Pub Date : 2024-05-31DOI: 10.1016/j.jco.2024.101868
Stefan Hain, Karsten Urban
We present a well-posed ultra-weak space-time variational formulation for the time-dependent version of the linear Schrödinger equation with an instationary Hamiltonian. We prove optimal inf-sup stability and introduce a space-time Petrov-Galerkin discretization with optimal discrete inf-sup stability.
We show norm-preservation of the ultra-weak formulation. The inf-sup optimal Petrov-Galerkin discretization is shown to be asymptotically norm-preserving, where the deviation is shown to be in the order of the discretization. In addition, we introduce a Galerkin discretization, which has suboptimal inf-sup stability but exact norm-preservation.
Numerical experiments underline the performance of the ultra-weak space-time variational formulation, especially for non-smooth initial data.
{"title":"An ultra-weak space-time variational formulation for the Schrödinger equation","authors":"Stefan Hain, Karsten Urban","doi":"10.1016/j.jco.2024.101868","DOIUrl":"https://doi.org/10.1016/j.jco.2024.101868","url":null,"abstract":"<div><p>We present a well-posed ultra-weak space-time variational formulation for the time-dependent version of the linear Schrödinger equation with an instationary Hamiltonian. We prove optimal inf-sup stability and introduce a space-time Petrov-Galerkin discretization with optimal discrete inf-sup stability.</p><p>We show norm-preservation of the ultra-weak formulation. The inf-sup optimal Petrov-Galerkin discretization is shown to be asymptotically norm-preserving, where the deviation is shown to be in the order of the discretization. In addition, we introduce a Galerkin discretization, which has suboptimal inf-sup stability but exact norm-preservation.</p><p>Numerical experiments underline the performance of the ultra-weak space-time variational formulation, especially for non-smooth initial data.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"85 ","pages":"Article 101868"},"PeriodicalIF":1.7,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0885064X24000451/pdfft?md5=1ddbc4d09e904e3ef47e872092642e90&pid=1-s2.0-S0885064X24000451-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141290844","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}
Pub Date : 2024-05-31DOI: 10.1016/j.jco.2024.101869
Peter Kritzer
We give an overview of certain aspects of tractability analysis of multivariate problems. This paper is not intended to give a complete account of the subject, but provides an insight into how the theory works for particular types of problems. We mainly focus on linear problems on Hilbert spaces, and mostly allow arbitrary linear information. In such cases, tractability analysis is closely linked to an analysis of the singular values of the operator under consideration. We also highlight the more recent developments regarding exponential and generalized tractability. The theoretical results are illustrated by several examples throughout the article.
{"title":"Selected aspects of tractability analysis","authors":"Peter Kritzer","doi":"10.1016/j.jco.2024.101869","DOIUrl":"https://doi.org/10.1016/j.jco.2024.101869","url":null,"abstract":"<div><p>We give an overview of certain aspects of tractability analysis of multivariate problems. This paper is not intended to give a complete account of the subject, but provides an insight into how the theory works for particular types of problems. We mainly focus on linear problems on Hilbert spaces, and mostly allow arbitrary linear information. In such cases, tractability analysis is closely linked to an analysis of the singular values of the operator under consideration. We also highlight the more recent developments regarding exponential and generalized tractability. The theoretical results are illustrated by several examples throughout the article.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"84 ","pages":"Article 101869"},"PeriodicalIF":1.7,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141250103","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}
Pub Date : 2024-05-13DOI: 10.1016/j.jco.2024.101867
Peter Mathé , Bernd Hofmann
We introduce a notion of tractability for ill-posed operator equations in Hilbert space. For such operator equations the asymptotics of the best possible rate of reconstruction in terms of the underlying noise level is known in many cases. However, the relevant question is, which level of discretization, again driven by the noise level, is required in order to achieve this best possible accuracy. The proposed concept adapts the one from Information-based Complexity. Several examples indicate the relevance of this concept in the light of the curse of dimensionality.
