Mathematics of Computation最新文献_第5页

Identifying the source term in the potential equation with weighted sparsity regularization 利用加权稀疏正则化识别电势方程中的源项

IF 2 2区数学 Q1 MATHEMATICS, APPLIED

Mathematics of Computation

Pub Date : 2024-01-17 DOI: 10.1090/mcom/3941

Ole Elvetun, Bjørn Nielsen

We explore the possibility for using boundary measurements to recover a sparse source term f ( x ) f(x) in the potential equation. Employing weighted sparsity regularization and standard results for subgradients, we derive simple-to-check criteria which assure that a number of sinks ( f ( x ) > 0 f(x)>0 ) and sources ( f ( x ) > 0 f(x)>0 ) can be identified. Furthermore, we present two cases for which these criteria always are fulfilled: (a) well-separated sources and sinks, and (b) many sources or sinks located at the boundary plus one interior source/sink. Our approach is such that the linearity of the associated forward operator is preserved in the discrete formulation. The theory is therefore conveniently developed in terms of Euclidean spaces, and it can be applied to a wide range of problems. In particular, it can be applied to both isotropic and anisotropic cases. We present a series of numerical experiments. This work is motivated by the observation that standard methods typically suggest that internal sinks and sources are located close to the boundary.

我们探讨了利用边界测量来恢复势方程中稀疏源项 f ( x ) f(x) 的可能性。利用加权稀疏正则化和子梯度的标准结果，我们推导出简单易查的标准，确保可以识别出若干汇（ f ( x ) > 0 f(x)>0 ）和源（ f ( x ) > 0 f(x)>0 ）。此外，我们还介绍了始终满足这些标准的两种情况：(a) 源和汇完全分离；(b) 许多源或汇位于边界加上一个内部源/汇。我们的方法是在离散表述中保留相关前向算子的线性。因此，该理论可以方便地在欧几里得空间中展开，并可应用于各种问题。特别是，它既可用于各向同性的情况，也可用于各向异性的情况。我们介绍了一系列数值实验。标准方法通常认为内部汇和源位于边界附近，而这一观察结果正是这项工作的动机。

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引用次数: 0

Convergence analysis of Laguerre approximations for analytic functions 解析函数拉盖尔近似的收敛性分析

IF 2 2区数学 Q1 MATHEMATICS, APPLIED

Mathematics of Computation

Pub Date : 2024-01-17 DOI: 10.1090/mcom/3942

Haiyong Wang

Laguerre spectral approximations play an important role in the development of efficient algorithms for problems in unbounded domains. In this paper, we present a comprehensive convergence rate analysis of Laguerre spectral approximations for analytic functions. By exploiting contour integral techniques from complex analysis, we prove that Laguerre projection and interpolation methods of degree n n converge at the root-exponential rate O ( exp ⁡ ( − 2 ρ n ) ) O(exp (-2rho sqrt {n})) with ρ > 0 rho >0 when the underlying function is analytic inside and on a parabola with focus at the origin and vertex at z = − ρ 2 z=-rho ^2 . As far as we know, this is the first rigorous proof of root-exponential convergence of Laguerre approximations for analytic functions. Several important applications of our analysis are also discussed, including Laguerre spectral differentiations, Gauss-Laguerre quadrature rules, the scaling factor and the Weeks method for the inversion of Laplace transform, and some sharp convergence rate estimates are derived. Numerical experiments are presented to verify the theoretical results.

拉盖尔谱近似在无界域问题的高效算法开发中发挥着重要作用。在本文中，我们对分析函数的拉盖尔谱近似进行了全面的收敛率分析。通过利用复分析中的等值线积分技术，我们证明了 n n 级的拉盖尔投影和插值方法以根指数收敛率 O ( exp ( - 2 ρ n ) ) O(exp (-2rho sqrt {n}))，其中 ρ > 0 rho >0，当基础函数在抛物线内部和抛物线上解析时，抛物线的焦点在原点，顶点在 z = - ρ 2 z=-rho ^2。据我们所知，这是第一次严格证明分析函数的拉盖尔近似的根指数收敛性。我们的分析还讨论了几个重要应用，包括拉盖尔谱微分、高斯-拉盖尔正交规则、拉普拉斯变换反演的缩放因子和威克斯方法，并推导出一些尖锐的收敛率估计值。还给出了数值实验来验证理论结果。

