073","DOIUrl":"https://doi.org/10.1515/mcma-2020-2073","url":null,"abstract":"Abstract This study deals with a narrow escape problem, a well-know difficult problem of evaluating the probability for a diffusing particle to reach a small part of a boundary far away from the starting position of the particle. A direct simulation of the diffusion trajectories would take an enormous computer simulation time. Instead, we use a different approach which drastically improves the efficiency of the diffusion trajectory tracking algorithm by introducing an artificial drift velocity directed to the target position. The method can be efficiently applied to solve narrow escape problems for domains of long extension in one direction which is the case in many practical problems in biology and chemistry. The algorithm is meshless both in space and time, and is well applied to solve high-dimensional problems in complicated domains. We present in this paper a detailed numerical analysis of the method for the case of a rectangular parallelepiped. Both stationary and transient diffusion problems are handled.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"26 1","pages":"177 - 191"},"PeriodicalIF":0.9,"publicationDate":"2020-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/mcma-2020-2073","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49180735","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We study the asymptotic stability of the semi-discrete (SD) numerical method for the approximation of stochastic differential equations. Recently, we examined the order of ℒ 2 {mathcal{L}^{2}} -convergence of the truncated SD method and showed that it can be arbitrarily close to 1 2 {frac{1}{2}} ; see [I. S. Stamatiou and N. Halidias, Convergence rates of the semi-discrete method for stochastic differential equations, Theory Stoch. Process. 24 2019, 2, 89–100]. We show that the truncated SD method is able to preserve the asymptotic stability of the underlying SDE. Motivated by a numerical example, we also propose a different SD scheme, using the Lamperti transformation to the original SDE. Numerical simulations support our theoretical findings.
{"title":"A note on the asymptotic stability of the semi-discrete method for stochastic differential equations","authors":"N. Halidias, I. Stamatiou","doi":"10.1515/mcma-2022-2102","DOIUrl":"https://doi.org/10.1515/mcma-2022-2102","url":null,"abstract":"Abstract We study the asymptotic stability of the semi-discrete (SD) numerical method for the approximation of stochastic differential equations. Recently, we examined the order of ℒ 2 {mathcal{L}^{2}} -convergence of the truncated SD method and showed that it can be arbitrarily close to 1 2 {frac{1}{2}} ; see [I. S. Stamatiou and N. Halidias, Convergence rates of the semi-discrete method for stochastic differential equations, Theory Stoch. Process. 24 2019, 2, 89–100]. We show that the truncated SD method is able to preserve the asymptotic stability of the underlying SDE. Motivated by a numerical example, we also propose a different SD scheme, using the Lamperti transformation to the original SDE. Numerical simulations support our theoretical findings.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"28 1","pages":"13 - 25"},"PeriodicalIF":0.9,"publicationDate":"2020-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48538212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Methods from integral geometry and geometric probability allow us to estimate geometric size measures indirectly. In this article, a Monte Carlo algorithm for simultaneous estimation of hyper-volumes and hyper-surface areas of a class of compact sets in Euclidean space is developed. The algorithm is based on Santalo’s formula and the Hadwiger formula from integral geometry, and employs a comparison principle to assign geometric probabilities. An essential component of the method is to be able to generate uniform sets of random lines on the sphere. We utilize an empirically established method to generate these random chords, and we describe a geometric randomness model associated with it. We verify our results by computing measures for hyper-ellipsoids and certain non-convex sets.
{"title":"On the density of lines and Santalo’s formula for computing geometric size measures","authors":"Khaldoun El-Khaldi, E. Saleeby","doi":"10.1515/mcma-2020-2071","DOIUrl":"https://doi.org/10.1515/mcma-2020-2071","url":null,"abstract":"Abstract Methods from integral geometry and geometric probability allow us to estimate geometric size measures indirectly. In this article, a Monte Carlo algorithm for simultaneous estimation of hyper-volumes and hyper-surface areas of a class of compact sets in Euclidean space is developed. The algorithm is based on Santalo’s formula and the Hadwiger formula from integral geometry, and employs a comparison principle to assign geometric probabilities. An essential component of the method is to be able to generate uniform sets of random lines on the sphere. We utilize an empirically established method to generate these random chords, and we describe a geometric randomness model associated with it. We verify our results by computing measures for hyper-ellipsoids and certain non-convex sets.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"26 1","pages":"315 - 323"},"PeriodicalIF":0.9,"publicationDate":"2020-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/mcma-2020-2071","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42959373","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This article develops a new estimator of the marginal likelihood that requires only a sample of the posterior distribution as the input from the analyst. This sample may come from any sampling scheme, such as Gibbs sampling or Metropolis–Hastings sampling. The presented approach can be implemented generically in almost any application of Bayesian modeling and significantly decreases the computational burdens associated with marginal likelihood estimation compared to existing techniques. The functionality of this method is demonstrated in the context of probit and logit regressions, on two mixtures of normals models, and also on a high-dimensional random intercept probit. Simulation results show that the simple approach presented here achieves excellent stability in low-dimensional models, and also clearly outperforms existing methods when the number of coefficients in the model increases.
