The recently introduced Genetic Column Generation (GenCol) algorithm has been numerically observed to efficiently and accurately compute high-dimensional optimal transport (OT) plans for general multi-marginal problems, but theoretical results on the algorithm have hitherto been lacking. The algorithm solves the OT linear program on a dynamically updated low-dimensional submanifold consisting of sparse plans. The submanifold dimension exceeds the sparse support of optimal plans only by a fixed factor βbeta. Here we prove that for β≥2beta geq 2 and in the two-marginal case, GenCol always converges to an exact solution, for arbitrary costs and marginals. The proof relies on the concept of c-cyclical monotonicity. As an offshoot, GenCol rigorously reduces the data complexity of numerically solving two-marginal OT problems from O(ℓ2)O(ell ^2) to O(ℓ)O(ell ) without any loss in accuracy, where ℓell is the number of discretization points for a single marginal. At the end of the paper we also present some insights into the convergence behavior in the multi-marginal case.
最近推出的遗传列生成（GenCol）算法已被数值观测到，可以高效、准确地计算一般多边际问题的高维最优运输（OT）计划，但迄今为止还缺乏有关该算法的理论成果。该算法在由稀疏计划组成的动态更新的低维子平面上求解 OT 线性程序。子平面的维度仅以固定系数 β beta 的方式超出最优计划的稀疏支持。在这里，我们将证明对于 β ≥ 2 beta geq 2 和双边际情况，GenCol 总是收敛于精确解，适用于任意成本和边际。证明依赖于 c 周期单调性的概念。作为一个分支，GenCol 严格地将数值求解双边际 OT 问题的数据复杂度从 O ( ℓ 2 ) O(ell ^2) 降低到 O ( ℓ ) O(ell)，并且没有任何精度损失，其中 ℓ ell 是单个边际的离散点数。在本文的最后，我们还提出了对多边际情况下收敛行为的一些见解。
{"title":"Convergence proof for the GenCol algorithm in the case of two-marginal optimal transport","authors":"Gero Friesecke, Maximilian Penka","doi":"10.1090/mcom/3968","DOIUrl":"https://doi.org/10.1090/mcom/3968","url":null,"abstract":"<p>The recently introduced Genetic Column Generation (GenCol) algorithm has been numerically observed to efficiently and accurately compute high-dimensional optimal transport (OT) plans for general multi-marginal problems, but theoretical results on the algorithm have hitherto been lacking. The algorithm solves the OT linear program on a dynamically updated low-dimensional submanifold consisting of sparse plans. The submanifold dimension exceeds the sparse support of optimal plans only by a fixed factor <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"beta\"> <mml:semantics> <mml:mi>β<!-- β --></mml:mi> <mml:annotation encoding=\"application/x-tex\">beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Here we prove that for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"beta greater-than-or-equal-to 2\"> <mml:semantics> <mml:mrow> <mml:mi>β<!-- β --></mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">beta geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and in the two-marginal case, GenCol always converges to an exact solution, for arbitrary costs and marginals. The proof relies on the concept of c-cyclical monotonicity. As an offshoot, GenCol rigorously reduces the data complexity of numerically solving two-marginal OT problems from <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis script l squared right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">O(ell ^2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis script l right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">O(ell )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> without any loss in accuracy, where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script l\"> <mml:semantics> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the number of discretization points for a single marginal. At the end of the paper we also present some insights into the convergence behavior in the multi-marginal case.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140935884","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}
Matthew de Courcy-Ireland, Maria Dostert, Maryna Viazovska
We prove that the Cohn–Elkies linear programming bound for sphere packing is not sharp in dimension 6. The proof uses duality and optimization over a space of modular forms, generalizing a construction of Cohn–Triantafillou [Math. Comp. 91 (2021), pp. 491–508] to the case of odd weight and non-trivial character.
