Special Issue "Advanced Numerical Methods in Applied Sciences"

A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: closed (30 April 2018)

Special Issue Editors

Guest Editor
Prof. Dr. Luigi Brugnano

Dipartimento di Matematica e Informatica "U.Dini" Università di Firenze, 50134 Firenze, Italy
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Interests: geometric numerical integration; numerical methods for differential equations; numerical linear algebra; parallel algorithms in Numerical Analysis
Guest Editor
Prof. Dr. Felice Iavernaro

Dipartimento di Matematica, Università degli Studi di Bari, 70121 Bari, Italy
Website | E-Mail
Interests: numerical solution of ODEs with initial and boundary conditions, and in particular Hamiltonian systems; numerical solution of algebraic differential equations (DAEs); sequential and parallel software for ODE and DAE; maintenance of the test set; numerical solution of elliptical partial differential equations (PDEs); history of olomorphic dynamics; geometric integration

Special Issue Information

Dear Colleagues,

The use of scientific computing tools is, nowadays, customary for solving problems at several complexity levels in Applied Sciences. The great need for reliable software in the scientific community conveys a continuous stimulus to develop new and more performing numerical methods, able to grasp the particular features of the problem at hand. This has been the case for many different settings of Numerical Analysis and this Special Issue aims at covering some important developments in various areas of application.

Prof. Dr. Luigi Brugnano
Prof. Dr. Felice Iavernaro
Guest Editors

Manuscript Submission Information

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Keywords

  • ordinary differential equations
  • partial differential equations
  • evolutionary problems
  • optimization problems
  • geometric integration

Published Papers (13 papers)

