Journal of Complexity最新文献_第4页

Linear Monte Carlo quadrature with optimal confidence intervals 具有最佳置信区间的线性蒙特卡罗正交法

IF 1.7 2区数学

Journal of Complexity Pub Date : 2024-03-18 DOI: 10.1016/j.jco.2024.101851

Robert J. Kunsch

{"title":"Linear Monte Carlo quadrature with optimal confidence intervals","authors":"Robert J. Kunsch","doi":"10.1016/j.jco.2024.101851","DOIUrl":"https://doi.org/10.1016/j.jco.2024.101851","url":null,"abstract":"<div><p>We study the numerical integration of functions from isotropic Sobolev spaces <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>(</mo><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> using finitely many function evaluations within randomized algorithms, aiming for the smallest possible probabilistic error guarantee <span><math><mi>ε</mi><mo>></mo><mn>0</mn></math></span> at confidence level <span><math><mn>1</mn><mo>−</mo><mi>δ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. For spaces consisting of continuous functions, non-linear Monte Carlo methods with optimal confidence properties have already been known, in few cases even linear methods that succeed in that respect. In this paper we promote a method called <em>stratified control variates</em> (SCV) and by it show that already linear methods achieve optimal probabilistic error rates in the high smoothness regime without the need to adjust algorithmic parameters to the uncertainty <em>δ</em>. We also analyse a version of SCV in the low smoothness regime where <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msubsup><mo>(</mo><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span> may contain functions with singularities. Here, we observe a polynomial dependence of the error on <span><math><msup><mrow><mi>δ</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> in contrast to the logarithmic dependence in the high smoothness regime.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0885064X24000281/pdfft?md5=6b29dfcc17f60ddbf6ee19d289e21700&pid=1-s2.0-S0885064X24000281-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140180612","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Heuristic approaches to obtain low-discrepancy point sets via subset selection 通过子集选择获得低差异点集的启发式方法

IF 1.7 2区数学

Journal of Complexity Pub Date : 2024-03-16 DOI: 10.1016/j.jco.2024.101852

François Clément , Carola Doerr , Luís Paquete

{"title":"Heuristic approaches to obtain low-discrepancy point sets via subset selection","authors":"François Clément , Carola Doerr , Luís Paquete","doi":"10.1016/j.jco.2024.101852","DOIUrl":"https://doi.org/10.1016/j.jco.2024.101852","url":null,"abstract":"<div><p>Building upon the exact methods presented in our earlier work (2022) <span>[5]</span>, we introduce a heuristic approach for the star discrepancy subset selection problem. The heuristic gradually improves the current-best subset by replacing one of its elements at a time. While it does not necessarily return an optimal solution, we obtain promising results for all tested dimensions. For example, for moderate sizes <span><math><mn>30</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>240</mn></math></span>, we obtain point sets in dimension 6 with <span><math><msub><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> star discrepancy up to 35% better than that of the first <em>n</em> points of the Sobol' sequence. Our heuristic works in all dimensions, the main limitation being the precision of the discrepancy calculation algorithms. We provide a comparison with an energy functional introduced by Steinerberger (2019) <span>[31]</span>, showing that our heuristic performs better on all tested instances. Finally, our results give further empirical information on inverse star discrepancy conjectures.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0885064X24000293/pdfft?md5=026f36f25d20579c91a0fc64a95356e5&pid=1-s2.0-S0885064X24000293-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140190621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Linear implicit approximations of invariant measures of semi-linear SDEs with non-globally Lipschitz coefficients 具有非全局 Lipschitz 系数的半线性 SDE 不变量的线性隐含近似值

