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3 pages, 200 KiB

Open AccessEditorial

Advanced Numerical Methods in Applied Sciences

by Luigi Brugnano and Felice Iavernaro

Axioms 2019, 8(1), 16; https://doi.org/10.3390/axioms8010016 - 31 Jan 2019

Cited by 1 | Viewed by 3025

Abstract

The use of scientific computing tools is, nowadays, customary for solving problems in Applied Sciences at several levels of complexity. The great need for reliable software in the scientific community conveys a continuous stimulus to develop new and more performing numerical methods which [...] Read more.

The use of scientific computing tools is, nowadays, customary for solving problems in Applied Sciences at several levels of complexity. The great need for reliable software in the scientific community conveys a continuous stimulus to develop new and more performing numerical methods which are 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. Full article

(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)

Research

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18 pages, 361 KiB

Open AccessArticle

The Generalized Schur Algorithm and Some Applications

by Teresa Laudadio, Nicola Mastronardi and Paul Van Dooren

Axioms 2018, 7(4), 81; https://doi.org/10.3390/axioms7040081 - 9 Nov 2018

Cited by 5 | Viewed by 4523

Abstract

The generalized Schur algorithm is a powerful tool allowing to compute classical decompositions of matrices, such as the

Q R

and

L U

factorizations. When applied to matrices with particular structures, the generalized Schur algorithm computes these factorizations with a complexity of one [...] Read more.

The generalized Schur algorithm is a powerful tool allowing to compute classical decompositions of matrices, such as the

Q R

and

L U

factorizations. When applied to matrices with particular structures, the generalized Schur algorithm computes these factorizations with a complexity of one order of magnitude less than that of classical algorithms based on Householder or elementary transformations. In this manuscript, we describe the main features of the generalized Schur algorithm. We show that it helps to prove some theoretical properties of the R factor of the

Q R

factorization of some structured matrices, such as symmetric positive definite Toeplitz and Sylvester matrices, that can hardly be proven using classical linear algebra tools. Moreover, we propose a fast implementation of the generalized Schur algorithm for computing the rank of Sylvester matrices, arising in a number of applications. Finally, we propose a generalized Schur based algorithm for computing the null-space of polynomial matrices. Full article

(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)

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26 pages, 5646 KiB

Open AccessArticle

Efficient Implementation of ADER Discontinuous Galerkin Schemes for a Scalable Hyperbolic PDE Engine

by Michael Dumbser, Francesco Fambri, Maurizio Tavelli, Michael Bader and Tobias Weinzierl

Axioms 2018, 7(3), 63; https://doi.org/10.3390/axioms7030063 - 1 Sep 2018

Cited by 41 | Viewed by 5222

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 [...] Read more.

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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20 pages, 993 KiB

Open AccessArticle

On a Class of Hermite-Obreshkov One-Step Methods with Continuous Spline Extension

by Francesca Mazzia and Alessandra Sestini

Axioms 2018, 7(3), 58; https://doi.org/10.3390/axioms7030058 - 20 Aug 2018

Cited by 12 | Viewed by 3407 | Correction

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. [...] Read more.

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

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13 pages, 4277 KiB

Open AccessArticle

A Convex Model for Edge-Histogram Specification with Applications to Edge-Preserving Smoothing

by Kelvin C. K. Chan, Raymond H. Chan and Mila Nikolova

Axioms 2018, 7(3), 53; https://doi.org/10.3390/axioms7030053 - 2 Aug 2018

Cited by 2 | Viewed by 3624

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 [...] Read more.

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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13 pages, 264 KiB

Open AccessArticle

Trees, Stumps, and Applications

by John C. Butcher

Axioms 2018, 7(3), 52; https://doi.org/10.3390/axioms7030052 - 1 Aug 2018

Cited by 4 | Viewed by 2937

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 [...] Read more.

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)

14 pages, 323 KiB

Open AccessArticle

A Gradient System for Low Rank Matrix Completion

by Carmela Scalone and Nicola Guglielmi

Axioms 2018, 7(3), 51; https://doi.org/10.3390/axioms7030051 - 24 Jul 2018

Cited by 4 | Viewed by 3849

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 [...] Read more.

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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29 pages, 887 KiB

Open AccessArticle

Block Generalized Locally Toeplitz Sequences: From the Theory to the Applications

by Carlo Garoni, Mariarosa Mazza and Stefano Serra-Capizzano

Axioms 2018, 7(3), 49; https://doi.org/10.3390/axioms7030049 - 19 Jul 2018

Cited by 28 | Viewed by 3662

Abstract

The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices

A_{n}

arising from virtually any kind of numerical discretization of differential equations (DEs). Indeed, when the mesh fineness parameter n tends to [...] Read more.

