SEMINARI DI ANALISI NUMERICA

         Prossimo seminario riportato in rosso


        Prof. Luca Dieci, Georgia Institute of Technology, Atlanta, USA
       Titolo: The boundary method for semi-discrete optimal transport and Wasserstein distance computations

We introduce the "boundary method," a new technique for semi-discrete optimal transport problems.  We give  theoretical justification, convergence results, and algorithmic development and testing.




        Dott. Andrea Bressan, University of Oslo, Norway
       Titolo: Quasi-interpolants for tensor-product B-splines over T-meshes

Quasi-interpolants are local approximation operators. They are used both as theoretical tool to derive approximation estimates and to construct well behaved approximation while avoiding the computational cost of an interpolation problem. Different extensions of the tensor-product construction that allow for local refinability were proposed in the last decades. In particular they are a prerequisite for adaptive approximation that can be applied both in shape description as  in analysis. It is thus of interest to study quasi-interpolants for these new constructions. The talk will focus on the stability of quasi-interpolant with respect of the mesh and the knot-vectors associated to the basis functions.



        Dott. Cesare Bracco, UniversitÓ di Firenze, Italy
       Titolo: Adaptive data fitting by quasi-interpolation in hierarchical spline spaces
      

The hierarchical splines have been introduced to address the issue of local refinement of splines spaces, and proved to be an effective tool in a variety of problems (see, e.g., [3, 4]). A natural application is data fitting, where an efficient approximation requires the use of different levels of resolution according to the local features of the data. In this talk, we will show how hierarchical spline spaces can be used both for gridded and scattered data, with techniques recently developed in [1, 2]. In both cases, we will construct quasi-interpolants (QIs), combining the general framework introduced in [5] with suitable local approximants. While in the gridded case, by exploiting the regular structure of the data, the QI is designed to use a higher resolution in the areas where approximating sharp or little details is required, for scattered data the QI must also adapt to the potentially very irregular distribution of the data. We will examine several examples to give an overview of the possible issues arising when approximating different types of data.

[1] C. Bracco, C. Giannelli, F. Mazzia and A. Sestini, Bivariate hierarchical Hermite spline quasi-interpolation, BIT 56 (2016), 1165-1188.
[2] C. Bracco, C. Giannelli and A. Sestini, Adaptive scattered data fitting by extension of local approximations to hierarchical splines, submitted for publication (2017).
[3] C. Giannelli, B. JĘuttler, S.K. Kleiss, A. Mantzaflaris, B. Simeon and J. ˇSpeh, THBsplines: An effective mathematical technology for adaptive refinement in geometric design and isogeometric analysis, Comput. Meth. Appl. Mech. Eng. 299 (2016), 337- 365.
[4] G. Kiss, C. Giannelli, U. Zore, B. JĘuttler, D. Gro▀mann and J. Barner, Adaptive CAD model (re-)construction with THB-splines, Graph. Models 76 (2014), 273-288.
[5] H. Speleers and C. Manni, Effortless quasi-interpolation in hierarchical spaces, Numer. Math. 132 (2016), 155-184.




        Dott. Gianmarco Gurioli, UniversitÓ di Firenze, Italy
       Titolo: Recent Developments about Discrete Line Integral Methods for Hamiltonian Problems. Energy and QUadratic Invariants Preserving methods (EQUIP).
      

One of the main features, when dealing with Hamiltonian problems, is the conservation of the Hamiltonian function along the numerical solution. It is for this reason that we study the family of Runge-Kutta energy-preserving methods named Hamiltonian Boundary Value Methods (HBVMs), discussing their order and preservation properties. The analysis of these methods, which have the advantage of preserving the Hamiltonian within round-off errors, predicts several numerical tests on the Keplero Problem. These tests are aimed at confirming the theoretical achievements and comparing HBVM methods with those that are usually used to solve differential problems. Sometimes conservative problems are not in Hamiltonian form and they may possess multiple independent invariants. For this purpose, we extend the approach basing it on a suitable discrete line integral, thereby achieving multiple invariants conservation. In so doing, passing through the definition of Enhanced Line Integral Methods (ELIMs), we consider the class of Energy and QUadratic Invariants Preserving methods (EQUIP), defined by a symplectic map (so that methods conserve all quadratic invariants) and, at the same time, able to yield energy conservation.





