Abstract In this paper we suggest randomized linear solvers with a focus on refinement issue to achieve a high precision while maintaining all the advantages of the Monte Carlo method for solving systems of large dimension with dense matrices. It is shown that each iterative refinement step reduces the error by one order of magnitude. The crucial point of the suggested method is, in contrast to the standard Monte Carlo method, that the randomized vector algorithm computes the entire solution column at once, rather than a single component. This makes it possible to efficiently construct the iterative refinement method. We apply the developed method for solving a system of elasticity equations.
{"title":"Randomized vector iterative linear solvers of high precision for large dense system","authors":"Karl K. Sabelfeld, Anastasiya Kireeva","doi":"10.1515/mcma-2023-2013","DOIUrl":"https://doi.org/10.1515/mcma-2023-2013","url":null,"abstract":"Abstract In this paper we suggest randomized linear solvers with a focus on refinement issue to achieve a high precision while maintaining all the advantages of the Monte Carlo method for solving systems of large dimension with dense matrices. It is shown that each iterative refinement step reduces the error by one order of magnitude. The crucial point of the suggested method is, in contrast to the standard Monte Carlo method, that the randomized vector algorithm computes the entire solution column at once, rather than a single component. This makes it possible to efficiently construct the iterative refinement method. We apply the developed method for solving a system of elasticity equations.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135548383","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 : 2023-09-01DOI: 10.1515/mcma-2023-frontmatter3
{"title":"Frontmatter","authors":"","doi":"10.1515/mcma-2023-frontmatter3","DOIUrl":"https://doi.org/10.1515/mcma-2023-frontmatter3","url":null,"abstract":"","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"81 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136310148","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, we will consider periodic EGARCH ( p , p ) {operatorname{EGARCH}(p,p)} (exponential generalized autoregressive conditional heteroscedastic) processes denoted by PEGARCH ( p , p ) {operatorname{PEGARCH}(p,p)} . These processes are similar to the standard EGARCH processes, but include seasonally varying coefficients. We examine the probabilistic structure of an EGARCH-type stochastic difference equation with periodically-varying parameters. We propose necessary and sufficient conditions ensuring the existence of stationary solutions (in a periodic sense) based on a Markovian representation. The closed forms of higher moments are, under these conditions, established. Furthermore, the expressions for the Kurtosis coefficient and the autocorrelations of squared observations are derived. The general theory is illustrated by considering special cases such as the symmetric and the asymmetric cases of the second order PEGARCH model.
{"title":"On the stationarity and existence of moments of the periodic EGARCH process","authors":"Ines Lescheb, Walid Slimani","doi":"10.1515/mcma-2023-2011","DOIUrl":"https://doi.org/10.1515/mcma-2023-2011","url":null,"abstract":"Abstract In this paper, we will consider periodic EGARCH ( p , p ) {operatorname{EGARCH}(p,p)} (exponential generalized autoregressive conditional heteroscedastic) processes denoted by PEGARCH ( p , p ) {operatorname{PEGARCH}(p,p)} . These processes are similar to the standard EGARCH processes, but include seasonally varying coefficients. We examine the probabilistic structure of an EGARCH-type stochastic difference equation with periodically-varying parameters. We propose necessary and sufficient conditions ensuring the existence of stationary solutions (in a periodic sense) based on a Markovian representation. The closed forms of higher moments are, under these conditions, established. Furthermore, the expressions for the Kurtosis coefficient and the autocorrelations of squared observations are derived. The general theory is illustrated by considering special cases such as the symmetric and the asymmetric cases of the second order PEGARCH model.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44711818","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 is known that the block-based version of the bootstrap method can be used for distributional parameter estimation of dependent data. One of the advantages of this method is that it improves mean square errors. The paper makes two contributions. First, we consider the moving blocking bootstrap method for estimation of parameters of the autoregressive model. For each block, the parameters are estimated based on the modified maximum likelihood method. Second, we provide a method for model selection, Vuong’s test and tracking interval, i.e. for selecting the optimal model for the innovation’s distribution. Our analysis provides analytic results on the asymptotic distribution of the bootstrap estimators and also computational results via simulations. Some properties of the moving blocking bootstrap method are investigated through Monte Carlo simulation. This simulation study shows that, sometimes, Vuong’s test based on the modified maximum likelihood method is not able to distinguish between the two models; Vuong’s test based on the moving blocking bootstrap selects one of the competing models as optimal model. We have studied real data, the S&P500 data, and select optimal model for this data based on the theoretical results.