IF 1.7 2区数学

Journal of Complexity Pub Date : 2024-03-13 DOI: 10.1016/j.jco.2024.101842

Chenxu Pang , Xiaojie Wang , Yue Wu

{"title":"Linear implicit approximations of invariant measures of semi-linear SDEs with non-globally Lipschitz coefficients","authors":"Chenxu Pang , Xiaojie Wang , Yue Wu","doi":"10.1016/j.jco.2024.101842","DOIUrl":"https://doi.org/10.1016/j.jco.2024.101842","url":null,"abstract":"<div><p>This article investigates the weak approximation towards the invariant measure of semi-linear stochastic differential equations (SDEs) under non-globally Lipschitz coefficients. For this purpose, we propose a linear-theta-projected Euler (LTPE) scheme, which also admits an invariant measure, to handle the potential influence of the linear stiffness. Under certain assumptions, both the SDE and the corresponding LTPE method are shown to converge exponentially to the underlying invariant measures, respectively. Moreover, with time-independent regularity estimates for the corresponding Kolmogorov equation, the weak error between the numerical invariant measure and the original one can be guaranteed with convergence of order one. In terms of computational complexity, the proposed ergodicity preserving scheme with the nonlinearity explicitly treated has a significant advantage over the ergodicity preserving implicit Euler method in the literature. Numerical experiments are provided to verify our theoretical findings.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0885064X24000190/pdfft?md5=196a33f1ce0b753c885d6d05ad1d70a4&pid=1-s2.0-S0885064X24000190-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140188091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Homogeneous algorithms and solvable problems on cones 锥体上的同质算法和可解问题

IF 1.7 2区数学

Journal of Complexity Pub Date : 2024-03-13 DOI: 10.1016/j.jco.2024.101840

David Krieg , Peter Kritzer

引用次数: 0

Radius of information for two intersected centered hyperellipsoids and implications in optimal recovery from inaccurate data 两个相交居中的超椭球体的信息半径及其对从不准确数据中优化恢复的影响

IF 1.7 2区数学

Journal of Complexity Pub Date : 2024-03-08 DOI: 10.1016/j.jco.2024.101841

Simon Foucart , Chunyang Liao

{"title":"Radius of information for two intersected centered hyperellipsoids and implications in optimal recovery from inaccurate data","authors":"Simon Foucart , Chunyang Liao","doi":"10.1016/j.jco.2024.101841","DOIUrl":"https://doi.org/10.1016/j.jco.2024.101841","url":null,"abstract":"<div><p>For objects belonging to a known model set and observed through a prescribed linear process, we aim at determining methods to recover linear quantities of these objects that are optimal from a worst-case perspective. Working in a Hilbert setting, we show that, if the model set is the intersection of two hyperellipsoids centered at the origin, then there is an optimal recovery method which is linear. It is specifically given by a constrained regularization procedure whose parameters can be precomputed by semidefinite programming. This general framework can be applied to several scenarios, including the two-space problem and problems involving <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-inaccurate data. It can also be applied to the problem of recovery from <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-inaccurate data. For the latter, we reach the conclusion of existence of an optimal recovery method which is linear, again given by constrained regularization, under a computationally verifiable sufficient condition.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140123263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A space-time adaptive low-rank method for high-dimensional parabolic partial differential equations 高维抛物偏微分方程的时空自适应低阶方法

IF 1.7 2区数学

Journal of Complexity Pub Date : 2024-02-09 DOI: 10.1016/j.jco.2024.101839

Markus Bachmayr, Manfred Faldum

引用次数: 0

Asymptotic analysis in multivariate worst case approximation with Gaussian kernels 用高斯核进行多变量最坏情况逼近的渐近分析

IF 1.7 2区数学

Journal of Complexity Pub Date : 2024-02-07 DOI: 10.1016/j.jco.2024.101838

A.A. Khartov , I.A. Limar

引用次数: 0

Thomas Jahn, Tino Ullrich and Felix Voigtlaender are the Winners of the 2023 Best Paper Award of the Journal of Complexity 托马斯-扬、蒂诺-乌尔里希和费利克斯-沃伊特兰德荣获《复杂性学报》2023 年度最佳论文奖