Mathematics of Computation最新文献

{"title":"Fourier optimization and Montgomery’s pair correlation conjecture","authors":"Emanuel Carneiro, Micah Milinovich, Antonio Pedro Ramos","doi":"10.1090/mcom/3990","DOIUrl":"https://doi.org/10.1090/mcom/3990","url":null,"abstract":"<p>Assuming the Riemann hypothesis, we improve the current upper and lower bounds for the average value of Montgomery’s function <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F left-parenthesis alpha comma upper T right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">F(alpha , T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over long intervals by means of a Fourier optimization framework. The function <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F left-parenthesis alpha comma upper T right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">F(alpha , T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is often used to study the pair correlation of the non-trivial zeros of the Riemann zeta-function. Two ideas play a central role in our approach: (i) the introduction of new averaging mechanisms in our conceptual framework and (ii) the full use of the class of test functions introduced by Cohn and Elkies for the sphere packing bounds, going beyond the usual class of bandlimited functions. We conclude that such an average value, that is conjectured to be <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1\"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding=\"application/x-tex\">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, lies between <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"0.9303\"> <mml:semantics> <mml:mn>0.9303</mml:mn> <mml:annotation encoding=\"application/x-tex\">0.9303</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1.3208\"> <mml:semantics> <mml:mn>1.3208</mml:mn> <mml:annotation encoding=\"application/x-tex\">1.3208</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Our Fourier optimization framework also yields an improvement on the current bounds for the analogous problem concerning the non-trivial zeros in the family of Dirichlet <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L\"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding=\"application/x-tex\">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-functions.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511719","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}

{"title":"Schinzel-type bounds for the Mahler measure on lemniscates","authors":"Ryan Looney, Igor Pritsker","doi":"10.1090/mcom/3985","DOIUrl":"https://doi.org/10.1090/mcom/3985","url":null,"abstract":"<p>We study the generalized Mahler measure on lemniscates, and prove a sharp lower bound for the measure of totally real integer polynomials that includes the classical result of Schinzel expressed in terms of the golden ratio. Moreover, we completely characterize many cases when this lower bound is attained. For example, we explicitly describe all lemniscates and the corresponding quadratic polynomials that achieve our lower bound for the generalized Mahler measure. It turns out that the extremal polynomials attaining the bound must have even degree. The main computational part of this work is related to finding many extremals of degree four and higher, which is a new feature compared to the original Schinzel’s theorem where only quadratic irreducible extremals are possible.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511722","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}

{"title":"Analysis of stochastic probing methods for estimating the trace of functions of sparse symmetric matrices","authors":"Andreas Frommer, Michele Rinelli, Marcel Schweitzer","doi":"10.1090/mcom/3984","DOIUrl":"https://doi.org/10.1090/mcom/3984","url":null,"abstract":"<p>We consider the problem of estimating the trace of a matrix function <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f left-parenthesis upper A right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">f(A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In certain situations, in particular if <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f left-parenthesis upper A right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">f(A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> cannot be well approximated by a low-rank matrix, combining probing methods based on graph colorings with stochastic trace estimation techniques can yield accurate approximations at moderate cost. So far, such methods have not been thoroughly analyzed, though, but were rather used as efficient heuristics by practitioners. In this manuscript, we perform a detailed analysis of stochastic probing methods and, in particular, expose conditions under which the expected approximation error in the stochastic probing method scales more favorably with the dimension of the matrix than the error in non-stochastic probing. Extending results from Aune, Simpson, and Eidsvik [Stat. Comput. 24 (2014), pp. 247–263], we also characterize situations in which using just one stochastic vector is always—not only in expectation—better than the deterministic probing method. Several numerical experiments illustrate our theory and compare with existing methods.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511721","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}