The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices

A_{n}

arising from virtually any kind of numerical discretization of differential equations (DEs). Indeed, when the mesh fineness parameter n tends to infinity, these matrices

A_{n}

give rise to a sequence

{A_{n}}_{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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19 pages, 629 KiB

Open AccessArticle

Optimal B-Spline Bases for the Numerical Solution of Fractional Differential Problems

by Francesca Pitolli

Axioms 2018, 7(3), 46; https://doi.org/10.3390/axioms7030046 - 2 Jul 2018

Cited by 15 | Viewed by 4033

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 [...] Read more.

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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14 pages, 302 KiB

Open AccessArticle

On Partial Cholesky Factorization and a Variant of Quasi-Newton Preconditioners for Symmetric Positive Definite Matrices

by Benedetta Morini

Axioms 2018, 7(3), 44; https://doi.org/10.3390/axioms7030044 - 1 Jul 2018

Cited by 1 | Viewed by 3630

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 [...] Read more.

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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25 pages, 6401 KiB

Open AccessArticle

Refinement Algorithms for Adaptive Isogeometric Methods with Hierarchical Splines

by Cesare Bracco, Carlotta Giannelli and Rafael Vázquez

Axioms 2018, 7(3), 43; https://doi.org/10.3390/axioms7030043 - 21 Jun 2018

Cited by 26 | Viewed by 5454

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 [...] Read more.

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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17 pages, 436 KiB

Open AccessArticle

Efficient BEM-Based Algorithm for Pricing Floating Strike Asian Barrier Options (with MATLAB^® Code)

by Alessandra Aimi, Lorenzo Diazzi and Chiara Guardasoni

Axioms 2018, 7(2), 40; https://doi.org/10.3390/axioms7020040 - 15 Jun 2018

Cited by 4 | Viewed by 4049

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 [...] Read more.

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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23 pages, 953 KiB

Open AccessArticle

On the Analysis of Mixed-Index Time Fractional Differential Equation Systems

by Kevin Burrage, Pamela Burrage, Ian Turner and Fanhai Zeng

Axioms 2018, 7(2), 25; https://doi.org/10.3390/axioms7020025 - 17 Apr 2018

Cited by 4 | Viewed by 4812

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 [...] Read more.

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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14 pages, 11136 KiB

Open AccessReview

Stability Issues for Selected Stochastic Evolutionary Problems: A Review

by Angelamaria Cardone, Dajana Conte, Raffaele D’Ambrosio and Beatrice Paternoster

Axioms 2018, 7(4), 91; https://doi.org/10.3390/axioms7040091 - 1 Dec 2018

Cited by 15 | Viewed by 3052

Abstract

We review some recent contributions of the authors regarding the numerical approximation of stochastic problems, mostly based on stochastic differential equations modeling random damped oscillators and stochastic Volterra integral equations. The paper focuses on the analysis of selected stability issues, i.e., the preservation [...] Read more.

We review some recent contributions of the authors regarding the numerical approximation of stochastic problems, mostly based on stochastic differential equations modeling random damped oscillators and stochastic Volterra integral equations. The paper focuses on the analysis of selected stability issues, i.e., the preservation of the long-term character of stochastic oscillators over discretized dynamics and the analysis of mean-square and asymptotic stability properties of

ϑ

-methods for Volterra integral equations. Full article

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19 pages, 335 KiB

Open AccessReview

Collocation Methods for Volterra Integral and Integro-Differential Equations: A Review

by Angelamaria Cardone, Dajana Conte, Raffaele D’Ambrosio and Beatrice Paternoster

Axioms 2018, 7(3), 45; https://doi.org/10.3390/axioms7030045 - 1 Jul 2018

Cited by 25 | Viewed by 4962

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 [...] Read more.

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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28 pages, 407 KiB

Open AccessReview

Line Integral Solution of Differential Problems

by Luigi Brugnano and Felice Iavernaro

Axioms 2018, 7(2), 36; https://doi.org/10.3390/axioms7020036 - 1 Jun 2018

Cited by 90 | Viewed by 5375

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 [...] Read more.

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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Other

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6 pages, 582 KiB

Open AccessCorrection

Correction to “On a Class of Hermite-Obreshkov One-Step Methods with Continuous Spline Extension” [Axioms 7(3), 58, 2018]

by Francesca Mazzia and Alessandra Sestini

Axioms 2019, 8(2), 59; https://doi.org/10.3390/axioms8020059 - 16 May 2019

Viewed by 2434

Abstract

The authors of the above mentioned paper specify that the considered class of one-step symmetric Hermite-Obreshkov methods satisfies the property of conjugate-symplecticity up to order

p + r,

where

r = 2

and p is the order of the method. This generalization [...] Read more.

The authors of the above mentioned paper specify that the considered class of one-step symmetric Hermite-Obreshkov methods satisfies the property of conjugate-symplecticity up to order

p + r,

where

r = 2

and p is the order of the method. This generalization of conjugate-symplecticity states that the methods conserve quadratic first integrals and the Hamiltonian function over time intervals of length

O (h^{- r})

. Theorem 1 of the above mentioned paper is then replaced by a new one. All the other results in the paper do not change. Two new figures related to the already considered Kepler problem are also added. Full article

(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)

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