Dolomites Research Notes on Approximation最新文献

英文中文

Positivity-preserving and elementary stable nonstandard method for a COVID-19 SIR model COVID-19 SIR模型的保正初等稳定非标准方法

IF 1.3 Q3 MATHEMATICS

Dolomites Research Notes on Approximation

Pub Date : 2022-01-01 DOI: 10.14658/pupj-drna-2022-5-7

D. Conte, N. Guarino, G. Pagano, B. Paternoster

The main purpose of this work is to build a numerical method for solving an epidemiological model that describes the spread of COVID-19 in some countries. The method is constructed using a NonStandard Finite Difference (NSFD) discretization for the analyzed model, in order to preserve its positivity and equilibrium points properties. Numerical simulations testify the best performance of the proposed scheme with respect to the related Standard Finite Difference (SFD) method, the famous explicit four-stage order-four Runge-Kutta known as RK4, and another positivity-preserving nonstandard method. © 2022, Padova University Press. All rights reserved.

引用次数: 3

Computation of the Bell-Laplace transforms 贝尔拉普拉斯变换的计算

IF 1.3 Q3 MATHEMATICS

Dolomites Research Notes on Approximation

Pub Date : 2021-01-01 DOI: 10.14658/PUPJ-DRNA-2021-1-7

D. Caratelli, C. Cesarano, P. Ricci

since it converts a function of a real variable t (often representing the time) to a function of a complex variable s (complex frequency). This transform is used for solving differential equations, since it transforms differential into algebraic equations and convolution into multiplication. It can be applied to local integrable functions on [0,+∞) and it converges in each half plane Re(s)> a, the constant a, also known as the convergece abscissa, depends on the growth behavior of f (t). A large number of LT can be found in the literature and, together with the respective antitransforms, are usually used in the solution of the most diverse differential problems. The numerical computation of the link between transforms and antitransforms was considered, for example, by F.G. Tricomi [22, 23], which highlighted the link with the series expansions in Laguerre polynomials. These results have been extended to more general expansions [17], however the results of Tricomi have been proven numerically more convenient by the point of view of numerical complexity. Recently, extensions of the LT have been considered in [16] and in [6] the numerical computation was carried out by approximating the respective kernels by means of expansions in a general Dirichlet series. Extensions of LT (called Laguerre-Laplace transforms) have been obtained in the first place [6] by replacing the exponential with the Laguerre-type exponentials, introduced in [8] and previously studied in [10], [11], [13]. Subsequently the kernel was replaced by an expansion whose coefficients are combination of Bell polynomials, exploiting a transformation, introduced in [16], which uses the Blissard formula, a typical tool of the umbral calculus [19, 20]. In order to validate the computational methods, some examples were taken into consideration, by the first author, with the aid of the computer algebra program Mathematica©. The rapid decay of the considered kernels allows to extend the integration interval to a right neighborhood of the origin. However, the higher computational complexity using the theoretical approach based on the generating function of the generalized Lucas polynomials suggested to approximate the original kernel by a truncation of a general Dirichlet’s series, and to use the matrix pencil method for evaluating the best coefficients. The results obtained confirm the correctness of the procedure introduced.

因为它将实变量t(通常表示时间)的函数转换为复变量s(复频率)的函数。这个变换用于求解微分方程，因为它将微分方程转化为代数方程，将卷积转化为乘法。它可以应用于[0，+∞)上的局部可积函数，它收敛于每个半平面Re(s)> a，常数a，也称为收敛横坐标，取决于f (t)的生长行为。在文献中可以找到大量的LT，并且与相应的反变换一起，通常用于解决最多样化的微分问题。例如，F.G. Tricomi[22,23]考虑了变换与反变换之间联系的数值计算，突出了与拉盖尔多项式级数展开的联系。这些结果已推广到更一般的展开[17]，但从数值复杂性的角度来看，Tricomi的结果在数值上更方便。最近，在[16]和[6]中考虑了LT的扩展，在一般的Dirichlet级数中，通过展开式逼近各自的核进行了数值计算。LT的扩展(称为Laguerre-Laplace变换)首先通过用laguerre型指数代替指数得到了[6]，该指数在[8]中引入，并在[10]，[11]，[13]中进行了研究。随后，核被一个展开式取代，其系数是贝尔多项式的组合，利用[16]中引入的一个变换，该变换使用了Blissard公式，这是一个典型的本影微积分工具[19,20]。为了验证计算方法，第一作者在计算机代数程序Mathematica©的帮助下考虑了一些例子。所考虑的核的快速衰减允许将积分区间扩展到原点的右邻域。然而，基于广义卢卡斯多项式生成函数的理论方法计算复杂度较高，建议通过对一般狄利克雷级数的截断来近似原始核，并使用矩阵铅笔法来评估最佳系数。所得结果证实了所介绍方法的正确性。