{"title":"Six-dimensional sphere packing and linear programming","authors":"Matthew de Courcy-Ireland, Maria Dostert, Maryna Viazovska","doi":"10.1090/mcom/3959","DOIUrl":"https://doi.org/10.1090/mcom/3959","url":null,"abstract":"<p>We prove that the Cohn–Elkies linear programming bound for sphere packing is not sharp in dimension 6. The proof uses duality and optimization over a space of modular forms, generalizing a construction of Cohn–Triantafillou [Math. Comp. 91 (2021), pp. 491–508] to the case of odd weight and non-trivial character.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140935684","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}
We study the convergence of Langevin-Simulated Annealing type algorithms with multiplicative noise, i.e. for V:Rd→RV : mathbb {R}^d to mathbb {R} a potential function to minimize, we consider the stochastic differential equation dYt=−σσ⊤∇V(Yt)dY_t = - sigma sigma ^top nabla V(Y_t)dt+a(t)σ(Yt)dWt+a(t)2Υ(Yt)dt
我们研究了带有乘法噪声的朗格文模拟退火算法的收敛性，即对于 V : R d → R V ：mathbb {R}^d to mathbb {R} 的势函数最小化、我们考虑随机微分方程 d Y t = - σ σ ⊤∇ V ( Y t ) dY_t = -V(Y_t) d t + a ( t ) σ ( Y t ) d W t + a ( t ) 2 Υ ( Y t ) d t dt + a(t)sigma (Y_t)dW_t + a(t)^2Upsilon (Y_t)dt 、其中 ( W t ) (W_t) 是布朗运动，其中 σ ： R d → M d ( R ) σ : mathbb {R}^d to mathcal {M}_d(mathbb {R}) 是一个自适应（乘法）噪声，其中 a : R + → R + a : mathbb {R}^+ to mathbb {R}^+ 是一个递减到 0 0 的函数，Υ Upsilon 是一个修正项。这种设置可以应用于机器学习中出现的优化问题；与经典的朗格文方程 d Y t = -∇ V ( Y t ) d t + σ d W t dY_t = -nabla V(Y_t)dt + sigma dW_t 相比，允许 σ sigma 取决于位置会带来更快的收敛速度。σ sigma 是常量矩阵的情况已被广泛研究，但对一般情况的研究却很少。我们证明了 Y t 的 L 1 L^1 - Wasserstein 距离的收敛性。我们证明了 Y t Y_t 和相关欧拉方案 Y ¯ t （bar {Y}_t）的瓦瑟斯坦距离收敛于某个由 argmin ( V ) operatorname {argmin}(V) 支持的度量 ν ⋆ nu ^star ，并给出了密度 ∝ exp ( - 2 V ( x ) / a ( t ) 2 ) 的瞬时吉布斯度量 ν a ( t ) nu _{a(t)} 的收敛速率。 propto exp (-2V(x)/a(t)^2) .为此，我们首先考虑 a a 是片断常数函数的情况。我们再次找到经典的时间表 a ( t ) = A log - 1 / 2 ( t ) a(t) = Alog ^{-1/2}(t) 。然后，我们利用遍历特性给出了步进常数情况下的瓦瑟斯坦距离的边界，从而证明了一般情况下的收敛性。
{"title":"Convergence of Langevin-simulated annealing algorithms with multiplicative noise","authors":"Pierre Bras, Gilles Pagès","doi":"10.1090/mcom/3899","DOIUrl":"https://doi.org/10.1090/mcom/3899","url":null,"abstract":"<p>We study the convergence of Langevin-Simulated Annealing type algorithms with multiplicative noise, i.e. for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper V colon double-struck upper R Superscript d Baseline right-arrow double-struck upper R\"> <mml:semantics> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>:</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo stretchy=\"false\">→<!-- → --></mml:mo> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">V : mathbb {R}^d to mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a potential function to minimize, we consider the stochastic differential equation <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d upper Y Subscript t Baseline equals minus sigma sigma Superscript down-tack Baseline nabla upper V left-parenthesis upper Y Subscript t Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi>σ<!-- σ --></mml:mi> <mml:msup> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi mathvariant=\"normal\">⊤<!-- ⊤ --></mml:mi> </mml:msup> <mml:mi mathvariant=\"normal\">∇<!-- ∇ --></mml:mi> <mml:mi>V</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">dY_t = - sigma sigma ^top nabla V(Y_t)</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=\"d t plus a left-parenthesis t right-parenthesis sigma left-parenthesis upper Y Subscript t Baseline right-parenthesis d upper W Subscript t plus a left-parenthesis t right-parenthesis squared normal upper Upsilon left-parenthesis upper Y Subscript t Baseline right-parenthesis d t\"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mo>+</mml:mo> <mml:mi>a</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mi>d</mml:mi> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>a</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>t</mml:mi> <mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mn>2</mml:mn> </mml:msup> <mml:mi mathvariant=\"normal\">Υ<!-- Υ --></mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> </mml:mro","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931328","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}