{"title":"Error bounds for Gauss–Jacobi quadrature of analytic functions on an ellipse","authors":"Hiroshi Sugiura, Takemitsu Hasegawa","doi":"10.1090/mcom/3977","DOIUrl":"https://doi.org/10.1090/mcom/3977","url":null,"abstract":"<p>For the Gauss–Jacobi quadrature on <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-bracket negative 1 comma 1 right-bracket\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">[</mml:mo> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=\"false\">]</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">[-1,1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the location is estimated where the kernel of the error functional for functions analytic on an ellipse and its interior in the complex plane attains its maximum modulus. For the Jacobi weight <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis 1 minus t right-parenthesis Superscript alpha Baseline left-parenthesis 1 plus t right-parenthesis Superscript beta\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>t</mml:mi> <mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mi>α</mml:mi> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>t</mml:mi> <mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mi>β</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(1-t)^alpha (1+t)^beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (<inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"alpha greater-than negative 1\"> <mml:semantics> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">alpha &gt;-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"beta greater-than negative 1\"> <mml:semantics> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">beta &gt;-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) except for the Gegenbauer weight (<inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"alpha equals beta\"> <mml:semantics> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mi>β</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">alpha =beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>), the location is the intersection point of the ellipse with the real axis in the complex plane. For the Gegenbauer weight, it is the intersection point(s) with either the real or the imaginary axis or other axes with angle <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"one fourth pi\"> <mml:semantics> <mml:mrow> <mml:mstyle displaystyle=\"","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140935842","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}

{"title":"Numerical solution of Poisson partial differential equation in high dimension using two-layer neural networks","authors":"Mathias Dus, Virginie Ehrlacher","doi":"10.1090/mcom/3971","DOIUrl":"https://doi.org/10.1090/mcom/3971","url":null,"abstract":"<p>The aim of this article is to analyze numerical schemes using two-layer neural networks with infinite width for the resolution of the high-dimensional Poisson partial differential equation with Neumann boundary condition. Using Barron’s representation of the solution [IEEE Trans. Inform. Theory 39 (1993), pp. 930–945] with a probability measure defined on the set of parameter values, the energy is minimized thanks to a gradient curve dynamic on the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding=\"application/x-tex\">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Wasserstein space of the set of parameter values defining the neural network. Inspired by the work from Bach and Chizat [On the global convergence of gradient descent for over-parameterized models using optimal transport, 2018; ICM–International Congress of Mathematicians, EMS Press, Berlin, 2023], we prove that if the gradient curve converges, then the represented function is the solution of the elliptic equation considered. Numerical experiments are given to show the potential of the method.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511724","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}

{"title":"Müntz Legendre polynomials: Approximation properties and applications","authors":"Tengteng Cui, Chuanju Xu","doi":"10.1090/mcom/3987","DOIUrl":"https://doi.org/10.1090/mcom/3987","url":null,"abstract":"<p>The Müntz Legendre polynomials are a family of generalized orthogonal polynomials, defined by contour integral associated with a complex sequence <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Lamda equals StartSet lamda 0 comma lamda 1 comma lamda 2 comma midline-horizontal-ellipsis EndSet\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"normal\">Λ</mml:mi> <mml:mo>=</mml:mo> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">Lambda ={lambda _{0},lambda _{1},lambda _{2},cdots }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this paper, we are interested in two subclasses of the Müntz Legendre polynomials. Precisely, we theoretically and numerically investigate the basic approximation properties of the Müntz Legendre polynomials for two sets of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Lamda\"> <mml:semantics> <mml:mi mathvariant=\"normal\">Λ</mml:mi> <mml:annotation encoding=\"application/x-tex\">Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> sequences: <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"lamda Subscript k Baseline equals lamda\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mi>k</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>λ</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">lambda _{k}=lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"lamda Subscript k Baseline equals k lamda plus q\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>k</mml:mi> <mml:mi>λ</mml:mi> <mml:mo>+</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">lambda _k=klambda +q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"lamda\"> <mml:semantics> <mml:mi>λ</mml:mi> <mml:annotation encoding=\"application/x-tex\">lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"q\"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding=\"applicat","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511717","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}

{"title":"Virtual element methods for Biot–Kirchhoff poroelasticity","authors":"Rekha Khot, David Mora, Ricardo Ruiz-Baier","doi":"10.1090/mcom/3983","DOIUrl":"https://doi.org/10.1090/mcom/3983","url":null,"abstract":"<p>This paper analyses conforming and nonconforming virtual element formulations of arbitrary polynomial degrees on general polygonal meshes for the coupling of solid and fluid phases in deformable porous plates. The governing equations consist of one fourth-order equation for the transverse displacement of the middle surface coupled with a second-order equation for the pressure head relative to the solid with mixed boundary conditions. We propose novel enrichment operators that connect nonconforming virtual element spaces of general degree to continuous Sobolev spaces. These operators satisfy additional orthogonal and best-approximation properties (referred to as conforming companion operators in the context of finite element methods), which play an important role in the nonconforming methods. This paper proves a priori error estimates in the best-approximation form, and derives residual–based reliable and efficient a posteriori error estimates in appropriate norms, and shows that these error bounds are robust with respect to the main model parameters. The computational examples illustrate the numerical behaviour of the suggested virtual element discretisations and confirm the theoretical findings on different polygonal meshes with mixed boundary conditions.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511720","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}