{"title":"Computation of the Bell-Laplace transforms","authors":"D. Caratelli, C. Cesarano, P. Ricci","doi":"10.14658/PUPJ-DRNA-2021-1-7","DOIUrl":"https://doi.org/10.14658/PUPJ-DRNA-2021-1-7","url":null,"abstract":"since it converts a function of a real variable t (often representing the time) to a function of a complex variable s (complex frequency). This transform is used for solving differential equations, since it transforms differential into algebraic equations and convolution into multiplication. It can be applied to local integrable functions on [0,+∞) and it converges in each half plane Re(s)> a, the constant a, also known as the convergece abscissa, depends on the growth behavior of f (t). A large number of LT can be found in the literature and, together with the respective antitransforms, are usually used in the solution of the most diverse differential problems. The numerical computation of the link between transforms and antitransforms was considered, for example, by F.G. Tricomi [22, 23], which highlighted the link with the series expansions in Laguerre polynomials. These results have been extended to more general expansions [17], however the results of Tricomi have been proven numerically more convenient by the point of view of numerical complexity. Recently, extensions of the LT have been considered in [16] and in [6] the numerical computation was carried out by approximating the respective kernels by means of expansions in a general Dirichlet series. Extensions of LT (called Laguerre-Laplace transforms) have been obtained in the first place [6] by replacing the exponential with the Laguerre-type exponentials, introduced in [8] and previously studied in [10], [11], [13]. Subsequently the kernel was replaced by an expansion whose coefficients are combination of Bell polynomials, exploiting a transformation, introduced in [16], which uses the Blissard formula, a typical tool of the umbral calculus [19, 20]. In order to validate the computational methods, some examples were taken into consideration, by the first author, with the aid of the computer algebra program Mathematica©. The rapid decay of the considered kernels allows to extend the integration interval to a right neighborhood of the origin. However, the higher computational complexity using the theoretical approach based on the generating function of the generalized Lucas polynomials suggested to approximate the original kernel by a truncation of a general Dirichlet’s series, and to use the matrix pencil method for evaluating the best coefficients. The results obtained confirm the correctness of the procedure introduced.","PeriodicalId":51943,"journal":{"name":"Dolomites Research Notes on Approximation","volume":"14 1","pages":"74-91"},"PeriodicalIF":1.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66792642","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}

引用次数: 2

Bernstein and Markov-type inequalities for polynomials on Lp(μ) spaces Lp（μ）空间上多项式的Bernstein和Markov型不等式

IF 1.3 Q3 MATHEMATICS

Dolomites Research Notes on Approximation

Pub Date : 2020-03-24 DOI: 10.14658/pupj-drna-2019-Special_Issue-4

M. Chatzakou, Y. Sarantopoulos

In this work, we discuss generalizations of the classical Bernstein and Markov type inequalities for polynomials and we present some new inequalities for the $k$th Frechet derivative of homogeneous polynomials on real and complex $L_{p}(mu)$ spaces. We also give applications to homogeneous polynomials and symmetric multilinear mappings in $L_{p}(mu)$ spaces. Finally, Bernstein's inequality for homogeneous polynomials on both real and complex Hilbert spaces has been discussed.

在这项工作中，我们讨论了多项式的经典Bernstein和Markov型不等式的推广，并给出了实和复$L_{p}（mu）$空间上齐次多项式的$k$th Frechet导数的一些新不等式。我们还给出了$L_{p}（mu）$空间中齐次多项式和对称多线性映射的应用。最后，讨论了实Hilbert空间和复Hilbert空间上齐次多项式的Bernstein不等式。

引用次数: 3

RBF-based tensor decomposition with applications to oenology 基于rbf的张量分解及其在酿酒学中的应用

IF 1.3 Q3 MATHEMATICS

Dolomites Research Notes on Approximation

Pub Date : 2020-01-01 DOI: 10.14658/PUPJ-DRNA-2020-1-5

E. Perracchione

As usually claimed, meshless methods work in any dimension and are easy to implement. However in practice, to preserve the convergence order when the dimension grows, they need a huge number of sampling points and both computational costs and memory turn out to be prohibitive. Moreover, when a large number of points is involved, the usual instability of the Radial Basis Function (RBF) approximants becomes evident. To partially overcome this drawback, we propose to apply tensor decomposition methods. This, together with rational RBFs, allows us to obtain efficient interpolation schemes for high dimensions. The effectiveness of our approach is also verified by an application to oenology.