In this paper we study the nonconvex constrained composition optimization, in which the objective contains a composition of two expected-value functions whose accurate information is normally expensive to calculate. We propose a STochastic nEsted Primal-dual (STEP) method for such problems. In each iteration, with an auxiliary variable introduced to track the inner layer function values we compute stochastic gradients of the nested function using a subsampling strategy. To alleviate difficulties caused by possibly nonconvex constraints, we construct a stochastic approximation to the linearized augmented Lagrangian function to update the primal variable, which further motivates to update the dual variable in a weighted-average way. Moreover, to better understand the asymptotic dynamics of the update schemes we consider a deterministic continuous-time system from the perspective of ordinary differential equation (ODE). We analyze the Karush-Kuhn-Tucker measure at the output by the STEP method with constant parameters and establish its iteration and sample complexities to find an ϵepsilon-stationary point, ensuring that expected stationarity, feasibility as well as complementary slackness are below accuracy ϵepsilon. To leverage the benefit of the (near) initial feasibility in the STEP method, we propose a two-stage framework incorporating a feasibility-seeking phase, aiming to locate a nearly feasible initial point. Moreover, to enhance the adaptivity of the STEP algorithm, we propose an adaptive variant by adaptively adjusting its parameters, along with a complexity analysis. Numerical results on a risk-averse portfolio optimization problem and an orthogonal nonnegative matrix decomposition reveal the effectiveness of the proposed algorithms.
{"title":"Stochastic nested primal-dual method for nonconvex constrained composition optimization","authors":"Lingzi Jin, Xiao Wang","doi":"10.1090/mcom/3965","DOIUrl":"https://doi.org/10.1090/mcom/3965","url":null,"abstract":"<p>In this paper we study the nonconvex constrained composition optimization, in which the objective contains a composition of two expected-value functions whose accurate information is normally expensive to calculate. We propose a STochastic nEsted Primal-dual (STEP) method for such problems. In each iteration, with an auxiliary variable introduced to track the inner layer function values we compute stochastic gradients of the nested function using a subsampling strategy. To alleviate difficulties caused by possibly nonconvex constraints, we construct a stochastic approximation to the linearized augmented Lagrangian function to update the primal variable, which further motivates to update the dual variable in a weighted-average way. Moreover, to better understand the asymptotic dynamics of the update schemes we consider a deterministic continuous-time system from the perspective of ordinary differential equation (ODE). We analyze the Karush-Kuhn-Tucker measure at the output by the STEP method with constant parameters and establish its iteration and sample complexities to find an <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon\"> <mml:semantics> <mml:mi>ϵ<!-- ϵ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">epsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-stationary point, ensuring that expected stationarity, feasibility as well as complementary slackness are below accuracy <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon\"> <mml:semantics> <mml:mi>ϵ<!-- ϵ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">epsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. To leverage the benefit of the (near) initial feasibility in the STEP method, we propose a two-stage framework incorporating a feasibility-seeking phase, aiming to locate a nearly feasible initial point. Moreover, to enhance the adaptivity of the STEP algorithm, we propose an adaptive variant by adaptively adjusting its parameters, along with a complexity analysis. Numerical results on a risk-averse portfolio optimization problem and an orthogonal nonnegative matrix decomposition reveal the effectiveness of the proposed algorithms.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931201","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}
Implicit-explicit Runge-Kutta (IMEX-RK) schemes are popular methods to treat multiscale equations that contain a stiff part and a non-stiff part, where the stiff part is characterized by a small parameter εvarepsilon. In this work, we prove rigorously the uniform stability and uniform accuracy of a class of IMEX-RK schemes for a linear hyperbolic system with stiff relaxation. The result we obtain is optimal in the sense that it holds regardless of the value of εvarepsilon and the order of accuracy is the same as the design order of the original scheme, i.e., there is no order reduction.