{"title":"Bootstrap choice of non-nested autoregressive model with non-normal innovations","authors":"Sedigheh Zamani Mehreyan","doi":"10.1515/mcma-2023-2010","DOIUrl":"https://doi.org/10.1515/mcma-2023-2010","url":null,"abstract":"Abstract It is known that the block-based version of the bootstrap method can be used for distributional parameter estimation of dependent data. One of the advantages of this method is that it improves mean square errors. The paper makes two contributions. First, we consider the moving blocking bootstrap method for estimation of parameters of the autoregressive model. For each block, the parameters are estimated based on the modified maximum likelihood method. Second, we provide a method for model selection, Vuong’s test and tracking interval, i.e. for selecting the optimal model for the innovation’s distribution. Our analysis provides analytic results on the asymptotic distribution of the bootstrap estimators and also computational results via simulations. Some properties of the moving blocking bootstrap method are investigated through Monte Carlo simulation. This simulation study shows that, sometimes, Vuong’s test based on the modified maximum likelihood method is not able to distinguish between the two models; Vuong’s test based on the moving blocking bootstrap selects one of the competing models as optimal model. We have studied real data, the S&P500 data, and select optimal model for this data based on the theoretical results.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46015548","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 : 2023-06-01DOI: 10.1515/mcma-2023-frontmatter2
{"title":"Frontmatter","authors":"","doi":"10.1515/mcma-2023-frontmatter2","DOIUrl":"https://doi.org/10.1515/mcma-2023-frontmatter2","url":null,"abstract":"","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135673784","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 The paper gives new results for the Milstein scheme of stochastic differential equations. We show that (i) the Milstein scheme holds as a weak approximation in total variation sense and is given by second-order polynomials of Brownian motion without using iterated integrals under non-commutative vector fields; (ii) the accuracy of the Milstein scheme is better than that of the Euler–Maruyama scheme in an asymptotic sense. In particular, we prove d TV ( X T ε , X ¯ T ε , Mil , ( n ) ) ≤ C ε 2 / n d_{mathrm{TV}}(X_{T}^{varepsilon},bar{X}_{T}^{varepsilon,mathrm{Mil},(n)})leq Cvarepsilon^{2}/n and d TV ( X T ε , X ¯ T ε , EM , ( n ) ) ≤ C ε / n d_{mathrm{TV}}(X_{T}^{varepsilon},bar{X}_{T}^{varepsilon,mathrm{EM},(n)})leq Cvarepsilon/n , where d TV d_{mathrm{TV}} is the total variation distance, X ε X^{varepsilon} is a solution of a stochastic differential equation with a small parameter 𝜀, and X ¯ ε , Mil , ( n ) bar{X}^{varepsilon,mathrm{Mil},(n)} and X ¯ ε , EM , ( n ) bar{X}^{varepsilon,mathrm{EM},(n)} are the Milstein scheme without iterated integrals and the Euler–Maruyama scheme, respectively. In computational aspect, the scheme is useful to estimate probability distribution functions by a simple simulation without Lévy area computation. Numerical examples demonstrate the validity of the method.
{"title":"Total variation bound for Milstein scheme without iterated integrals","authors":"Toshihiro Yamada","doi":"10.2139/ssrn.4333285","DOIUrl":"https://doi.org/10.2139/ssrn.4333285","url":null,"abstract":"Abstract The paper gives new results for the Milstein scheme of stochastic differential equations. We show that (i) the Milstein scheme holds as a weak approximation in total variation sense and is given by second-order polynomials of Brownian motion without using iterated integrals under non-commutative vector fields; (ii) the accuracy of the Milstein scheme is better than that of the Euler–Maruyama scheme in an asymptotic sense. In particular, we prove d TV ( X T ε , X ¯ T ε , Mil , ( n ) ) ≤ C ε 2 / n d_{mathrm{TV}}(X_{T}^{varepsilon},bar{X}_{T}^{varepsilon,mathrm{Mil},(n)})leq Cvarepsilon^{2}/n and d TV ( X T ε , X ¯ T ε , EM , ( n ) ) ≤ C ε / n d_{mathrm{TV}}(X_{T}^{varepsilon},bar{X}_{T}^{varepsilon,mathrm{EM},(n)})leq Cvarepsilon/n , where d TV d_{mathrm{TV}} is the total variation distance, X ε X^{varepsilon} is a solution of a stochastic differential equation with a small parameter 𝜀, and X ¯ ε , Mil , ( n ) bar{X}^{varepsilon,mathrm{Mil},(n)} and X ¯ ε , EM , ( n ) bar{X}^{varepsilon,mathrm{EM},(n)} are the Milstein scheme without iterated integrals and the Euler–Maruyama scheme, respectively. In computational aspect, the scheme is useful to estimate probability distribution functions by a simple simulation without Lévy area computation. Numerical examples demonstrate the validity of the method.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47464087","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}