正如通常所说的那样，无网格方法可以在任何维度上工作，并且易于实现。然而，在实际应用中，为了在维数增加时保持收敛顺序，它们需要大量的采样点，计算成本和内存都令人望而却步。此外，当涉及大量点时，径向基函数(RBF)近似的通常不稳定性变得明显。为了部分克服这一缺点，我们建议应用张量分解方法。这与理性rbf一起，使我们能够获得高维的有效插值方案。我们的方法的有效性也通过酿酒学的应用得到了验证。

引用次数: 1

Hopf bifurcation analysis of the fast subsystem of a polynomial phantom burster model 多项式虚影爆发模型快速子系统的Hopf分岔分析

IF 1.3 Q3 MATHEMATICS

Dolomites Research Notes on Approximation

Pub Date : 2018-01-01 DOI: 10.14658/PUPJ-DRNA-2018-3-2

I. M. Bulai, M. G. Pedersen

Phantom bursters were introduced to explain bursting electrical activity in β -cells with different periods. We study a polynomial version of the phantom bursting model. In particular we analyse the fast subsystem, where the slowest variable is assumed constant. We find the equilibrium points of the fast subsystem and analyse their stability. Furthermore an analytical analysis of the existence of Hopf bifurcation points and the stability of the resulting periodics is performed by studying the sign of the first Lyapunov coefficient.

引入幻相爆发来解释不同时期β细胞的爆发电活动。我们研究了一个多项式版本的虚影爆破模型。我们特别分析了快速子系统，其中最慢的变量被假设为常数。找到了快速子系统的平衡点，并对其稳定性进行了分析。此外，通过研究第一Lyapunov系数的符号，对Hopf分岔点的存在性和周期的稳定性进行了解析分析。

引用次数: 2

Non standard properties of m-subharmonic functions m次调和函数的非标准性质

IF 1.3 Q3 MATHEMATICS

Dolomites Research Notes on Approximation

Pub Date : 2018-01-01 DOI: 10.14658/PUPJ-DRNA-2018-4-4

S. Dinew, S. Kołodziej

We survey elements of the nonlinear potential theory associated to m-subharmonic functions and the complex Hessian equation. We focus on properties which distinguish m-subharmonic functions from plurisubharmonic ones. Introduction Plurisubharmonic functions arose as multidimensional generalizations of subharmonic functions in the complex plane (see [LG]). Thus it is not surprising that these two classes of functions share many similarities. There are however many subtler properties which make a plurisubharmonic function in Cn, n > 1 differ from a general subharmonic function. Below we list some of the basic ones: Liouville type properties. it is known ([LG]) that an entire plurisubharmonic function cannot be bounded from above unless it is constant. The function u(z) = −1 ||z||2n−2 in C n, n> 1 is an example that this is not true for subharmonic ones; Integrability. Any plurisubarmonic function belongs to L loc for any 1≤ p <∞. For subharmonic functions this is true only for p < n n−1 as the function u above shows. Symmetries. Any holomorphic mapping preserves plurisubharmonic functions in the sense that a composition of a plurisubharmonic function with a holomporhic mapping is still plurisubharmonic. This does not hold for subharmonic functions in Cn, n> 1. The notion of m-subharmonic function (see [Bl1], [DK2, DK1]) interpolates between subharmonicity and plurisubharmonicity. It is thus expected that the corresponding nonlinear potential theory will share the joint properties of potential and pluripotential theories. Indeed in the works of Li, Blocki, Chinh, Abdullaev and Sadullaev, Dhouib and Elkhadhra, Nguyen and many others the m-subharmonic potential theory was thoroughly developed. In particular S. Y. Li [Li] solved the associated smooth Dirichlet problem under suitable assumptions, proving thus an analogue of the Caffarelli-Nirenberg-Spruck theorem [CNS] who dealt with the real setting. Z. Blocki [Bl1, Bl3] noted that the Bedford-Taylor apparatus from [BT1] and [BT2] can be adapted to m-subharmonic setting. He also described the domain of definition of the complex Hessian operator. L. H. Chinh developed the variational apporach to the complex Hessian equation [Chi1] and studied the associated viscosity theory of weak solutions in [Chi3]. He also developed the theory of m-subharmonic Cegrell classes [Chi1, Chi2]. Abdullaev and Sadullaev in [AS] defined the corresponding m-capacities (this was done also independently by Chinh in [Chi2] and the authors in [DK2]). A. Dhouib and F. Elkhadhra investigated m-subharmonicity with respect to a current [DE] and noticed several interesting phenomena. N. C. Nguyen in [N] investigated existence of solutions to the Hessian equations if a subsolution exists. Arguably the most interesting part of the theory is the one that differs from its pluripotential counterpart. This involves not only new phenomena but also requires new tools. Obviously there are good reasons for such a discrepancy. The very notion o