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引用次数: 0

On median filters for motion by mean curvature 关于平均曲率运动的中值滤波器

IF 2 2区数学 Q1 MATHEMATICS, APPLIED

Mathematics of Computation

Pub Date : 2024-01-17 DOI: 10.1090/mcom/3940

Selim Esedoḡlu, Jiajia Guo, David Li

The median filter scheme is an elegant, monotone discretization of the level set formulation of motion by mean curvature. It turns out to evolve every level set of the initial condition precisely by another class of methods known as threshold dynamics. Median filters are, in other words, the natural level set versions of threshold dynamics algorithms. Exploiting this connection, we revisit median filters in light of recent progress on the threshold dynamics method. In particular, we give a variational formulation of, and exhibit a Lyapunov function for, median filters, resulting in energy based unconditional stability properties. The connection also yields analogues of median filters in the multiphase setting of mean curvature flow of networks. These new multiphase level set methods do not require frequent redistancing, and can accommodate a wide range of surface tensions.

中值滤波方案是对平均曲率运动的水平集表述的一种优雅的单调离散化。事实证明，它可以通过另一类称为阈值动力学的方法精确地演化初始条件的每个水平集。换句话说，中值滤波器是阈值动力学算法的自然水平集版本。利用这种联系，我们根据阈值动力学方法的最新进展重新审视了中值滤波器。特别是，我们给出了中值滤波器的变分公式，并展示了中值滤波器的 Lyapunov 函数，从而获得了基于能量的无条件稳定性。这种联系还产生了中值滤波器在网络平均曲率流的多相设置中的类比。这些新的多相水平集方法不需要频繁地重新调整，并能适应广泛的表面张力。

引用次数: 0

Faster truncated integer multiplication 更快的截断整数乘法

IF 2 2区数学 Q1 MATHEMATICS, APPLIED

Mathematics of Computation

Pub Date : 2024-01-10 DOI: 10.1090/mcom/3939

David Harvey

We present new algorithms for computing the low n n bits or the high n n bits of the product of two n n -bit integers. We show that these problems may be solved in asymptotically 75 75% of the time required to compute the full 2 n 2n -bit product, assuming that the underlying integer multiplication algorithm relies on computing cyclic convolutions of sequences of real numbers.

我们提出了计算两个 n n 位整数乘积的低 n n 位或高 n n 位的新算法。我们的研究表明，假设底层整数乘法算法依赖于计算实数序列的循环卷积，那么这些问题可以在计算全部 2 n 2n 位乘积所需时间的 75% 左右的时间内得到解决。

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引用次数: 0

Rational group algebras of generalized strongly monomial groups: Primitive idempotents and units 广义强单项式群的有理群代数：原始幂等数和单位

IF 2 2区数学 Q1 MATHEMATICS, APPLIED

Mathematics of Computation

Pub Date : 2024-01-03 DOI: 10.1090/mcom/3937

Gurmeet Bakshi, Jyoti Garg, Gabriela Olteanu

We present a method to explicitly compute a complete set of orthogonal primitive idempotents in a simple component with Schur index 1 of a rational group algebra Q G mathbb {Q}G for G G a finite generalized strongly monomial group. For the same groups with no exceptional simple components in Q G mathbb {Q}G , we describe a subgroup of finite index in the group of units U ( Z G ) mathcal {U}(mathbb {Z}G) of the integral group ring Z G mathbb {Z}G that is generated by three nilpotent groups for which we give explicit description of their generators. We exemplify the theoretical constructions with a detailed concrete example to illustrate the theory. We also show that the Frobenius groups of odd order with a cyclic complement are a class of generalized strongly monomial groups where the theory developed in this paper is applicable.

对于有限广义强单项式群 G G，我们提出了一种明确计算有理群代数 Q G mathbb {Q}G 中舒尔指数为 1 的简单分量中完整的正交原始幂级数的方法。对于 Q G mathbb {Q}G 中没有特殊简单成分的相同群，我们描述了积分群环 Z G mathbb {Z}G 的单位群 U ( Z G ) mathcal {U}(mathbb {Z}G) 中的有限指数子群，该子群由三个零能群生成，我们给出了它们的生成子的明确描述。我们用一个详细的具体例子来举例说明理论构造。我们还证明了具有循环补集的奇阶弗罗贝纽斯群是一类广义强单项式群，本文所建立的理论适用于这类群。