{"title":"Uniform accuracy of implicit-explicit Runge-Kutta (IMEX-RK) schemes for hyperbolic systems with relaxation","authors":"Jingwei Hu, Ruiwen Shu","doi":"10.1090/mcom/3967","DOIUrl":"https://doi.org/10.1090/mcom/3967","url":null,"abstract":"<p>Implicit-explicit Runge-Kutta (IMEX-RK) schemes are popular methods to treat multiscale equations that contain a stiff part and a non-stiff part, where the stiff part is characterized by a small 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<!-- ε --></mml:mi> <mml:annotation encoding=\"application/x-tex\">varepsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this work, we prove rigorously the uniform stability and uniform accuracy of a class of IMEX-RK schemes for a linear hyperbolic system with stiff relaxation. The result we obtain is optimal in the sense that it holds regardless of the value of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon\"> <mml:semantics> <mml:mi>ε<!-- ε --></mml:mi> <mml:annotation encoding=\"application/x-tex\">varepsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the order of accuracy is the same as the design order of the original scheme, i.e., there is no order reduction.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140942540","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}
We study the Mahler measures of the polynomial family Qk(x,y)=x3+y3+1−kxyQ_k(x,y) = x^3+y^3+1-kxy using the method previously developed by the authors. An algorithm is implemented to search for complex multiplication points with class numbers ⩽3leqslant 3, we employ these points to derive interesting formulas that link the Mahler measures of Qk(x,y)Q_k(x,y) to LL-values of modular forms. As by-products, some conjectural identities of Samart are confirmed, one of them involves the modified Mahler measure
我们利用作者之前开发的方法研究了多项式族 Q k ( x , y ) = x 3 + y 3 + 1 - k x y Q_k(x,y) = x^3+y^3+1-kxy 的马勒度量。我们利用这些点推导出有趣的公式，将 Q k ( x , y ) Q_k(x,y) 的马勒度量与模块形式的 L L 值联系起来。作为副产品，萨马特的一些猜想得到了证实，其中之一涉及萨马特最近引入的修正马勒度量 n ~ ( k ) （tilde {n}(k)）。对于 k = 729 ± 405 3 3 k=sqrt [3]{729pm 405sqrt {3}} ，我们也证明了表示 n ~ ( k ) 的等式。 对于 k = 729 ± 405 3 3 k= （sqrt [3]{729pm 405sqrt {3}}，我们还证明了一个等式，该等式将 Q k ( x , y ) 的马勒度量 Q_k(x,y)的 2 × 2 2 （times 2 ）行列式表达为 Q ( 3 ) 上两个同源椭圆曲线的 L L - 值的某个倍数（mathbb {Q}(sqrt {3})）。
{"title":"CM points, class numbers, and the Mahler measures of 𝑥³+𝑦³+1-𝑘𝑥𝑦","authors":"Zhengyu Tao, Xuejun Guo","doi":"10.1090/mcom/3961","DOIUrl":"https://doi.org/10.1090/mcom/3961","url":null,"abstract":"<p>We study the Mahler measures of the polynomial family <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Q Subscript k Baseline left-parenthesis x comma y right-parenthesis equals x cubed plus y cubed plus 1 minus k x y\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>Q</mml:mi>\u0000 <mml:mi>k</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>y</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:msup>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mn>3</mml:mn>\u0000 </mml:msup>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:msup>\u0000 <mml:mi>y</mml:mi>\u0000 <mml:mn>3</mml:mn>\u0000 </mml:msup>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mi>y</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Q_k(x,y) = x^3+y^3+1-kxy</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> using the method previously developed by the authors. An algorithm is implemented to search for complex multiplication points with class numbers <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"less-than-or-slanted-equals 3\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo>⩽<!-- ⩽ --></mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">leqslant 3</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, we employ these points to derive interesting formulas that link the Mahler measures of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Q Subscript k Baseline left-parenthesis x comma y right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>Q</mml:mi>\u0000 <mml:mi>k</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>y</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Q_k(x,y)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> to <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L\">\u0000 <mml:semantics>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">L</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-values of modular forms. As by-products, some conjectural identities of Samart are confirmed, one of them involves the modified Mahler measure <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"ModifyingAbove n With tilde left-parenthesis k right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140249718","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}
Radial basis functions (RBFs) are prominent examples for reproducing kernels with associated reproducing kernel Hilbert spaces (RKHSs). The convergence theory for the kernel-based interpolation in that space is well understood and optimal rates for the whole RKHS are often known. Schaback added the doubling trick [Math. Comp. 68 (1999), pp. 201–216], which shows that functions having double the smoothness required by the RKHS (along with specific, albeit complicated boundary behavior) can be approximated with higher convergence rates than the optimal rates for the whole space. Other advances allowed interpolation of target functions which are less smooth, and different norms which measure interpolation error. The current state of the art of error analysis for RBF interpolation treats target functions having smoothness up to twice that of the native space, but error measured in norms which are weaker than that required for membership in the RKHS.