我们研究了与m-次谐波函数和复Hessian方程相关的非线性势理论的要素。我们着重于区分m次谐波函数和多次谐波函数的性质。复数次谐波函数是复数平面上次谐波函数的多维推广(参见[LG])。因此，这两类函数有许多相似之处也就不足为奇了。然而，有许多更微妙的性质使得Cn, n > 1中的多次谐波函数不同于一般的次谐波函数。下面我们列出了一些基本的:Liouville类型的属性。众所周知([LG])，除非整个多次谐波函数是常数，否则它不能从上面有界。函数u(z) = - 1 ||z||2n - 2在C n n, n> 1中是一个例子这对于次谐波是不成立的;可积性。对于任意1≤p 1，任何多次调和函数都属于lloc。m-次谐波函数的概念(参见[Bl1]， [DK2, DK1])插补在次谐波和多次次谐波之间。因此，期望相应的非线性势理论具有势理论和多势理论的联合性质。实际上，在Li、Blocki、Chinh、Abdullaev和Sadullaev、Dhouib和Elkhadhra、Nguyen和其他许多人的著作中，m-次谐波势理论得到了彻底的发展。特别是S. Y. Li [Li]在适当的假设下解决了相关的光滑Dirichlet问题，从而证明了处理实际情况的Caffarelli-Nirenberg-Spruck定理[CNS]的类似。Z. Blocki [Bl1, Bl3]注意到[BT1]和[BT2]中的Bedford-Taylor装置可以适应m-次谐波设置。他还描述了复Hessian算子的定义域。L. H. Chinh发展了复Hessian方程的变分方法[Chi1]，并在[Chi3]中研究了弱解的相关黏性理论。他还发展了m-次谐波Cegrell类理论[Chi1, Chi2]。Abdullaev和Sadullaev在[AS]中定义了相应的m-容量(Chinh在[Chi2]和作者在[DK2]中也独立完成了这一工作)。a. Dhouib和F. Elkhadhra研究了电流的m-次谐波[DE]，并注意到几个有趣的现象。N. C. Nguyen [N]研究了Hessian方程的子解存在性。可以说，这个理论中最有趣的部分是它与多能理论的对应部分不同。这不仅涉及新的现象，而且需要新的工具。显然，这种差异是有充分理由的。多元次谐波的概念是独立于Kähler度规的，与m次谐波形成鲜明对比。m- hessian方程的基本解为- 1 |z|2 n m- 2，因此在原点处存在强于对数的奇点，函数在无穷远处有界。而且它仅在p < nm n−m时是Lp可积的。本调查笔记的目的是收集m-次谐波函数的这种独特的结果。我们的选择当然是主观的，我们不涉及许多重要的问题，如紧流形上的m极集或m次调和函数。首先我们处理m-sh函数的对称性。我们特别证明了这些对称的集合与任意1< m< n的全纯和反全纯正交仿射映射的集合重合，与边界情况形成鲜明对比。我们还研究了Lelong数的上水平集的类似物。在Harvey和Lawson ([HL1])和Chu ([Ch])的论证之后，我们提出了一个惊人的事实的证明，即上层水平集在m< n时是离散的。这再次与Siu定理暗示当m= n时这些集合的可解析性的多次谐波情况截然不同。雅盖隆大学，Kraków，波兰。Dinew·Kołodziej 36笔记组织如下:第1节列出了基本概念和工具。特别地，我们已经涵盖了m-sh函数的线性代数和势理论性质。我们还包括一个相当简短的小节，专门讨论一般椭圆偏微分方程的弱解。第2节的第一部分专门讨论m-sh函数的对称性。在第二种方法中，我们构造一个特殊的非线性算子Pm。我们证明了所有的m-sh函数都是Pm的子解，更重要的是，Pm与m-Hessian算子具有相同的基本解。我们希望指出，Pm是Harvey和Lawson定义的具有相同Riesz特征的一致椭圆算子的更一般构造的一个例子(见[HL1])。最后，在第3节中，我们研究了m-sh函数的Lelong数的类似物的上水平集。本节依赖于Harvey和Lawson ([HL1, HL2])以及Chu ([Ch])的一般性论点。当我们处理m-sh函数的具体情况时，我们的论点稍微简单一些，但主要思想是相同的。奉献精神。我们很高兴把这篇文章献给诺姆，一位伟大的朋友和数学家。Aknowledgements。两位作者均获得NCN基金2013/08/A/ST1/00312的资助。在本节中，我们回顾出现在m-次谐波函数势理论中的概念和工具。1.1线性代数。用Mn表示所有厄米对称n× n矩阵的集合。固定矩阵M∈Mn。用λ(M) = (λ1，λ2，…，λn)表示其特征值按降序排列。定义1.1。与M相关的第M个对称多项式定义为Sm(M) = Sm(λ(M)) =∑0< j10，···，Sm(λ)>}。(1.1)这些锥的以下两个性质是经典的:(Maclaurin不等式)如果λ∈Γm则(S j (j)) 1 j≥(Si (i)) 1 i对于1≤j≤i≤m;2. (g<s:1> rding不等式，[Ga]) Γm对于任意m都是凸锥，函数s1m m在约束于Γm时是凹的;关于这些锥的进一步性质，我们参考读者[Bl1]或[W]。1.2 m-次谐波函数的势理论方面。我们只考虑一个相对紧凑的域Ω∧Cn。下面我们假设n≥2。用d =∂+∂'和d c:= i(∂'−∂)表示标准的外部微分算子。用β:= dd c |z|2表示Cn中的正则Kähler形式。我们现在定义光滑的m次谐波函数。定义1.2。给定一个C2(Ω)函数u，我们称它为Ω中的m次谐波，如果对于任意z∈Ω， Hessian矩阵∂2u∂zi∂z´j (z)具有特征值，在圆锥的闭包中形成一个向量Γm。特征值向量的几何性质可以用微分形式的语言更解析地表述:u是m次调和的当且仅当下列不等式成立:(dd u)∧β n−k≥0,k = 1，···，m。注意，如果n−k≥1，这些不等式依赖于背景Kähler形式β。因此，定义相对于一般Kähler形式ω的次调和性是有意义的(详见[DK1])。然而，在本调查中，我们将只处理标准的Kähler形式β。在([Bl1]) Z. Błocki证明中，我们可以放宽对u的光滑性要求，并开发出Hessian算子的非线性势理论，就像Bedford和Taylor在多次谐波函数([BT1]， [BT2])的情况下所做的那样。一般来说，m-sh函数定义如下:定义1.3。设u是定义域Ω∈Cn上的次调和函数。如果对于任意c2 -光滑m-sh函数v1，···，vm−1的集合，不等式dd u∧dd c v1∧···∧dd c vm−1∧β n−m≥0在弱电流意义下成立，则u称为m次谐波(简称m-sh)。所有m−Ω -sh函数的集合用SHm(Ω)表示。备注1。在m = n的情况下，m-sh函数是简单的多次谐波函数。同样，在m-sh次多项式集合上测试u的m次谐波性也足够了(见[Bl1])。利用定义中的近似序列u j，可以根据[BT2]中给出的由局部有界m-sh函数给出的电流楔积的Bedford和Taylor构造。它们被归纳定义为dd∧∧···∧dd up∧