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引用次数: 0

Optimal transportation for electrical impedance tomography 电阻抗断层成像的最佳传输

2区数学 Q1 MATHEMATICS, APPLIED

Mathematics of Computation

Pub Date : 2023-11-13 DOI: 10.1090/mcom/3919

Gang Bao, Yixuan Zhang

This work establishes a framework for solving inverse boundary problems with the geodesic-based quadratic Wasserstein distance ( W 2 W_{2} ). A general form of the Fréchet gradient is systematically derived from the optimal transportation (OT) theory. In addition, a fast algorithm based on the new formulation of OT on S 1 mathbb {S}^{1} is developed to solve the corresponding optimal transport problem. The computational complexity of the algorithm is reduced to O ( N ) O(N) from O ( N 3 ) O(N^{3}) of the traditional method. Combining with the adjoint-state method, this framework provides a new computational approach for solving the challenging electrical impedance tomography problem. Numerical examples are presented to illustrate the effectiveness of our method.

本文建立了求解基于测地线的二次Wasserstein距离(w2w_{2})逆边界问题的框架。从最优运输理论出发，系统地导出了fr切特梯度的一般形式。此外，基于s1 mathbb {S}^{1}上新的OT公式，提出了求解相应最优运输问题的快速算法。该算法的计算复杂度由传统方法的O(N 3) O(N^{3})降低到O(N) O(N)。结合伴随状态法，该框架为解决具有挑战性的电阻抗层析成像问题提供了一种新的计算方法。数值算例说明了该方法的有效性。

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引用次数: 0

Using aromas to search for preserved measures and integrals in Kahan’s method 在Kahan的方法中使用香味来搜索保留的度量和积分

2区数学 Q1 MATHEMATICS, APPLIED

Mathematics of Computation

Pub Date : 2023-11-08 DOI: 10.1090/mcom/3921

Geir Bogfjellmo, Elena Celledoni, Robert McLachlan, Brynjulf Owren, G. Quispel

The numerical method of Kahan applied to quadratic differential equations is known to often generate integrable maps in low dimensions and can in more general situations exhibit preserved measures and integrals. Computerized methods based on discrete Darboux polynomials have recently been used for finding these measures and integrals. However, if the differential system contains many parameters, this approach can lead to highly complex results that can be difficult to interpret and analyse. But this complexity can in some cases be substantially reduced by using aromatic series. These are a mathematical tool introduced independently by Chartier and Murua and by Iserles, Quispel and Tse. We develop an algorithm for this purpose and derive some necessary conditions for the Kahan map to have preserved measures and integrals expressible in terms of aromatic functions. An important reason for the success of this method lies in the equivariance of the map from vector fields to their aromatic functions. We demonstrate the algorithm on a number of examples showing a great reduction in complexity compared to what had been obtained by a fixed basis such as monomials.

应用于二次微分方程的Kahan数值方法通常产生低维的可积映射，并且可以在更一般的情况下表现出保留测度和积分。基于离散达布多项式的计算机化方法最近被用于寻找这些度量和积分。然而，如果微分系统包含许多参数，这种方法可能导致高度复杂的结果，难以解释和分析。但在某些情况下，这种复杂性可以通过使用芳香系列大大降低。这些是由Chartier和Murua以及Iserles, Quispel和Tse独立引入的数学工具。我们为此目的开发了一种算法，并推导了Kahan映射具有可用芳族函数表示的保留测度和积分的必要条件。该方法成功的一个重要原因是矢量场到其芳族函数的映射的等方差。我们在许多例子上演示了该算法，与固定基(如单项式)相比，该算法的复杂性大大降低。

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引用次数: 0

Ideal Solutions in the Prouhet-Tarry-Escott problem 普鲁赫-塔里-埃斯科特问题的理想解

2区数学 Q1 MATHEMATICS, APPLIED

Mathematics of Computation

Pub Date : 2023-11-06 DOI: 10.1090/mcom/3917

Don Coppersmith, Michael Mossinghoff, Danny Scheinerman, Jeffrey VanderKam

For given positive integers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</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="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m greater-than n"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>></mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">m>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the <italic>Prouhet–Tarry–Escott problem</italic> asks if there exist two disjoint multisets of integers of size <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> having identical <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>th moments for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 less-than-or-equal-to k less-than-or-equal-to m"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>k</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>m</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1leq kleq m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; in the <italic>ideal</italic> case one requires <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m equals n minus 1"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m=n-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which is maximal. We describe some searches for ideal solutions to the Prouhet–Tarry–Escott problem, especially solutions possessing a particular symmetry, both over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">mathbb {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and over the ring of intege