{"title":"Estimating marginal likelihoods from the posterior draws through a geometric identity","authors":"Johannes Reichl","doi":"10.1515/mcma-2020-2068","DOIUrl":"https://doi.org/10.1515/mcma-2020-2068","url":null,"abstract":"Abstract This article develops a new estimator of the marginal likelihood that requires only a sample of the posterior distribution as the input from the analyst. This sample may come from any sampling scheme, such as Gibbs sampling or Metropolis–Hastings sampling. The presented approach can be implemented generically in almost any application of Bayesian modeling and significantly decreases the computational burdens associated with marginal likelihood estimation compared to existing techniques. The functionality of this method is demonstrated in the context of probit and logit regressions, on two mixtures of normals models, and also on a high-dimensional random intercept probit. Simulation results show that the simple approach presented here achieves excellent stability in low-dimensional models, and also clearly outperforms existing methods when the number of coefficients in the model increases.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"26 1","pages":"205 - 221"},"PeriodicalIF":0.9,"publicationDate":"2020-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/mcma-2020-2068","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45898188","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, to construct a confidence interval (general and shortest) for quantiles of normal distribution in one population, we present a pivotal quantity that has non-central t distribution. In the case of two independent normal populations, we propose a confidence interval for the ratio of quantiles based on the generalized pivotal quantity, and we introduce a simple method for extracting its percentiles, based on which a shorter confidence interval can be created. Also, we provide general and shorter confidence intervals using the method of variance estimate recovery. The performance of five proposed methods will be examined by using simulation and examples.
{"title":"Constructing a confidence interval for the ratio of normal distribution quantiles","authors":"A. Malekzadeh, S. Mahmoudi","doi":"10.1515/mcma-2020-2070","DOIUrl":"https://doi.org/10.1515/mcma-2020-2070","url":null,"abstract":"Abstract In this paper, to construct a confidence interval (general and shortest) for quantiles of normal distribution in one population, we present a pivotal quantity that has non-central t distribution. In the case of two independent normal populations, we propose a confidence interval for the ratio of quantiles based on the generalized pivotal quantity, and we introduce a simple method for extracting its percentiles, based on which a shorter confidence interval can be created. Also, we provide general and shorter confidence intervals using the method of variance estimate recovery. The performance of five proposed methods will be examined by using simulation and examples.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"26 1","pages":"325 - 334"},"PeriodicalIF":0.9,"publicationDate":"2020-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46385615","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A crude Monte Carlo (MC) method allows to calculate integrals over a d-dimensional cube. As the number N of integration nodes becomes large, the rate of probable error of the MC method decreases as O ( 1 / N ) {O(1/sqrt{N})} . The use of quasi-random points instead of random points in the MC algorithm converts it to the quasi-Monte Carlo (QMC) method. The asymptotic error estimate of QMC integration of d-dimensional functions contains a multiplier 1 / N {1/N} . However, the multiplier ( ln N ) d {(ln N)^{d}} is also a part of the error estimate, which makes it virtually useless. We have proved that, in the general case, the QMC error estimate is not limited to the factor 1 / N {1/N} . However, our numerical experiments show that using quasi-random points of Sobol sequences with N = 2 m {N=2^{m}} with natural m makes the integration error approximately proportional to 1 / N {1/N} . In our numerical experiments, d ≤ 15 {dleq 15} , and we used N ≤ 2 40 {Nleq 2^{40}} points generated by the SOBOLSEQ16384 code published in 2011. In this code, d ≤ 2 14 {dleq 2^{14}} and N ≤ 2 63 {Nleq 2^{63}} .
{"title":"QMC integration errors and quasi-asymptotics","authors":"I. Sobol, B. Shukhman","doi":"10.1515/mcma-2020-2067","DOIUrl":"https://doi.org/10.1515/mcma-2020-2067","url":null,"abstract":"Abstract A crude Monte Carlo (MC) method allows to calculate integrals over a d-dimensional cube. As the number N of integration nodes becomes large, the rate of probable error of the MC method decreases as O ( 1 / N ) {O(1/sqrt{N})} . The use of quasi-random points instead of random points in the MC algorithm converts it to the quasi-Monte Carlo (QMC) method. The asymptotic error estimate of QMC integration of d-dimensional functions contains a multiplier 1 / N {1/N} . However, the multiplier ( ln N ) d {(ln N)^{d}} is also a part of the error estimate, which makes it virtually useless. We have proved that, in the general case, the QMC error estimate is not limited to the factor 1 / N {1/N} . However, our numerical experiments show that using quasi-random points of Sobol sequences with N = 2 m {N=2^{m}} with natural m makes the integration error approximately proportional to 1 / N {1/N} . In our numerical experiments, d ≤ 15 {dleq 15} , and we used N ≤ 2 40 {Nleq 2^{40}} points generated by the SOBOLSEQ16384 code published in 2011. In this code, d ≤ 2 14 {dleq 2^{14}} and N ≤ 2 63 {Nleq 2^{63}} .","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"26 1","pages":"171 - 176"},"PeriodicalIF":0.9,"publicationDate":"2020-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/mcma-2020-2067","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45152742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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