{"title":"Extensible grid sampling for quantile estimation","authors":"Jingyu Tan, Zhijian He, Xiaoqun Wang","doi":"10.1090/mcom/3986","DOIUrl":"https://doi.org/10.1090/mcom/3986","url":null,"abstract":"<p>Quantiles are used as a measure of risk in many stochastic systems. We study the estimation of quantiles with the Hilbert space-filling curve (HSFC) sampling scheme that transforms specifically chosen one-dimensional points into high dimensional stratified samples while still remaining the extensibility. We study the convergence and asymptotic normality for the estimate based on HSFC. By a generalized Dvoretzky–Kiefer–Wolfowitz inequality for independent but not identically distributed samples, we establish the strong consistency for such an estimator. We find that under certain conditions, the distribution of the quantile estimator based on HSFC is asymptotically normal. The asymptotic variance is of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis n Superscript negative 1 minus 1 slash d Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">O(n^{-1-1/d})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when using <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> HSFC-based quadrature points in dimension <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d\"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding=\"application/x-tex\">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which is more efficient than the Monte Carlo sampling and the Latin hypercube sampling. Since the asymptotic variance does not admit an explicit form, we establish an asymptotically valid confidence interval by the batching method. We also prove a Bahadur representation for the quantile estimator based on HSFC. Numerical experiments show that the quantile estimator is asymptotically normal with a comparable mean squared error rate of randomized quasi-Monte Carlo (RQMC) sampling. Moreover, the coverage of the confidence intervals constructed with HSFC is better than that with RQMC.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511723","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}

{"title":"A random active set method for strictly convex quadratic problem with simple bounds","authors":"Ran Gu, Bing Gao","doi":"10.1090/mcom/3982","DOIUrl":"https://doi.org/10.1090/mcom/3982","url":null,"abstract":"<p>The active set method aims at finding the correct active set of the optimal solution and it is a powerful method for solving strictly convex quadratic problems with bound constraints. To guarantee the finite step convergence, existing active set methods all need strict conditions or some additional strategies, which can significantly impact the efficiency of the algorithm. In this paper, we propose a random active set method that introduces randomness in the active set’s update process. We prove that the algorithm can converge in a finite number of iterations with probability one, without any extra conditions on the problem or any supplementary strategies. At last, numerical experiments show that the algorithm can obtain the correct active set within a few iterations, and it has better efficiency and robustness than the existing methods.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141511718","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}

{"title":"Uniform error estimate of an asymptotic preserving scheme for the Lévy-Fokker-Planck equation","authors":"Weiran Sun, Li Wang","doi":"10.1090/mcom/3975","DOIUrl":"https://doi.org/10.1090/mcom/3975","url":null,"abstract":"<p>We establish a uniform-in-scaling error estimate for the asymptotic preserving (AP) scheme proposed by Xu and Wang [Commun. Math. Sci. 21 (2023), pp. 1–23] for the Lévy-Fokker-Planck (LFP) equation. The main difficulties stem not only from the interplay between the scaling and numerical parameters but also the slow decay of the tail of the equilibrium state. We tackle these problems by separating the parameter domain according to the relative size of the scaling parameter <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon\"> <mml:semantics> <mml:mi>ε</mml:mi> <mml:annotation encoding=\"application/x-tex\">varepsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula>: in the regime where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon\"> <mml:semantics> <mml:mi>ε</mml:mi> <mml:annotation encoding=\"application/x-tex\">varepsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is large, we design a weighted norm to mitigate the issue caused by the fat tail, while in the regime where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon\"> <mml:semantics> <mml:mi>ε</mml:mi> <mml:annotation encoding=\"application/x-tex\">varepsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is small, we prove a strong convergence of LFP towards its fractional diffusion limit with an explicit convergence rate. This method extends the traditional AP estimates to cases where uniform bounds are unavailable. Our result applies to any dimension and to the whole span of the fractional power.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140942482","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}