{"title":"Non standard properties of m-subharmonic functions","authors":"S. Dinew, S. Kołodziej","doi":"10.14658/PUPJ-DRNA-2018-4-4","DOIUrl":"https://doi.org/10.14658/PUPJ-DRNA-2018-4-4","url":null,"abstract":"We survey elements of the nonlinear potential theory associated to m-subharmonic functions and the complex Hessian equation. We focus on properties which distinguish m-subharmonic functions from plurisubharmonic ones. Introduction Plurisubharmonic functions arose as multidimensional generalizations of subharmonic functions in the complex plane (see [LG]). Thus it is not surprising that these two classes of functions share many similarities. There are however many subtler properties which make a plurisubharmonic function in Cn, n > 1 differ from a general subharmonic function. Below we list some of the basic ones: Liouville type properties. it is known ([LG]) that an entire plurisubharmonic function cannot be bounded from above unless it is constant. The function u(z) = −1 ||z||2n−2 in C n, n> 1 is an example that this is not true for subharmonic ones; Integrability. Any plurisubarmonic function belongs to L loc for any 1≤ p <∞. For subharmonic functions this is true only for p < n n−1 as the function u above shows. Symmetries. Any holomorphic mapping preserves plurisubharmonic functions in the sense that a composition of a plurisubharmonic function with a holomporhic mapping is still plurisubharmonic. This does not hold for subharmonic functions in Cn, n> 1. The notion of m-subharmonic function (see [Bl1], [DK2, DK1]) interpolates between subharmonicity and plurisubharmonicity. It is thus expected that the corresponding nonlinear potential theory will share the joint properties of potential and pluripotential theories. Indeed in the works of Li, Blocki, Chinh, Abdullaev and Sadullaev, Dhouib and Elkhadhra, Nguyen and many others the m-subharmonic potential theory was thoroughly developed. In particular S. Y. Li [Li] solved the associated smooth Dirichlet problem under suitable assumptions, proving thus an analogue of the Caffarelli-Nirenberg-Spruck theorem [CNS] who dealt with the real setting. Z. Blocki [Bl1, Bl3] noted that the Bedford-Taylor apparatus from [BT1] and [BT2] can be adapted to m-subharmonic setting. He also described the domain of definition of the complex Hessian operator. L. H. Chinh developed the variational apporach to the complex Hessian equation [Chi1] and studied the associated viscosity theory of weak solutions in [Chi3]. He also developed the theory of m-subharmonic Cegrell classes [Chi1, Chi2]. Abdullaev and Sadullaev in [AS] defined the corresponding m-capacities (this was done also independently by Chinh in [Chi2] and the authors in [DK2]). A. Dhouib and F. Elkhadhra investigated m-subharmonicity with respect to a current [DE] and noticed several interesting phenomena. N. C. Nguyen in [N] investigated existence of solutions to the Hessian equations if a subsolution exists. Arguably the most interesting part of the theory is the one that differs from its pluripotential counterpart. This involves not only new phenomena but also requires new tools. Obviously there are good reasons for such a discrepancy. The very notion o","PeriodicalId":51943,"journal":{"name":"Dolomites Research Notes on Approximation","volume":"11 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66792660","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}

