{"title":"Corrigendum to \"Convergence of adaptive, discontinuous Galerkin methods\"","authors":"C. Kreuzer, E. Georgoulis","doi":"10.1090/mcom/3611","DOIUrl":"https://doi.org/10.1090/mcom/3611","url":null,"abstract":"","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90222268","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}
We consider acoustic scattering in heterogeneous media with piecewise constant wave number. The discretization is carried out using a Galerkin boundary element method in space and Runge-Kutta convolution quadrature in time. We prove well-posedness of the scheme and provide a priori estimates for the convergence in space and time.
{"title":"Time domain boundary integral equations and convolution quadrature for scattering by composite media","authors":"A. Rieder, F. Sayas, J. Melenk","doi":"10.1090/mcom/3730","DOIUrl":"https://doi.org/10.1090/mcom/3730","url":null,"abstract":"We consider acoustic scattering in heterogeneous media with piecewise constant wave number. The discretization is carried out using a Galerkin boundary element method in space and Runge-Kutta convolution quadrature in time. We prove well-posedness of the scheme and provide a priori estimates for the convergence in space and time.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86494174","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}
We prove a generalized Faulhaber inequality to bound the sums of the jj-th powers of the first nn (possibly shifted) natural numbers. With the help of this inequality we are able to improve the known bounds for bracketing numbers of dd-dimensional axis-parallel boxes anchored in 00 (or, put differently, of lower left orthants intersected with the dd-dimensional unit cube [0,1]d[0,1]^d). We use these improved bracketing numbers to establish new bounds for the star-discrepancy of negatively dependent random point sets and its expectation. We apply our findings also to the weighted star-discrepancy.
{"title":"A generalized Faulhaber inequality, improved bracketing covers, and applications to discrepancy","authors":"M. Gnewuch, Hendrik Pasing, Christian Weiss","doi":"10.1090/mcom/3666","DOIUrl":"https://doi.org/10.1090/mcom/3666","url":null,"abstract":"<p>We prove a generalized Faulhaber inequality to bound the sums of the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"j\">\u0000 <mml:semantics>\u0000 <mml:mi>j</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">j</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-th powers of the first <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\u0000 <mml:semantics>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> (possibly shifted) natural numbers. With the help of this inequality we are able to improve the known bounds for bracketing numbers of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d\">\u0000 <mml:semantics>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">d</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-dimensional axis-parallel boxes anchored in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"0\">\u0000 <mml:semantics>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> (or, put differently, of lower left orthants intersected with the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d\">\u0000 <mml:semantics>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">d</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-dimensional unit cube <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-bracket 0 comma 1 right-bracket Superscript d\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">[</mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:msup>\u0000 <mml:mo stretchy=\"false\">]</mml:mo>\u0000 <mml:mi>d</mml:mi>\u0000 </mml:msup>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">[0,1]^d</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>). We use these improved bracketing numbers to establish new bounds for the star-discrepancy of negatively dependent random point sets and its expectation. We apply our findings also to the weighted star-discrepancy.</p>","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78783459","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}
Approximation algorithms like sum-up rounding that allow to compute integer-valued approximations of the continuous controls in a weak∗ sense have attracted interest recently. They allow to approximate (optimal) feasible solutions of continuous relaxations of mixed-integer control problems (MIOCPs) with integer controls arbitrarily close. To this end, they use compactness properties of the underlying state equation, a feature that is tied to the infinite-dimensional vantage point. In this work, we consider a class of MIOCPs that are constrained by pointwise mixed state-control constraints. We show that a continuous relaxation that involves so-called vanishing constraints has beneficial properties for the described approximation methodology. Moreover, we complete recent work on a variant of the sum-up rounding algorithm for this problem class. In particular, we prove that the observed infeasibility of the produced integer-valued controls vanishes in an L∞-sense with respect to the considered relaxation. Moreover, we improve the bound on the control approximation error to a value that is asymptotically tight.
