Pub Date : 2019-09-08DOI: 10.1515/9783110652581-009
L. Brandolini, G. Travaglini
Counting integer points in large convex bodies with smooth boundaries containing isolated flat points is oftentimes an intermediate case between balls (or convex bodies with smooth boundaries having everywhere positive curvature) and cubes (or convex polytopes). In this paper, we provide a detailed description of several discrepancy problems in the particular planar case where the boundary coincides locally with the graph of the function ℝ ∋ t -> |t|^γ, with γ > 2. We consider both integer points problems and irregularities of distribution problems. The above “restriction” to a particular family of convex bodies is compensated by the fact that many proofs are elementary. The paper is entirely self-contained.
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Pub Date : 2019-03-29DOI: 10.1515/9783110652581-004
T. Goda, Kosuke Suzuki
In this article we review some of recent results on higher order quasi-Monte Carlo (HoQMC) methods. After a seminal work by Dick (2007, 2008) who originally introduced the concept of HoQMC, there have been significant theoretical progresses on HoQMC in terms of discrepancy as well as multivariate numerical integration. Moreover, several successful and promising applications of HoQMC to partial differential equations with random coefficients and Bayesian estimation/inversion problems have been reported recently. In this article we start with standard quasi-Monte Carlo methods based on digital nets and sequences in the sense of Niederreiter, and then move onto their higher order version due to Dick. The Walsh analysis of smooth functions plays a crucial role in developing the theory of HoQMC, and the aim of this article is to provide a unified picture on how the Walsh analysis enables recent developments of HoQMC both for discrepancy and numerical integration.
{"title":"4. Recent advances in higher order quasi-Monte Carlo methods","authors":"T. Goda, Kosuke Suzuki","doi":"10.1515/9783110652581-004","DOIUrl":"https://doi.org/10.1515/9783110652581-004","url":null,"abstract":"In this article we review some of recent results on higher order quasi-Monte Carlo (HoQMC) methods. After a seminal work by Dick (2007, 2008) who originally introduced the concept of HoQMC, there have been significant theoretical progresses on HoQMC in terms of discrepancy as well as multivariate numerical integration. Moreover, several successful and promising applications of HoQMC to partial differential equations with random coefficients and Bayesian estimation/inversion problems have been reported recently. In this article we start with standard quasi-Monte Carlo methods based on digital nets and sequences in the sense of Niederreiter, and then move onto their higher order version due to Dick. The Walsh analysis of smooth functions plays a crucial role in developing the theory of HoQMC, and the aim of this article is to provide a unified picture on how the Walsh analysis enables recent developments of HoQMC both for discrepancy and numerical integration.","PeriodicalId":310493,"journal":{"name":"Discrepancy Theory","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122809425","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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