引用次数: 6

Recent advancements in preconditioning techniques for large size linear systems suited for high performance computing 适用于高性能计算的大型线性系统预处理技术的最新进展

IF 1.3 Q3 MATHEMATICS

Dolomites Research Notes on Approximation

Pub Date : 2018-01-01 DOI: 10.14658/pupj-drna-2018-3-3

A. Franceschini, M. Ferronato, C. Janna, V. Magri

The numerical simulations of real-world engineering problems create models with several millions or even billions of degrees of freedom. Most of these simulations are centered on the solution of systems of non-linear equations, that, once linearized, become a sequence of linear systems, whose solution is often the most time-demanding task. Thus, in order to increase the capability of modeling larger cases, it is of paramount importance to exploit the resources of High Performance Computing architectures. In this framework, the development of new algorithms to accelerate the solution of linear systems for many-core architectures is a really active research field. Our main focus is algebraic preconditioning and, among the various options, we elect to develop approximate inverses for symmetric and positive definite (SPD) linear systems [22], both as stand-alone preconditioner or smoother for AMG techniques. This choice is mainly supported by the almost perfect parallelism that intrinsically characterizes these algorithms. As basic kernel, the Factorized Sparse Approximate Inverse (FSAI) developed in its adaptive form by Janna and Ferronato [18] is selected. Recent developments are i) a robust multilevel approach for SPD problems based on FSAI preconditioning, which eliminates the chance of algorithmic breakdowns independently of the preconditioner sparsity [14] and ii) a novel AMG approach featuring the adaptive FSAI method as a flexible smoother as well as new approaches to adaptively compute the prolongation operator. In this latter work, a new technique to build the prolongation is also presented.

现实世界工程问题的数值模拟创建了具有数百万甚至数十亿自由度的模型。这些模拟大多集中在非线性方程组的解上，这些方程组一旦线性化，就会变成一系列线性系统，其解通常是最耗时的任务。因此，为了增加对大型案例建模的能力，利用高性能计算体系结构的资源是至关重要的。在这个框架下，开发新的算法来加速求解多核体系结构的线性系统是一个非常活跃的研究领域。我们的主要重点是代数预处理，在各种选项中，我们选择开发对称和正定(SPD)线性系统[22]的近似逆，作为AMG技术的独立预处理或平滑。这种选择主要是由这些算法固有的近乎完美的并行性所支持的。选取Janna和Ferronato[18]以自适应形式提出的分解稀疏近似逆(FSAI)作为基本核。最近的研究进展包括:1)基于FSAI预处理的SPD问题鲁棒多级方法，该方法消除了独立于预处理稀疏度[14]的算法故障的机会;2)一种新颖的AMG方法，其特点是自适应FSAI方法具有灵活的平滑性，以及自适应计算扩展算子的新方法。在后一项工作中，还提出了一种构建延拓的新技术。