{"title":"Approximation properties of sum-up rounding in the presence of vanishing constraints","authors":"Paul Manns, C. Kirches, F. Lenders","doi":"10.1090/mcom/3606","DOIUrl":"https://doi.org/10.1090/mcom/3606","url":null,"abstract":"Approximation algorithms like sum-up rounding that allow to compute integer-valued approximations of the continuous controls in a weak∗ sense have attracted interest recently. They allow to approximate (optimal) feasible solutions of continuous relaxations of mixed-integer control problems (MIOCPs) with integer controls arbitrarily close. To this end, they use compactness properties of the underlying state equation, a feature that is tied to the infinite-dimensional vantage point. In this work, we consider a class of MIOCPs that are constrained by pointwise mixed state-control constraints. We show that a continuous relaxation that involves so-called vanishing constraints has beneficial properties for the described approximation methodology. Moreover, we complete recent work on a variant of the sum-up rounding algorithm for this problem class. In particular, we prove that the observed infeasibility of the produced integer-valued controls vanishes in an L∞-sense with respect to the considered relaxation. Moreover, we improve the bound on the control approximation error to a value that is asymptotically tight.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86743200","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":"Imaginary multiquadratic number fields with class group of exponent 3 and 5","authors":"Jürgen Klüners, T. Komatsu","doi":"10.1090/mcom/3609","DOIUrl":"https://doi.org/10.1090/mcom/3609","url":null,"abstract":"","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83103469","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}
L2 stable discontinuous Galerkin method with a family of numerical fluxes was studied for the one-dimensional wave equation by Cheng, Chou, Li, and Xing in [Math. Comp. 86 (2017), pp. 121–155]. Although optimal convergence rates were numerically observed with wide choices of parameters in the numerical fluxes, their error estimates were only proved for a sub-family with the construction of a local projection. In this paper, we first complete the one-dimensional analysis by providing optimal error estimates that match all numerical observations in that paper. The key ingredient is to construct an optimal global projection with the characteristic decomposition. We then extend the analysis on optimal error estimate to multidimensions by constructing a global projection on unstructured meshes, which can be considered as a perturbation away from the local projection studied by Cockburn, Gopalakrishnan, and Sayas in [Math. Comp. 79 (2010), pp. 1351–1367] for hybridizable discontinuous Galerkin methods. As a main contribution, we use a novel energy argument to prove the optimal approximation property of the global projection. This technique does not require explicit assembly of the matrix for the perturbed terms and hence can be easily used for unstructured meshes in multidimensions. Finally, numerical tests in two dimensions are provided to validate our analysis is sharp and at least one of the unknowns will degenerate to suboptimal rates if the assumptions are not satisfied.
《数学》杂志Cheng、Chou、Li和Xing研究了一维波动方程的具有一组数值通量的L2稳定不连续Galerkin方法。[p. 86 (2017)， pp. 121-155]。虽然在数值通量参数选择范围广泛的情况下，在数值上观察到最优的收敛速率，但它们的误差估计仅通过构造局部投影来证明。在本文中，我们首先通过提供与文中所有数值观测相匹配的最佳误差估计来完成一维分析。关键是利用特征分解构造最优的全局投影。然后，我们通过在非结构化网格上构造一个全局投影，将最优误差估计的分析扩展到多维，该投影可以被认为是远离Cockburn, Gopalakrishnan和Sayas在[Math]中研究的局部投影的扰动。Comp. 79 (2010)， pp. 1351-1367]杂交不连续Galerkin方法。作为主要贡献，我们使用了一个新的能量参数来证明全局投影的最优逼近性质。该技术不需要对扰动项进行矩阵的显式装配，因此可以很容易地用于多维非结构化网格。最后，给出了二维的数值实验来验证我们的分析是敏锐的，如果假设不满足，至少有一个未知数会退化到次优速率。
{"title":"Optimal error estimates of discontinuous Galerkin methods with generalized fluxes for wave equations on unstructured meshes","authors":"Zheng Sun, Y. Xing","doi":"10.1090/mcom/3605","DOIUrl":"https://doi.org/10.1090/mcom/3605","url":null,"abstract":"L2 stable discontinuous Galerkin method with a family of numerical fluxes was studied for the one-dimensional wave equation by Cheng, Chou, Li, and Xing in [Math. Comp. 86 (2017), pp. 121–155]. Although optimal convergence rates were numerically observed with wide choices of parameters in the numerical fluxes, their error estimates were only proved for a sub-family with the construction of a local projection. In this paper, we first complete the one-dimensional analysis by providing optimal error estimates that match all numerical observations in that paper. The key ingredient is to construct an optimal global projection with the characteristic decomposition. We then extend the analysis on optimal error estimate to multidimensions by constructing a global projection on unstructured meshes, which can be considered as a perturbation away from the local projection studied by Cockburn, Gopalakrishnan, and Sayas in [Math. Comp. 79 (2010), pp. 1351–1367] for hybridizable discontinuous Galerkin methods. As a main contribution, we use a novel energy argument to prove the optimal approximation property of the global projection. This technique does not require explicit assembly of the matrix for the perturbed terms and hence can be easily used for unstructured meshes in multidimensions. Finally, numerical tests in two dimensions are provided to validate our analysis is sharp and at least one of the unknowns will degenerate to suboptimal rates if the assumptions are not satisfied.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80436843","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}
In this paper we present an efficient algorithm to produce a provably dense sample of a smooth compact variety. The procedure is partly based on computing $textit{bottlenecks}$ of the variety. Using geometric information such as the bottlenecks and the $textit{local reach}$ we also provide bounds on the density of the sample needed in order to guarantee that the homology of the variety can be recovered from the sample. An implementation of the algorithm is provided together with numerical experiments and a computational comparison to the algorithm by Dufresne et. al.