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Research

Jump to: Review

Open AccessArticle Efficient Implementation of ADER Discontinuous Galerkin Schemes for a Scalable Hyperbolic PDE Engine
Received: 21 May 2018 / Revised: 21 August 2018 / Accepted: 22 August 2018 / Published: 1 September 2018
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Abstract
In this paper we discuss a new and very efficient implementation of high order accurate arbitrary high order schemes using derivatives discontinuous Galerkin (ADER-DG) finite element schemes on modern massively parallel supercomputers. The numerical methods apply to a very broad class of nonlinear
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In this paper we discuss a new and very efficient implementation of high order accurate arbitrary high order schemes using derivatives discontinuous Galerkin (ADER-DG) finite element schemes on modern massively parallel supercomputers. The numerical methods apply to a very broad class of nonlinear systems of hyperbolic partial differential equations. ADER-DG schemes are by construction communication-avoiding and cache-blocking, and are furthermore very well-suited for vectorization, and so they appear to be a good candidate for the future generation of exascale supercomputers. We introduce the numerical algorithm and show some applications to a set of hyperbolic equations with increasing levels of complexity, ranging from the compressible Euler equations over the equations of linear elasticity and the unified Godunov-Peshkov-Romenski (GPR) model of continuum mechanics to general relativistic magnetohydrodynamics (GRMHD) and the Einstein field equations of general relativity. We present strong scaling results of the new ADER-DG schemes up to 180,000 CPU cores. To our knowledge, these are the largest runs ever carried out with high order ADER-DG schemes for nonlinear hyperbolic PDE systems. We also provide a detailed performance comparison with traditional Runge-Kutta DG schemes. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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Open AccessArticle On a Class of Conjugate Symplectic Hermite-Obreshkov One-Step Methods with Continuous Spline Extension
Received: 23 May 2018 / Revised: 13 August 2018 / Accepted: 15 August 2018 / Published: 20 August 2018
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Abstract
The class of A-stable symmetric one-step Hermite–Obreshkov (HO) methods introduced by F. Loscalzo in 1968 for dealing with initial value problems is analyzed. Such schemes have the peculiarity of admitting a multiple knot spline extension collocating the differential equation at the mesh points.
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The class of A-stable symmetric one-step Hermite–Obreshkov (HO) methods introduced by F. Loscalzo in 1968 for dealing with initial value problems is analyzed. Such schemes have the peculiarity of admitting a multiple knot spline extension collocating the differential equation at the mesh points. As a new result, it is shown that these maximal order schemes are conjugate symplectic, which is a benefit when the methods have to be applied to Hamiltonian problems. Furthermore, a new efficient approach for the computation of the spline extension is introduced, adopting the same strategy developed for the BS linear multistep methods. The performances of the schemes are tested in particular on some Hamiltonian benchmarks and compared with those of the Gauss–Runge–Kutta schemes and Euler–Maclaurin formulas of the same order. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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Open AccessArticle A Convex Model for Edge-Histogram Specification with Applications to Edge-Preserving Smoothing
Received: 18 June 2018 / Revised: 1 August 2018 / Accepted: 1 August 2018 / Published: 2 August 2018
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Abstract
The goal of edge-histogram specification is to find an image whose edge image has a histogram that matches a given edge-histogram as much as possible. Mignotte has proposed a non-convex model for the problem in 2012. In his work, edge magnitudes of an
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The goal of edge-histogram specification is to find an image whose edge image has a histogram that matches a given edge-histogram as much as possible. Mignotte has proposed a non-convex model for the problem in 2012. In his work, edge magnitudes of an input image are first modified by histogram specification to match the given edge-histogram. Then, a non-convex model is minimized to find an output image whose edge-histogram matches the modified edge-histogram. The non-convexity of the model hinders the computations and the inclusion of useful constraints such as the dynamic range constraint. In this paper, instead of considering edge magnitudes, we directly consider the image gradients and propose a convex model based on them. Furthermore, we include additional constraints in our model based on different applications. The convexity of our model allows us to compute the output image efficiently using either Alternating Direction Method of Multipliers or Fast Iterative Shrinkage-Thresholding Algorithm. We consider several applications in edge-preserving smoothing including image abstraction, edge extraction, details exaggeration, and documents scan-through removal. Numerical results are given to illustrate that our method successfully produces decent results efficiently. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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Open AccessArticle Trees, Stumps, and Applications
Received: 23 May 2018 / Revised: 16 July 2018 / Accepted: 16 July 2018 / Published: 1 August 2018
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Abstract
The traditional derivation of Runge–Kutta methods is based on the use of the scalar test problem y(x)=f(x,y(x)). However, above order 4, this gives less restrictive order conditions than
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The traditional derivation of Runge–Kutta methods is based on the use of the scalar test problem y(x)=f(x,y(x)). However, above order 4, this gives less restrictive order conditions than those obtained from a vector test problem using a tree-based theory. In this paper, stumps, or incomplete trees, are introduced to explain the discrepancy between the two alternative theories. Atomic stumps can be combined multiplicatively to generate all trees. For the scalar test problem, these quantities commute, and certain sets of trees form isomeric classes. There is a single order condition for each class, whereas for the general vector-based problem, for which commutation of atomic stumps does not occur, there is exactly one order condition for each tree. In the case of order 5, the only nontrivial isomeric class contains two trees, and the number of order conditions reduces from 17 to 16 for scalar problems. A method is derived that satisfies the 16 conditions for scalar problems but not the complete set based on 17 trees. Hence, as a practical numerical method, it has order 4 for a general initial value problem, but this increases to order 5 for a scalar problem. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
Open AccessArticle A Gradient System for Low Rank Matrix Completion
Received: 7 May 2018 / Revised: 18 July 2018 / Accepted: 18 July 2018 / Published: 24 July 2018
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Abstract
In this article we present and discuss a two step methodology to find the closest low rank completion of a sparse large matrix. Given a large sparse matrix M, the method consists of fixing the rank to r and then looking for
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In this article we present and discuss a two step methodology to find the closest low rank completion of a sparse large matrix. Given a large sparse matrix M, the method consists of fixing the rank to r and then looking for the closest rank-r matrix X to M, where the distance is measured in the Frobenius norm. A key element in the solution of this matrix nearness problem consists of the use of a constrained gradient system of matrix differential equations. The obtained results, compared to those obtained by different approaches show that the method has a correct behaviour and is competitive with the ones available in the literature. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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Open AccessArticle Block Generalized Locally Toeplitz Sequences: From the Theory to the Applications