{"title":"Recent advancements in preconditioning techniques for large size linear systems suited for high performance computing","authors":"A. Franceschini, M. Ferronato, C. Janna, V. Magri","doi":"10.14658/pupj-drna-2018-3-3","DOIUrl":"https://doi.org/10.14658/pupj-drna-2018-3-3","url":null,"abstract":"The numerical simulations of real-world engineering problems create models with several millions or even billions of degrees of freedom. Most of these simulations are centered on the solution of systems of non-linear equations, that, once linearized, become a sequence of linear systems, whose solution is often the most time-demanding task. Thus, in order to increase the capability of modeling larger cases, it is of paramount importance to exploit the resources of High Performance Computing architectures. In this framework, the development of new algorithms to accelerate the solution of linear systems for many-core architectures is a really active research field. Our main focus is algebraic preconditioning and, among the various options, we elect to develop approximate inverses for symmetric and positive definite (SPD) linear systems [22], both as stand-alone preconditioner or smoother for AMG techniques. This choice is mainly supported by the almost perfect parallelism that intrinsically characterizes these algorithms. As basic kernel, the Factorized Sparse Approximate Inverse (FSAI) developed in its adaptive form by Janna and Ferronato [18] is selected. Recent developments are i) a robust multilevel approach for SPD problems based on FSAI preconditioning, which eliminates the chance of algorithmic breakdowns independently of the preconditioner sparsity [14] and ii) a novel AMG approach featuring the adaptive FSAI method as a flexible smoother as well as new approaches to adaptively compute the prolongation operator. In this latter work, a new technique to build the prolongation is also presented.","PeriodicalId":51943,"journal":{"name":"Dolomites Research Notes on Approximation","volume":"11 1","pages":"11-22"},"PeriodicalIF":1.3,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66792535","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}

引用次数: 2

On the metric space of pluriregular sets 关于多正则集的度量空间

IF 1.3 Q3 MATHEMATICS

Dolomites Research Notes on Approximation

Pub Date : 2018-01-01 DOI: 10.14658/PUPJ-DRNA-2018-4-5

M. Klimek, M. Kosek

The metric space of pluriregular sets was introduced over two decades ago but to this day most of its topological properties remain a mystery. The purpose of this short survey is to present the cur ...

多正则集的度量空间在二十多年前被引入，但直到今天，它的大部分拓扑性质仍然是一个谜。这个简短调查的目的是展示当前……

引用次数: 3

Spectral filtering for the reduction of the Gibbs phenomenon for polynomial approximation methods on Lissajous curves with applications in MPI 利萨曲线上多项式近似方法的吉布斯现象谱滤波及其在MPI中的应用

IF 1.3 Q3 MATHEMATICS

Dolomites Research Notes on Approximation

Pub Date : 2017-01-01 DOI: 10.14658/PUPJ-DRNA-2017-SPECIAL_ISSUE-13

S. Marchi, W. Erb, F. Marchetti

Polynomial interpolation and approximation methods on sampling points along Lissajous curves using Chebyshev series is an effective way for a fast image reconstruction in Magnetic Particle Imaging. Due to the nature of spectral methods, a Gibbs phenomenon occurs in the reconstructed image if the underlying function has discontinuities. A possible solution for this problem are spectral filtering methods acting on the coefficients of the approximating polynomial. In this work, after a description of the Gibbs phenomenon and classical filtering techniques in one and several dimensions, we present an adaptive spectral filtering process for the resolution of this phenomenon and for an improved approximation of the underlying function or image. In this adaptive filtering technique, the spectral filter depends on the distance of a spatial point to the nearest discontinuity. We show the effectiveness of this filtering approach in theory, in numerical simulations as well as in the application in Magnetic Particle Imaging.