对于给定正整数m m和n n，其中m ＆gt;n m＆gt;n, Prouhet-Tarry-Escott问题问是否存在两个大小为n n的不相交的多整数集，它们在1≤k≤m1 leq k leq m时具有相同的k个矩;理想情况下，需要m=n-1 m=n-1，这是最大值。本文描述了在Z mathbb Z{和若干虚二次域的整数环上对prouet - tarry - escott问题理想解的若干搜索，特别是具有特定对称性的解。在Z }mathbb Z{上，我们显著地扩展了尺寸为9,9,10,11,11和12,12的对称理想解的搜索，并且我们首次在最大尺寸为16,16的情况下进行了广泛的搜索。对于二次数域的情况，我们在高斯整数中找到了尺寸为10 - 10和12 - 12的新理想解，在Z[i 2] }mathbb Z{[i }sqrt 2{]中找到了尺寸为9 - 9和尺寸为12 - 12的新理想解。}

引用次数: 0

Approximation of stochastic Volterra equations with kernels of completely monotone type 具有完全单调型核的随机Volterra方程的逼近

2区数学 Q1 MATHEMATICS, APPLIED

Mathematics of Computation

Pub Date : 2023-11-02 DOI: 10.1090/mcom/3911

Aurélien Alfonsi, Ahmed Kebaier

In this work, we develop a multifactor approximation for d d -dimensional Stochastic Volterra Equations (SVE) with Lipschitz coefficients and kernels of completely monotone type that may be singular. First, we prove an L 2 L^2 -estimation between two SVEs with different kernels, which provides a quantification of the error between the SVE and any multifactor Stochastic Differential Equation (SDE) approximation. For the particular rough kernel case with Hurst parameter lying in ( 0 , 1 / 2 ) (0,1/2) , we propose various approximating multifactor kernels, state their rates of convergence and illustrate their efficiency for the rough Bergomi model. Second, we study a Euler discretization of the multifactor SDE and establish a convergence result towards the SVE that is uniform with respect to the approximating multifactor kernels. These obtained results lead us to build a new multifactor Euler scheme that reduces significantly the computational cost in an asymptotic way compared to the Euler scheme for SVEs. Finally, we show that our multifactor Euler scheme outperforms the Euler scheme for SVEs for option pricing in the rough Heston model.

在这项工作中，我们开发了具有Lipschitz系数和核可能是奇异的完全单调型的d -d维随机Volterra方程(SVE)的多因子逼近。首先，我们证明了两个具有不同核的随机微分方程(SVE)之间的l2l ^2估计，它提供了SVE与任何多因素随机微分方程(SDE)近似之间误差的量化。对于Hurst参数位于(0,1/2)(0,1/2)的特殊粗糙核情况，我们提出了各种近似的多因子核，说明了它们的收敛速度，并说明了它们对粗糙Bergomi模型的效率。其次，我们研究了多因子SDE的欧拉离散化，并建立了SVE对近似多因子核一致的收敛结果。这些得到的结果使我们建立了一个新的多因素欧拉格式，与SVEs的欧拉格式相比，该格式以渐近的方式显着降低了计算成本。最后，我们证明了我们的多因素欧拉方案在粗糙赫斯顿模型中优于SVEs的欧拉方案。

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引用次数: 0

On the computation of modular forms on noncongruence subgroups 非同余子群上模形式的计算

2区数学 Q1 MATHEMATICS, APPLIED

Mathematics of Computation

Pub Date : 2023-10-30 DOI: 10.1090/mcom/3903

David Berghaus, Hartmut Monien, Danylo Radchenko

We present two approaches that can be used to compute modular forms on noncongruence subgroups. The first approach uses Hejhal’s method for which we improve the arbitrary precision solving techniques so that the algorithm becomes about up to two orders of magnitude faster in practical computations. This allows us to obtain high precision numerical estimates of the Fourier coefficients from which the algebraic expressions can be identified using the LLL algorithm. The second approach is restricted to genus zero subgroups and uses efficient methods to compute the Belyi map from which the modular forms can be constructed.

我们提出了两种可用于计算非同余子群上模形式的方法。第一种方法使用了Hejhal的方法，我们改进了任意精度求解技术，使算法在实际计算中速度提高了两个数量级。这使我们能够获得傅立叶系数的高精度数值估计，从中可以使用LLL算法识别代数表达式。第二种方法被限制为零属子群，并使用有效的方法来计算Belyi映射，从Belyi映射可以构造模形式。

引用次数: 1