Received: 9 May 2018 / Revised: 6 July 2018 / Accepted: 16 July 2018 / Published: 19 July 2018
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Abstract
The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices An arising from virtually any kind of numerical discretization of differential equations (DEs). Indeed, when the mesh fineness parameter n tends to
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The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices An arising from virtually any kind of numerical discretization of differential equations (DEs). Indeed, when the mesh fineness parameter n tends to infinity, these matrices An give rise to a sequence {An}n, which often turns out to be a GLT sequence or one of its “relatives”, i.e., a block GLT sequence or a reduced GLT sequence. In particular, block GLT sequences are encountered in the discretization of systems of DEs as well as in the higher-order finite element or discontinuous Galerkin approximation of scalar DEs. Despite the applicative interest, a solid theory of block GLT sequences has been developed only recently, in 2018. The purpose of the present paper is to illustrate the potential of this theory by presenting a few noteworthy examples of applications in the context of DE discretizations. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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Open AccessArticle Optimal B-Spline Bases for the Numerical Solution of Fractional Differential Problems
Received: 14 May 2018 / Revised: 17 June 2018 / Accepted: 22 June 2018 / Published: 2 July 2018
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Abstract
Efficient numerical methods to solve fractional differential problems are particularly required in order to approximate accurately the nonlocal behavior of the fractional derivative. The aim of the paper is to show how optimal B-spline bases allow us to construct accurate numerical methods that
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Efficient numerical methods to solve fractional differential problems are particularly required in order to approximate accurately the nonlocal behavior of the fractional derivative. The aim of the paper is to show how optimal B-spline bases allow us to construct accurate numerical methods that have a low computational cost. First of all, we describe in detail how to construct optimal B-spline bases on bounded intervals and recall their main properties. Then, we give the analytical expression of their derivatives of fractional order and use these bases in the numerical solution of fractional differential problems. Some numerical tests showing the good performances of the bases in solving a time-fractional diffusion problem by a collocation–Galerkin method are also displayed. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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Open AccessArticle On Partial Cholesky Factorization and a Variant of Quasi-Newton Preconditioners for Symmetric Positive Definite Matrices
Received: 23 April 2018 / Revised: 19 June 2018 / Accepted: 20 June 2018 / Published: 1 July 2018
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Abstract
This work studies limited memory preconditioners for linear symmetric positive definite systems of equations. Connections are established between a partial Cholesky factorization from the literature and a variant of Quasi-Newton type preconditioners. Then, a strategy for enhancing the Quasi-Newton preconditioner via available information
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This work studies limited memory preconditioners for linear symmetric positive definite systems of equations. Connections are established between a partial Cholesky factorization from the literature and a variant of Quasi-Newton type preconditioners. Then, a strategy for enhancing the Quasi-Newton preconditioner via available information is proposed. Numerical experiments show the behaviour of the resulting preconditioner. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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Open AccessArticle Refinement Algorithms for Adaptive Isogeometric Methods with Hierarchical Splines
Received: 11 May 2018 / Revised: 16 June 2018 / Accepted: 18 June 2018 / Published: 21 June 2018
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Abstract
The construction of suitable mesh configurations for spline models that provide local refinement capabilities is one of the fundamental components for the analysis and development of adaptive isogeometric methods. We investigate the design and implementation of refinement algorithms for hierarchical B-spline spaces that
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The construction of suitable mesh configurations for spline models that provide local refinement capabilities is one of the fundamental components for the analysis and development of adaptive isogeometric methods. We investigate the design and implementation of refinement algorithms for hierarchical B-spline spaces that enable the construction of locally graded meshes. The refinement rules properly control the interaction of basis functions at different refinement levels. This guarantees a bounded number of nonvanishing (truncated) hierarchical B-splines on any mesh element. The performances of the algorithms are validated with standard benchmark problems. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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Open AccessArticle Efficient BEM-Based Algorithm for Pricing Floating Strike Asian Barrier Options (with MATLAB® Code)
Received: 11 May 2018 / Revised: 11 June 2018 / Accepted: 12 June 2018 / Published: 15 June 2018
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Abstract
This paper aims to illustrate how SABO (Semi-Analytical method for Barrier Option pricing) is easily applicable for pricing floating strike Asian barrier options with a continuous geometric average. Recently, this method has been applied in the Black–Scholes framework to European vanilla barrier options
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This paper aims to illustrate how SABO (Semi-Analytical method for Barrier Option pricing) is easily applicable for pricing floating strike Asian barrier options with a continuous geometric average. Recently, this method has been applied in the Black–Scholes framework to European vanilla barrier options with constant and time-dependent parameters or barriers and to geometric Asian barrier options with a fixed strike price. The greater efficiency of SABO with respect to classical finite difference methods is clearly evident in numerical simulations. For the first time, a user-friendly MATLAB® code is made available here. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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Open AccessArticle On the Analysis of Mixed-Index Time Fractional Differential Equation Systems
Received: 13 February 2018 / Revised: 11 April 2018 / Accepted: 11 April 2018 / Published: 17 April 2018
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Abstract
In this paper, we study the class of mixed-index time fractional differential equations in which different components of the problem have different time fractional derivatives on the left-hand side. We prove a theorem on the solution of the linear system of equations, which
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In this paper, we study the class of mixed-index time fractional differential equations in which different components of the problem have different time fractional derivatives on the left-hand side. We prove a theorem on the solution of the linear system of equations, which collapses to the well-known Mittag–Leffler solution in the case that the indices are the same and also generalises the solution of the so-called linear sequential class of time fractional problems. We also investigate the asymptotic stability properties of this class of problems using Laplace transforms and show how Laplace transforms can be used to write solutions as linear combinations of generalised Mittag–Leffler functions in some cases. Finally, we illustrate our results with some numerical simulations. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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Review

