Dolomites Research Notes on Approximation最新文献

{"title":"Positivity-preserving and elementary stable nonstandard method for a COVID-19 SIR model","authors":"D. Conte, N. Guarino, G. Pagano, B. Paternoster","doi":"10.14658/pupj-drna-2022-5-7","DOIUrl":"https://doi.org/10.14658/pupj-drna-2022-5-7","url":null,"abstract":"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.","PeriodicalId":51943,"journal":{"name":"Dolomites Research Notes on Approximation","volume":"15 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66793013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"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}

{"title":"Bernstein and Markov-type inequalities for polynomials on Lp(μ) spaces","authors":"M. Chatzakou, Y. Sarantopoulos","doi":"10.14658/pupj-drna-2019-Special_Issue-4","DOIUrl":"https://doi.org/10.14658/pupj-drna-2019-Special_Issue-4","url":null,"abstract":"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.","PeriodicalId":51943,"journal":{"name":"Dolomites Research Notes on Approximation","volume":"12 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2020-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42937428","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"RBF-based tensor decomposition with applications to oenology","authors":"E. Perracchione","doi":"10.14658/PUPJ-DRNA-2020-1-5","DOIUrl":"https://doi.org/10.14658/PUPJ-DRNA-2020-1-5","url":null,"abstract":"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.","PeriodicalId":51943,"journal":{"name":"Dolomites Research Notes on Approximation","volume":"13 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66792429","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hopf bifurcation analysis of the fast subsystem of a polynomial phantom burster model","authors":"I. M. Bulai, M. G. Pedersen","doi":"10.14658/PUPJ-DRNA-2018-3-2","DOIUrl":"https://doi.org/10.14658/PUPJ-DRNA-2018-3-2","url":null,"abstract":"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.","PeriodicalId":51943,"journal":{"name":"Dolomites Research Notes on Approximation","volume":"11 1","pages":"3-10"},"PeriodicalIF":1.3,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66792399","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"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}

{"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}

{"title":"On the metric space of pluriregular sets","authors":"M. Klimek, M. Kosek","doi":"10.14658/PUPJ-DRNA-2018-4-5","DOIUrl":"https://doi.org/10.14658/PUPJ-DRNA-2018-4-5","url":null,"abstract":"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 ...","PeriodicalId":51943,"journal":{"name":"Dolomites Research Notes on Approximation","volume":"11 1","pages":"51-61"},"PeriodicalIF":1.3,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66792734","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Spectral filtering for the reduction of the Gibbs phenomenon for polynomial approximation methods on Lissajous curves with applications in MPI","authors":"S. Marchi, W. Erb, F. Marchetti","doi":"10.14658/PUPJ-DRNA-2017-SPECIAL_ISSUE-13","DOIUrl":"https://doi.org/10.14658/PUPJ-DRNA-2017-SPECIAL_ISSUE-13","url":null,"abstract":"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.","PeriodicalId":51943,"journal":{"name":"Dolomites Research Notes on Approximation","volume":"10 1","pages":"128-137"},"PeriodicalIF":1.3,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66792250","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Beyond B-splines: exponential pseudo-splines and subdivision schemes reproducing exponential polynomials","authors":"C. Conti, M. Cotronei, Lucia Romani","doi":"10.14658/PUPJ-DRNA-2017-SPECIAL_ISSUE-6","DOIUrl":"https://doi.org/10.14658/PUPJ-DRNA-2017-SPECIAL_ISSUE-6","url":null,"abstract":"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","PeriodicalId":51943,"journal":{"name":"Dolomites Research Notes on Approximation","volume":"10 1","pages":"31-42"},"PeriodicalIF":1.3,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66791816","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}