{"title":"Total variation bound for Milstein scheme without iterated integrals","authors":"Toshihiro Yamada","doi":"10.1515/mcma-2023-2007","DOIUrl":"https://doi.org/10.1515/mcma-2023-2007","url":null,"abstract":"Abstract The paper gives new results for the Milstein scheme of stochastic differential equations. We show that (i) the Milstein scheme holds as a weak approximation in total variation sense and is given by second-order polynomials of Brownian motion without using iterated integrals under non-commutative vector fields; (ii) the accuracy of the Milstein scheme is better than that of the Euler–Maruyama scheme in an asymptotic sense. In particular, we prove <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:msub> <m:mi>d</m:mi> <m:mi>TV</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msubsup> <m:mi>X</m:mi> <m:mi>T</m:mi> <m:mi>ε</m:mi> </m:msubsup> <m:mo>,</m:mo> <m:msubsup> <m:mover accent=\"true\"> <m:mi>X</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mi>T</m:mi> <m:mrow> <m:mi>ε</m:mi> <m:mo>,</m:mo> <m:mi>Mil</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:msubsup> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>≤</m:mo> <m:mrow> <m:mrow> <m:mi>C</m:mi> <m:mo></m:mo> <m:msup> <m:mi>ε</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mo>/</m:mo> <m:mi>n</m:mi> </m:mrow> </m:mrow> </m:math> d_{mathrm{TV}}(X_{T}^{varepsilon},bar{X}_{T}^{varepsilon,mathrm{Mil},(n)})leq Cvarepsilon^{2}/n and <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:msub> <m:mi>d</m:mi> <m:mi>TV</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msubsup> <m:mi>X</m:mi> <m:mi>T</m:mi> <m:mi>ε</m:mi> </m:msubsup> <m:mo>,</m:mo> <m:msubsup> <m:mover accent=\"true\"> <m:mi>X</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mi>T</m:mi> <m:mrow> <m:mi>ε</m:mi> <m:mo>,</m:mo> <m:mi>EM</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:msubsup> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>≤</m:mo> <m:mrow> <m:mrow> <m:mi>C</m:mi> <m:mo></m:mo> <m:mi>ε</m:mi> </m:mrow> <m:mo>/</m:mo> <m:mi>n</m:mi> </m:mrow> </m:mrow> </m:math> d_{mathrm{TV}}(X_{T}^{varepsilon},bar{X}_{T}^{varepsilon,mathrm{EM},(n)})leq Cvarepsilon/n , where <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>d</m:mi> <m:mi>TV</m:mi> </m:msub> </m:math> d_{mathrm{TV}} is the total variation distance, <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>X</m:mi> <m:mi>ε</m:mi> </m:msup> </m:math> X^{varepsilon} is a solution of a stochastic differential equation with a small parameter 𝜀, and <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mover accent=\"true\"> <m:mi>X</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mrow> <m:mi>ε</m:mi> <m:mo>,</m:mo> <m:mi>Mil</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:msup> </m:math> bar{X}^{varepsilon,mathrm{Mil},(n)} and <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mover acc","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135996653","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 Finite-dimensional (FD) models X d ( t ) X_{d}(t) , i.e., deterministic functions of time and finite sets of 𝑑 random variables, are constructed for stationary and nonstationary Gaussian processes X ( t ) X(t) with continuous samples defined on a bounded time interval [ 0 , τ ] [0,tau] . The basis functions of these FD models are finite sets of eigenfunctions of the correlation functions of X ( t ) X(t) and of trigonometric functions. Numerical illustrations are presented for a stationary Gaussian process X ( t ) X(t) with exponential correlation function and a nonstationary version of this process obtained by time distortion. It was found that the FD models are consistent with the theoretical results in the sense that their samples approach the target samples as the stochastic dimension is increased.