{"title":"Tractability of linear ill-posed problems in Hilbert space","authors":"Peter Mathé , Bernd Hofmann","doi":"10.1016/j.jco.2024.101867","DOIUrl":"10.1016/j.jco.2024.101867","url":null,"abstract":"<div><p>We introduce a notion of tractability for ill-posed operator equations in Hilbert space. For such operator equations the asymptotics of the best possible rate of reconstruction in terms of the underlying noise level is known in many cases. However, the relevant question is, which level of discretization, again driven by the noise level, is required in order to achieve this best possible accuracy. The proposed concept adapts the one from Information-based Complexity. Several examples indicate the relevance of this concept in the light of the curse of dimensionality.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"84 ","pages":"Article 101867"},"PeriodicalIF":1.7,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061797","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}
Pub Date : 2024-05-06DOI: 10.1016/j.jco.2024.101866
Artūras Dubickas
<div><p>In this paper, we consider the problem of finding how close two sums of <em>m</em>th roots can be to each other. For integers <span><math><mi>m</mi><mo>≥</mo><mn>2</mn></math></span>, <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span> and <span><math><mn>0</mn><mo>≤</mo><mi>s</mi><mo>≤</mo><mi>k</mi></math></span>, let <span><math><msub><mrow><mi>e</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo><mo>></mo><mn>0</mn></math></span> and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo><mo>></mo><mn>0</mn></math></span> be the largest exponents such that for infinitely many integers <em>N</em> there exist <em>k</em> positive integers <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>≤</mo><mi>N</mi></math></span> for which two sums of their <em>m</em>th roots <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mroot><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mrow><mi>m</mi></mrow></mroot></math></span> and <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mi>s</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>k</mi></mrow></msubsup><mroot><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mrow><mi>m</mi></mrow></mroot></math></span> are distinct but not further than <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span> from each other, or they are distinct modulo 1 but not further than <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span> from each other modulo 1. Some upper bounds on <span><math><msub><mrow><mi>e</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> can be derived by a Liouville-type argument, while lower bounds are usually difficult to obtain. We prove that <span><math><msub><mrow><mi>e</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo><mo>≥</mo><mi>min</mi><mo></mo><mo>(</mo><mn>2</mn><mi>s</mi><mo>,</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>2</mn><mi>k</mi><mo>−</mo><mn>2</mn><mi>s</mi><mo>)</mo><mo>−</mo><mn>1</mn><mo>/</mo><mi>m</mi></math></span> for <span><math><mn>1</mn><mo>≤</mo><mi>s</mi><mo><</mo><mi>k</mi></math></span> and that <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi
{"title":"Approximate equality for two sums of roots","authors":"Artūras Dubickas","doi":"10.1016/j.jco.2024.101866","DOIUrl":"https://doi.org/10.1016/j.jco.2024.101866","url":null,"abstract":"<div><p>In this paper, we consider the problem of finding how close two sums of <em>m</em>th roots can be to each other. For integers <span><math><mi>m</mi><mo>≥</mo><mn>2</mn></math></span>, <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span> and <span><math><mn>0</mn><mo>≤</mo><mi>s</mi><mo>≤</mo><mi>k</mi></math></span>, let <span><math><msub><mrow><mi>e</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo><mo>></mo><mn>0</mn></math></span> and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo><mo>></mo><mn>0</mn></math></span> be the largest exponents such that for infinitely many integers <em>N</em> there exist <em>k</em> positive integers <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>≤</mo><mi>N</mi></math></span> for which two sums of their <em>m</em>th roots <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mroot><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mrow><mi>m</mi></mrow></mroot></math></span> and <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mi>s</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>k</mi></mrow></msubsup><mroot><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow><mrow><mi>m</mi></mrow></mroot></math></span> are distinct but not further than <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span> from each other, or they are distinct modulo 1 but not further than <span><math><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span> from each other modulo 1. Some upper bounds on <span><math><msub><mrow><mi>e</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> can be derived by a Liouville-type argument, while lower bounds are usually difficult to obtain. We prove that <span><math><msub><mrow><mi>e</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo><mo>≥</mo><mi>min</mi><mo></mo><mo>(</mo><mn>2</mn><mi>s</mi><mo>,</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>2</mn><mi>k</mi><mo>−</mo><mn>2</mn><mi>s</mi><mo>)</mo><mo>−</mo><mn>1</mn><mo>/</mo><mi>m</mi></math></span> for <span><math><mn>1</mn><mo>≤</mo><mi>s</mi><mo><</mo><mi>k</mi></math></span> and that <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"84 ","pages":"Article 101866"},"PeriodicalIF":1.7,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140905544","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}
Pub Date : 2024-04-30DOI: 10.1016/j.jco.2024.101858
Cheng Wang , Jun Fan
Functional data analysis offers a set of statistical methods concerned with extracting insights from intrinsically infinite-dimensional data and has attracted considerable amount of attentions in the past few decades. In this paper, we study robust functional linear regression model with a scalar response and a functional predictor in the framework of reproducing kernel Hilbert spaces. A gradient descent algorithm with early stopping is introduced to solve the corresponding empirical risk minimization problem associated with robust loss functions. By appropriately selecting the early stopping rule and the scaling parameter of the robust losses, the convergence of the proposed algorithm is established when the response variable is bounded or satisfies a moment condition. Explicit learning rates with respect to both estimation and prediction error are provided in terms of regularity of the regression function and eigenvalue decay rate of the integral operator induced by the reproducing kernel and covariance function.