Motivated by the fact that the kernels and the approximants they generate are smoother than required by the native space, this article extends the doubling trick to error which measures higher smoothness. This extension holds for a family of kernels satisfying easily checked hypotheses which we describe in this article, and includes many prominent RBFs. In the course of the proof, new convergence rates are obtained for the abstract operator considered by Devore and Ron in [Trans. Amer. Math. Soc. 362 (2010), pp. 6205–6229], and new Bernstein estimates are obtained relating high order smoothness norms to the native space norm.
{"title":"Extending error bounds for radial basis function interpolation to measuring the error in higher order Sobolev norms","authors":"T. Hangelbroek, C. Rieger","doi":"10.1090/mcom/3960","DOIUrl":"https://doi.org/10.1090/mcom/3960","url":null,"abstract":"<p>Radial basis functions (RBFs) are prominent examples for reproducing kernels with associated reproducing kernel Hilbert spaces (RKHSs). The convergence theory for the kernel-based interpolation in that space is well understood and optimal rates for the whole RKHS are often known. Schaback added the doubling trick [Math. Comp. 68 (1999), pp. 201–216], which shows that functions having double the smoothness required by the RKHS (along with specific, albeit complicated boundary behavior) can be approximated with higher convergence rates than the optimal rates for the whole space. Other advances allowed interpolation of target functions which are less smooth, and different norms which measure interpolation error. The current state of the art of error analysis for RBF interpolation treats target functions having smoothness up to twice that of the native space, but error measured in norms which are weaker than that required for membership in the RKHS.</p> <p>Motivated by the fact that the kernels and the approximants they generate are smoother than required by the native space, this article extends the doubling trick to error which measures higher smoothness. This extension holds for a family of kernels satisfying easily checked hypotheses which we describe in this article, and includes many prominent RBFs. In the course of the proof, new convergence rates are obtained for the abstract operator considered by Devore and Ron in [Trans. Amer. Math. Soc. 362 (2010), pp. 6205–6229], and new Bernstein estimates are obtained relating high order smoothness norms to the native space norm.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931238","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}
The ground states of Bose–Einstein condensates in a rotating frame can be described as constrained minimizers of the Gross–Pitaevskii energy functional with an angular momentum term. In this paper we consider the corresponding discrete minimization problem in Lagrange finite element spaces of arbitrary polynomial order and we investigate the approximation properties of discrete ground states. In particular, we prove a priori error estimates of optimal order in the L2L^2- and H1H^1-norm, as well as for the ground state energy and the corresponding chemical potential. A central issue in the analysis of the problem is the missing uniqueness of ground states, which is mainly caused by the invariance of the energy functional under complex phase shifts. Our error analysis is therefore based on an Euler–Lagrange functional that we restrict to certain tangent spaces in which we have local uniqueness of ground states. This gives rise to an error decomposition that is ultimately used to derive the desired a priori error estimates. We also present numerical experiments to illustrate various aspects of the problem structure.