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Open AccessReview Collocation Methods for Volterra Integral and Integro-Differential Equations: A Review
Received: 27 April 2018 / Revised: 18 June 2018 / Accepted: 19 June 2018 / Published: 1 July 2018
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Abstract
We present a collection of recent results on the numerical approximation of Volterra integral equations and integro-differential equations by means of collocation type methods, which are able to provide better balances between accuracy and stability demanding. We consider both exact and discretized one-step
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We present a collection of recent results on the numerical approximation of Volterra integral equations and integro-differential equations by means of collocation type methods, which are able to provide better balances between accuracy and stability demanding. We consider both exact and discretized one-step and multistep collocation methods, and illustrate main convergence results, making some comparisons in terms of accuracy and efficiency. Some numerical experiments complete the paper. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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Open AccessReview Line Integral Solution of Differential Problems
Received: 4 May 2018 / Revised: 27 May 2018 / Accepted: 28 May 2018 / Published: 1 June 2018
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Abstract
In recent years, the numerical solution of differential problems, possessing constants of motion, has been attacked by imposing the vanishing of a corresponding line integral. The resulting methods have been, therefore, collectively named (discrete) line integral methods, where it is
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In recent years, the numerical solution of differential problems, possessing constants of motion, has been attacked by imposing the vanishing of a corresponding line integral. The resulting methods have been, therefore, collectively named (discrete) line integral methods, where it is taken into account that a suitable numerical quadrature is used. The methods, at first devised for the numerical solution of Hamiltonian problems, have been later generalized along several directions and, actually, the research is still very active. In this paper we collect the main facts about line integral methods, also sketching various research trends, and provide a comprehensive set of references. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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