{"title":"On the convergence of gradient descent for robust functional linear regression","authors":"Cheng Wang , Jun Fan","doi":"10.1016/j.jco.2024.101858","DOIUrl":"https://doi.org/10.1016/j.jco.2024.101858","url":null,"abstract":"<div><p>Functional data analysis offers a set of statistical methods concerned with extracting insights from intrinsically infinite-dimensional data and has attracted considerable amount of attentions in the past few decades. In this paper, we study robust functional linear regression model with a scalar response and a functional predictor in the framework of reproducing kernel Hilbert spaces. A gradient descent algorithm with early stopping is introduced to solve the corresponding empirical risk minimization problem associated with robust loss functions. By appropriately selecting the early stopping rule and the scaling parameter of the robust losses, the convergence of the proposed algorithm is established when the response variable is bounded or satisfies a moment condition. Explicit learning rates with respect to both estimation and prediction error are provided in terms of regularity of the regression function and eigenvalue decay rate of the integral operator induced by the reproducing kernel and covariance function.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"84 ","pages":"Article 101858"},"PeriodicalIF":1.7,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140823062","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}
Pub Date : 2024-04-22DOI: 10.1016/j.jco.2024.101857
Erich Novak
{"title":"David Krieg is the winner of the 2024 Joseph F. Traub Prize for Achievement in Information-Based Complexity","authors":"Erich Novak","doi":"10.1016/j.jco.2024.101857","DOIUrl":"https://doi.org/10.1016/j.jco.2024.101857","url":null,"abstract":"","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"83 ","pages":"Article 101857"},"PeriodicalIF":1.7,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0885064X24000347/pdfft?md5=2ff01990fe6f5661ee24005dadc016f8&pid=1-s2.0-S0885064X24000347-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140633006","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}
Pub Date : 2024-04-20DOI: 10.1016/j.jco.2024.101856
Onyekachi Emenike , Fred J. Hickernell , Peter Kritzer
A large literature specifies conditions under which the information complexity for a sequence of numerical problems defined for dimensions grows at a moderate rate, i.e., the sequence of problems is tractable. Here, we focus on the situation where the space of available information consists of all linear functionals, and the problems are defined as linear operator mappings between Hilbert spaces. We unify the proofs of known tractability results and generalize a number of existing results. These generalizations are expressed as five theorems that provide equivalent conditions for (strong) tractability in terms of sums of functions of the singular values of the solution operators.
{"title":"A unified treatment of tractability for approximation problems defined on Hilbert spaces","authors":"Onyekachi Emenike , Fred J. Hickernell , Peter Kritzer","doi":"10.1016/j.jco.2024.101856","DOIUrl":"10.1016/j.jco.2024.101856","url":null,"abstract":"<div><p>A large literature specifies conditions under which the information complexity for a sequence of numerical problems defined for dimensions <span><math><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo></math></span> grows at a moderate rate, i.e., the sequence of problems is <em>tractable</em>. Here, we focus on the situation where the space of available information consists of all linear functionals, and the problems are defined as linear operator mappings between Hilbert spaces. We unify the proofs of known tractability results and generalize a number of existing results. These generalizations are expressed as five theorems that provide equivalent conditions for (strong) tractability in terms of sums of functions of the singular values of the solution operators.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"84 ","pages":"Article 101856"},"PeriodicalIF":1.7,"publicationDate":"2024-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140799561","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}
Pub Date : 2024-04-17DOI: 10.1016/j.jco.2024.101854
Santhosh George, Indra Bate, Muniyasamy M, Chandhini G, Kedarnath Senapati
Ezquerro and Hernandez (2009) studied a modified Chebyshev's method to solve a nonlinear equation approximately in the Banach space setting where the convergence analysis utilizes Taylor series expansion and hence requires the existence of at least fourth-order Fréchet derivative of the involved operator. No error estimate on the error distance was given in their work. In this paper, we obtained the convergence order and error estimate of the error distance without Taylor series expansion. We have made assumptions only on the involved operator and its first and second Fréchet derivative. So, we extend the applicability of the modified Chebyshev's method. Further, we compare the modified Chebyshev method's efficiency index and dynamics with other similar methods. Numerical examples validate the theoretical results.