旋转框架中玻色-爱因斯坦凝聚态的基态可以描述为带有角动量项的格罗斯-皮塔耶夫斯基能量函数的约束最小化。本文考虑了任意多项式阶拉格朗日有限元空间中相应的离散最小化问题，并研究了离散基态的近似特性。特别是，我们证明了 L 2 L^2 - 和 H 1 H^1 - 规范中最优阶的先验误差估计，以及基态能量和相应化学势的先验误差估计。问题分析中的一个核心问题是基态唯一性的缺失，这主要是由于复相移下能量函数的不变性造成的。因此，我们的误差分析基于欧拉-拉格朗日函数，并将其限制在具有局部唯一基态的特定切空间。这就产生了误差分解，最终用于推导所需的先验误差估计。我们还通过数值实验来说明问题结构的各个方面。
{"title":"On discrete ground states of rotating Bose–Einstein condensates","authors":"Patrick Henning, Mahima Yadav","doi":"10.1090/mcom/3962","DOIUrl":"https://doi.org/10.1090/mcom/3962","url":null,"abstract":"<p>The ground states of Bose–Einstein condensates in a rotating frame can be described as constrained minimizers of the Gross–Pitaevskii energy functional with an angular momentum term. In this paper we consider the corresponding discrete minimization problem in Lagrange finite element spaces of arbitrary polynomial order and we investigate the approximation properties of discrete ground states. In particular, we prove a priori error estimates of optimal order in the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L squared\"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">L^2</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=\"upper H Superscript 1\"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">H^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-norm, as well as for the ground state energy and the corresponding chemical potential. A central issue in the analysis of the problem is the missing uniqueness of ground states, which is mainly caused by the invariance of the energy functional under complex phase shifts. Our error analysis is therefore based on an Euler–Lagrange functional that we restrict to certain tangent spaces in which we have local uniqueness of ground states. This gives rise to an error decomposition that is ultimately used to derive the desired a priori error estimates. We also present numerical experiments to illustrate various aspects of the problem structure.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140942064","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}
We study the GMRES algorithm applied to linear systems of equations involving a scaled and shifted N×NNtimes N matrix whose entries are independent complex Gaussians. When the right-hand side of this linear system is independent of this random matrix, the N→∞Nto infty behavior of the GMRES residual error can be determined exactly. To handle cases where the right hand side depends on the random matrix, we study the pseudospectra and numerical range of Ginibre matrices and prove a restricted version of Crouzeix’s conjecture.
我们研究了应用于线性方程组的 GMRES 算法，该方程组涉及一个经过缩放和移位的 N × N N 次矩阵，该矩阵的条目是独立的复高斯。当这个线性方程组的右边独立于这个随机矩阵时，GMRES残余误差的 N → ∞ Nto infty 行为就可以精确地确定。为了处理右手侧依赖于随机矩阵的情况，我们研究了吉尼布雷矩阵的伪谱和数值范围，并证明了克鲁齐猜想的限制版本。
{"title":"GMRES, pseudospectra, and Crouzeix’s conjecture for shifted and scaled Ginibre matrices","authors":"Tyler Chen, Anne Greenbaum, Thomas Trogdon","doi":"10.1090/mcom/3963","DOIUrl":"https://doi.org/10.1090/mcom/3963","url":null,"abstract":"<p>We study the GMRES algorithm applied to linear systems of equations involving a scaled and shifted <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N times upper N\"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>×<!-- × --></mml:mo> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">Ntimes N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> matrix whose entries are independent complex Gaussians. When the right-hand side of this linear system is independent of this random matrix, the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N right-arrow normal infinity\"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo stretchy=\"false\">→<!-- → --></mml:mo> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">Nto infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> behavior of the GMRES residual error can be determined exactly. To handle cases where the right hand side depends on the random matrix, we study the pseudospectra and numerical range of Ginibre matrices and prove a restricted version of Crouzeix’s conjecture.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931458","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}
This is a brief report on some recent large-scale Monte Carlo simulations for approximating the density of zeros in character tables of large symmetric groups. Previous computations suggested that a large fraction of zeros cannot be explained by classical vanishing results. Our computations eclipse previous ones and suggest that the opposite is true. In fact, we find empirically that almost all of the zeros are of a single classical type.
{"title":"Large-scale Monte Carlo simulations for zeros in character tables of symmetric groups","authors":"Alexander Miller, Danny Scheinerman","doi":"10.1090/mcom/3964","DOIUrl":"https://doi.org/10.1090/mcom/3964","url":null,"abstract":"<p>This is a brief report on some recent large-scale Monte Carlo simulations for approximating the density of zeros in character tables of large symmetric groups. Previous computations suggested that a large fraction of zeros cannot be explained by classical vanishing results. Our computations eclipse previous ones and suggest that the opposite is true. In fact, we find empirically that almost all of the zeros are of a single classical type.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206155","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}
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