{"title":"Sampling and homology via bottlenecks","authors":"S. Rocco, David Eklund, Oliver Gäfvert","doi":"10.1090/mcom/3757","DOIUrl":"https://doi.org/10.1090/mcom/3757","url":null,"abstract":"In this paper we present an efficient algorithm to produce a provably dense sample of a smooth compact variety. The procedure is partly based on computing $textit{bottlenecks}$ of the variety. Using geometric information such as the bottlenecks and the $textit{local reach}$ we also provide bounds on the density of the sample needed in order to guarantee that the homology of the variety can be recovered from the sample. An implementation of the algorithm is provided together with numerical experiments and a computational comparison to the algorithm by Dufresne et. al.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91444765","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}
Sualeh Asif, Francesc Fit'e, Dylan Pentland, Andrew V. Sutherland
We study the sequence of zeta functions $Z(C_p,T)$ of a generic Picard curve $C:y^3=f(x)$ defined over $mathbb{Q}$ at primes $p$ of good reduction for $C$. By determining the density of the set of primes of ordinary reduction, we prove that, for all but a density zero subset of primes, the Zeta function $Z(C_p,T)$ is uniquely determined by the Cartier--Manin matrix $A_p$ of $C$ modulo $p$, the irreducibility of $f$ modulo $p$ (or the failure thereof), and the exponent of the Jacobian of $C$ modulo $p$; we also show that for primes $equiv 1 pmod{3}$ the matrix $A_p$ suffices and that for primes $equiv 2 pmod{3}$ the genericity assumption on $C$ is unnecessary. By combining this with recent work of Sutherland, we obtain a practical probabilistic algorithm of Las Vegas type that computes $Z(C_p,T)$ for almost all primes $p le N$ using $Nlog(N)^{3+o(1)}$ expected bit operations. This is the first practical result of this type for curves of genus greater than 2.
我们研究了一类广义Picard曲线$C:y^3=f(x)$的zeta函数$Z(C_p,T)$序列，该曲线在$mathbb{Q}$上定义在质数$p$上，对$C$有很好的约简。通过确定普通约简素数集合的密度，证明了除了密度为零的素数子集外，Zeta函数$Z(C_p,T)$是由$C$模$p$的Cartier—Manin矩阵$A_p$、$f$模$p$的不可约性(或其失效)和$C$模$p$的雅可比矩阵指数唯一决定的;我们还证明，对于质数$equiv 1 pmod{3}$，矩阵$A_p$是足够的，对于质数$equiv 2 pmod{3}$，在$C$上的一般性假设是不必要的。通过将此与Sutherland最近的工作相结合，我们获得了一个实用的拉斯维加斯类型的概率算法，该算法使用$Nlog(N)^{3+o(1)}$期望位操作来计算几乎所有质数$p le N$的$Z(C_p,T)$。这是该类型对大于2的曲线的第一个实际结果。
{"title":"Computing L-polynomials of Picard curves from Cartier-Manin matrices","authors":"Sualeh Asif, Francesc Fit'e, Dylan Pentland, Andrew V. Sutherland","doi":"10.1090/mcom/3675","DOIUrl":"https://doi.org/10.1090/mcom/3675","url":null,"abstract":"We study the sequence of zeta functions $Z(C_p,T)$ of a generic Picard curve $C:y^3=f(x)$ defined over $mathbb{Q}$ at primes $p$ of good reduction for $C$. By determining the density of the set of primes of ordinary reduction, we prove that, for all but a density zero subset of primes, the Zeta function $Z(C_p,T)$ is uniquely determined by the Cartier--Manin matrix $A_p$ of $C$ modulo $p$, the irreducibility of $f$ modulo $p$ (or the failure thereof), and the exponent of the Jacobian of $C$ modulo $p$; we also show that for primes $equiv 1 pmod{3}$ the matrix $A_p$ suffices and that for primes $equiv 2 pmod{3}$ the genericity assumption on $C$ is unnecessary. By combining this with recent work of Sutherland, we obtain a practical probabilistic algorithm of Las Vegas type that computes $Z(C_p,T)$ for almost all primes $p le N$ using $Nlog(N)^{3+o(1)}$ expected bit operations. This is the first practical result of this type for curves of genus greater than 2.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81918599","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}
Hittmeir recently presented a deterministic algorithm that provably computes the prime factorisation of a positive integer $N$ in $N^{2/9+o(1)}$ bit operations. Prior to this breakthrough, the best known complexity bound for this problem was $N^{1/4+o(1)}$, a result going back to the 1970s. In this paper we push Hittmeir's techniques further, obtaining a rigorous, deterministic factoring algorithm with complexity $N^{1/5+o(1)}$.
{"title":"An exponent one-fifth algorithm for deterministic integer factorisation","authors":"David Harvey","doi":"10.1090/MCOM/3658","DOIUrl":"https://doi.org/10.1090/MCOM/3658","url":null,"abstract":"Hittmeir recently presented a deterministic algorithm that provably computes the prime factorisation of a positive integer $N$ in $N^{2/9+o(1)}$ bit operations. Prior to this breakthrough, the best known complexity bound for this problem was $N^{1/4+o(1)}$, a result going back to the 1970s. In this paper we push Hittmeir's techniques further, obtaining a rigorous, deterministic factoring algorithm with complexity $N^{1/5+o(1)}$.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77977199","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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