{"title":"Enhancing the applicability of Chebyshev-like method","authors":"Santhosh George, Indra Bate, Muniyasamy M, Chandhini G, Kedarnath Senapati","doi":"10.1016/j.jco.2024.101854","DOIUrl":"https://doi.org/10.1016/j.jco.2024.101854","url":null,"abstract":"<div><p>Ezquerro and Hernandez (2009) studied a modified Chebyshev's method to solve a nonlinear equation approximately in the Banach space setting where the convergence analysis utilizes Taylor series expansion and hence requires the existence of at least fourth-order Fréchet derivative of the involved operator. No error estimate on the error distance was given in their work. In this paper, we obtained the convergence order and error estimate of the error distance without Taylor series expansion. We have made assumptions only on the involved operator and its first and second Fréchet derivative. So, we extend the applicability of the modified Chebyshev's method. Further, we compare the modified Chebyshev method's efficiency index and dynamics with other similar methods. Numerical examples validate the theoretical results.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"83 ","pages":"Article 101854"},"PeriodicalIF":1.7,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140619311","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}
Pub Date : 2024-04-16DOI: 10.1016/j.jco.2024.101855
Michael Gnewuch
We improve the best known upper bound for the bracketing number of d-dimensional axis-parallel boxes anchored in 0 (or, put differently, of lower left orthants intersected with the d-dimensional unit cube ). More precisely, we provide a better estimate for the cardinality of an algorithmic bracketing cover construction due to Eric Thiémard, which forms the core of his algorithm to approximate the star discrepancy of arbitrary point sets from Thiémard (2001) [22]. Moreover, the new upper bound for the bracketing number of anchored axis-parallel boxes yields an improved upper estimate for the bracketing number of arbitrary axis-parallel boxes in . In our upper bounds all constants are fully explicit.
我们改进了锚定在 0 的 d 维轴平行盒（或者换句话说，与 d 维单位立方体 [0,1]d 相交的左下正交）的已知括弧数上限。更确切地说，我们为埃里克-蒂埃玛尔（Eric Thiémard）提出的括号盖构造的心数提供了一个更好的估计，该构造构成了蒂埃玛尔（Thiémard）（2001）[22]中近似任意点集星形差异算法的核心。此外，锚定轴平行盒的括弧数的新上界可以改进[0,1]d 中任意轴平行盒的括弧数的上界估计。在我们的上界中，所有常数都是完全明确的。
{"title":"Improved bounds for the bracketing number of orthants or revisiting an algorithm of Thiémard to compute bounds for the star discrepancy","authors":"Michael Gnewuch","doi":"10.1016/j.jco.2024.101855","DOIUrl":"https://doi.org/10.1016/j.jco.2024.101855","url":null,"abstract":"<div><p>We improve the best known upper bound for the bracketing number of <em>d</em>-dimensional axis-parallel boxes anchored in 0 (or, put differently, of lower left orthants intersected with the <em>d</em>-dimensional unit cube <span><math><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>d</mi></mrow></msup></math></span>). More precisely, we provide a better estimate for the cardinality of an algorithmic bracketing cover construction due to Eric Thiémard, which forms the core of his algorithm to approximate the star discrepancy of arbitrary point sets from Thiémard (2001) <span>[22]</span>. Moreover, the new upper bound for the bracketing number of anchored axis-parallel boxes yields an improved upper estimate for the bracketing number of arbitrary axis-parallel boxes in <span><math><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>d</mi></mrow></msup></math></span>. In our upper bounds all constants are fully explicit.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"83 ","pages":"Article 101855"},"PeriodicalIF":1.7,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0885064X24000323/pdfft?md5=23c928b5ffffc6732ad1f4739311a07b&pid=1-s2.0-S0885064X24000323-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140619310","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}
Pub Date : 2024-03-24DOI: 10.1016/j.jco.2024.101853
Markus Holzleitner , Sergei V. Pereverzyev
This article offers a comprehensive treatment of polynomial functional regression, culminating in the establishment of a novel finite sample bound. This bound encompasses various aspects, including general smoothness conditions, capacity conditions, and regularization techniques. In doing so, it extends and generalizes several findings from the context of linear functional regression as well. We also provide numerical evidence that using higher order polynomial terms can lead to an improved performance.
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