{"title":"Monte Carlo estimates of extremes of stationary/nonstationary Gaussian processes","authors":"M. Grigoriu","doi":"10.1515/mcma-2023-2006","DOIUrl":"https://doi.org/10.1515/mcma-2023-2006","url":null,"abstract":"Abstract Finite-dimensional (FD) models X d ( t ) X_{d}(t) , i.e., deterministic functions of time and finite sets of 𝑑 random variables, are constructed for stationary and nonstationary Gaussian processes X ( t ) X(t) with continuous samples defined on a bounded time interval [ 0 , τ ] [0,tau] . The basis functions of these FD models are finite sets of eigenfunctions of the correlation functions of X ( t ) X(t) and of trigonometric functions. Numerical illustrations are presented for a stationary Gaussian process X ( t ) X(t) with exponential correlation function and a nonstationary version of this process obtained by time distortion. It was found that the FD models are consistent with the theoretical results in the sense that their samples approach the target samples as the stochastic dimension is increased.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"29 1","pages":"127 - 142"},"PeriodicalIF":0.9,"publicationDate":"2023-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44816176","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, we construct stochastic simulation algorithms for solving an elastostatics problem governed by the Lamé equation. Two different stochastic simulation methods are suggested: (1) a method based on a random walk on spheres, which is iteratively applied to anisotropic diffusion equations that are related through the mixed second-order derivatives (this method is meshless and can be applied to boundary value problems for complicated domains); (2) a randomized algorithm for solving large systems of linear algebraic equations that is the core of this method. It needs a mesh formation, but even for very fine grids, the algorithm shows a high efficiency. Both methods are scalable and can be easily parallelized.
{"title":"Two stochastic algorithms for solving elastostatics problems governed by the Lamé equation","authors":"A. Kireeva, Ivan Aksyuk, K. Sabelfeld","doi":"10.1515/mcma-2023-2008","DOIUrl":"https://doi.org/10.1515/mcma-2023-2008","url":null,"abstract":"Abstract In this paper, we construct stochastic simulation algorithms for solving an elastostatics problem governed by the Lamé equation. Two different stochastic simulation methods are suggested: (1) a method based on a random walk on spheres, which is iteratively applied to anisotropic diffusion equations that are related through the mixed second-order derivatives (this method is meshless and can be applied to boundary value problems for complicated domains); (2) a randomized algorithm for solving large systems of linear algebraic equations that is the core of this method. It needs a mesh formation, but even for very fine grids, the algorithm shows a high efficiency. Both methods are scalable and can be easily parallelized.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"29 1","pages":"143 - 160"},"PeriodicalIF":0.9,"publicationDate":"2023-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47878109","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, we introduce a new algorithm for estimating the lower bounds for the star discrepancy of any arbitrary point sets in [ 0 , 1 ] s [0,1]^{s} . Computing the exact star discrepancy is known to be an NP-hard problem, so we have been looking for effective approximation algorithms. The star discrepancy can be thought of as the maximum of a function called the local discrepancy, and we will develop approximation algorithms to maximize this function. Our algorithm is analogous to the random walk algorithm described in one of our previous papers [M. Alsolami and M. Mascagni, A random walk algorithm to estimate a lower bound of the star discrepancy, Monte Carlo Methods Appl. 28 (2022), 4, 341–348.]. We add a statistical technique to the random walk algorithm by implementing the Metropolis algorithm in random walks on each chosen dimension to accept or reject this movement. We call this Metropolis random walk algorithm. In comparison to all previously known techniques, our new algorithm is superior, especially in high dimensions. Also, it can quickly determine the precise value of the star discrepancy in most of our data sets of various sizes and dimensions, or at least the lower bounds of the star discrepancy.
{"title":"A Metropolis random walk algorithm to estimate a lower bound of the star discrepancy","authors":"Maryam Alsolami, M. Mascagni","doi":"10.1515/mcma-2023-2005","DOIUrl":"https://doi.org/10.1515/mcma-2023-2005","url":null,"abstract":"Abstract In this paper, we introduce a new algorithm for estimating the lower bounds for the star discrepancy of any arbitrary point sets in [ 0 , 1 ] s [0,1]^{s} . Computing the exact star discrepancy is known to be an NP-hard problem, so we have been looking for effective approximation algorithms. The star discrepancy can be thought of as the maximum of a function called the local discrepancy, and we will develop approximation algorithms to maximize this function. Our algorithm is analogous to the random walk algorithm described in one of our previous papers [M. Alsolami and M. Mascagni, A random walk algorithm to estimate a lower bound of the star discrepancy, Monte Carlo Methods Appl. 28 (2022), 4, 341–348.]. We add a statistical technique to the random walk algorithm by implementing the Metropolis algorithm in random walks on each chosen dimension to accept or reject this movement. We call this Metropolis random walk algorithm. In comparison to all previously known techniques, our new algorithm is superior, especially in high dimensions. Also, it can quickly determine the precise value of the star discrepancy in most of our data sets of various sizes and dimensions, or at least the lower bounds of the star discrepancy.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":"29 1","pages":"161 - 171"},"PeriodicalIF":0.9,"publicationDate":"2023-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44369916","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：