利用切比雪夫级数对Lissajous曲线上的采样点进行多项式插值和逼近是磁粒子成像中快速重建图像的有效方法。由于光谱方法的性质，如果底层函数具有不连续，则在重建图像中会出现吉布斯现象。这个问题的一个可能的解决方案是谱滤波方法作用于近似多项式的系数。在这项工作中，在描述了吉布斯现象和经典的一维和多维滤波技术之后，我们提出了一种自适应光谱滤波过程，用于解决这种现象，并改进了对底层函数或图像的近似。在这种自适应滤波技术中，光谱滤波取决于空间点到最近不连续点的距离。我们从理论上、数值模拟以及在磁粉成像中的应用上证明了这种滤波方法的有效性。

引用次数: 26

Beyond B-splines: exponential pseudo-splines and subdivision schemes reproducing exponential polynomials 超越b样条:指数伪样条和再现指数多项式的细分方案

IF 1.3 Q3 MATHEMATICS

Dolomites Research Notes on Approximation

Pub Date : 2017-01-01 DOI: 10.14658/PUPJ-DRNA-2017-SPECIAL_ISSUE-6

C. Conti, M. Cotronei, Lucia Romani

The main goal of this paper is to present some generalizations of polynomial B-splines, which include exponential B-splines and the larger family of exponential pseudo-splines. We especially focus on their connections to subdivision schemes. In addition, we generalize a well-known result on the approximation order of exponential pseudo-splines, providing conditions to establish the approximation order of nonstationary subdivision schemes reproducing spaces of exponential polynomial functions. 2010 MSC: 65D17, 65D15, 41A25

本文的主要目的是给出多项式b样条的一些推广，其中包括指数b样条和更大的指数伪样条族。我们特别关注它们与细分方案的联系。此外，我们推广了关于指数伪样条近似阶的一个著名结果，为建立再现指数多项式函数空间的非平稳细分格式的近似阶提供了条件。2010 msc: 65d17, 65d15, 41a25

引用次数: 10

下一页尾页

类型

全部化学•材料生命科学医学物理工程技术环境•农林材料科学地球科学法学管理学化学环境科学与生态学计算机科学教育学经济学农林科学人文科学生物学数学物理与天体物理心理学综合性期刊其他工业工程理学历史学农学文学信息工程

数据库

全部 ACS Publications Elsevier ieeexplore Springer The Royal Society of Chemistry Wiley

期刊

Dolomites Research Notes on Approximation

全部 Acc. Chem. Res. ACS Applied Bio Materials ACS Appl. Electron. Mater. ACS Appl. Energy Mater. ACS Appl. Mater. Interfaces ACS Appl. Nano Mater. ACS Appl. Polym. Mater. ACS BIOMATER-SCI ENG ACS Catal. ACS Cent. Sci. ACS Chem. Biol. ACS Chemical Health & Safety ACS Chem. Neurosci. ACS Comb. Sci. ACS Earth Space Chem. ACS Energy Lett. ACS Infect. Dis. ACS Macro Lett. ACS Mater. Lett. ACS Med. Chem. Lett. ACS Nano ACS Omega ACS Photonics ACS Sens. ACS Sustainable Chem. Eng. ACS Synth. Biol. Anal. Chem. BIOCHEMISTRY-US Bioconjugate Chem. BIOMACROMOLECULES Chem. Res. Toxicol. Chem. Rev. Chem. Mater. CRYST GROWTH DES ENERG FUEL Environ. Sci. Technol. Environ. Sci. Technol. Lett. Eur. J. Inorg. Chem. IND ENG CHEM RES Inorg. Chem. J. Agric. Food. Chem. J. Chem. Eng. Data J. Chem. Educ. J. Chem. Inf. Model. J. Chem. Theory Comput. J. Med. Chem. J. Nat. Prod. J PROTEOME RES J. Am. Chem. Soc. LANGMUIR MACROMOLECULES Mol. Pharmaceutics Nano Lett. Org. Lett. ORG PROCESS RES DEV ORGANOMETALLICS J. Org. Chem. J. Phys. Chem. J. Phys. Chem. A J. Phys. Chem. B J. Phys. Chem. C J. Phys. Chem. Lett. Analyst Anal. Methods Biomater. Sci. Catal. Sci. Technol. Chem. Commun. Chem. Soc. Rev. CHEM EDUC RES PRACT CRYSTENGCOMM Dalton Trans. Energy Environ. Sci. ENVIRON SCI-NANO ENVIRON SCI-PROC IMP ENVIRON SCI-WAT RES Faraday Discuss. Food Funct. Green Chem. Inorg. Chem. Front. Integr. Biol. J. Anal. At. Spectrom. J. Mater. Chem. A J. Mater. Chem. B J. Mater. Chem. C Lab Chip Mater. Chem. Front. Mater. Horiz. MEDCHEMCOMM Metallomics Mol. Biosyst. Mol. Syst. Des. Eng. Nanoscale Nanoscale Horiz. Nat. Prod. Rep. New J. Chem. Org. Biomol. Chem. Org. Chem. Front. PHOTOCH PHOTOBIO SCI PCCP Polym. Chem.

﹀