        Prof. Philippe Toint, University of Namur, Belgio
       Titolo: Complexity in nonlinear optimization made (quite) simple
      

The talk will consider the question of the worst-case evaluation complexity of finding approximate first-order critical points in nonlinear (nonconvex) smooth optimization using p-th order models.
A remarkably simple proof, based on a standard regularization algorithm, will be given that at worst O(epsilon^{-(p+1)/p}) evaluations of the objective function and its derivatives are needed to compute an epsilon-approximate critical point for unconstrained and convexly-constrained cases. A two-phases framework will also be described for handling the case where constraints are fully general (equalities and inequalities) and the evaluation complexity shown to be at worst O(epsilon^{-(p+2)/p}) in this case.





        Prof. Natasa Krejic, University of Novi Sad, Serbia
        Titolo: Spectral projected gradient method for stochastic optimization


We consider the Spectral Projected Gradient method for solving constrained optimization porblems with the objective function in the form of mathematical expectation. It is assumed that the feasible set is  convex, closed and easy to project on. The objective function is approximated by a sequence of Sample Average Approximation func- tions with different sample sizes. The sample size update is based on two error estimates - SAA error and approximate solution error. The Spectral Projected Gradient method combined with a nonmonotone line search is used. The almost sure convergence results are achieved without imposing explicit sample growth condition. Numerical results show the efficiency of the proposed method.
Key words: spectral projected gradient, constrained stochastic problems, sample average approximation, variable sample size.




        Prof. Tom Lyche, UniversitÓ di Roma Tor Vergata, Italy
        Titolo: Splines on the Powell-Sabin 12-split
 

Piecewise polynomials or splines defined over triangulations form an indispensable tool in the sciences, with applications ranging from scattered data fitting to finding numerical solutions to partial differential equations. In applications like geometric modeling and solving PDEs by isogeometric methods one often desires a low degree spline with C^1, C^2 or C^3 smoothness. For a general triangulation, it is known that the minimal degree of a triangular C^r element is 4r+1, e.g., degrees 5; 9; 13 for the classes C^1, C^2 or C^3. To obtain smooth splines of lower degree one can split each triangle in the triangulation into several subtriangles. One such split that we consider here is the Powell-Sabin 12-split of a triangle.
Once a space is chosen one determines its dimension. The spaces of C^1 quadratics and C^3 quintics on the 12-split of a single triangle have dimension 12 and 39, respectively.
Over a general triangulation T of a polygonal domain we can 12-split each triangle in T to obtain a triangulation T12. The dimensions of the corresponding C^1 quadratics and  C^2 quintics spaces (the latter with C^3 supersmoothness at the vertices and the interior edges of each macro triangle) are 3V+E and 10V+3E, respectively, where V and E are  the number of vertices and edges in T. Moreover, in addition to giving C^1 and C^2 spaces on any triangulation these spaces are suitable for multiresolution analysis, see for example [2].
To compute with these spaces one needs a suitable basis. In the univariate case the B-spline basis is an obvious choice. In this talk we consider a bivariate
generalization known as simplex splines. These are the natural generalization of B-splines to the multivariate case, see [4].
After a brief introduction to simplex splines we present simplex splines bases for C^1 quadratics and C^3 cubics on the 12-split of a single triangle. We give several Simplex spline bases for the C^3 element. These bases form a nonnegative partition of unity, satisfy a Marsden-like identity, and the restriction of each basis element to the boundary  edges of the macro element reduces to a standard univariate quinitc B-spline. The C^3 piecewise polynomial on one triangle can be combined with neighboring elements to form a C^2 representation on any triangulation.

[1]  Elaine Cohen, Tom Lyche and Richard F. Riesenfeld, A B-spline like basis for the Powell-Sabin 12-split based on simplex splines, Mathematics of Computation, Vol. 82 (2013), 1667—1707.
[2] Tom Lyche and Georg Muntingh, A Hermite interpolatory subdivision scheme for C^2-quintics on the Powell-Sabin 12-split, Comput. Aided Geom. Design Vol. 31(2014), no. 7--8, 464--474.
[3] Tom Lyche and Georg Muntingh, Stable simplex spline bases for C^3 quintics on the Powell-Sabin 12-split, available at http://arxiv.org/abs/1504.02628.
[4] Charles A. Micchelli, On a numerically efficient method for computing multivariate B-splines, in "Multivariate approximation theory", Walter Schempp and Karl Zeller (eds.), International Series of Numerical Mathematics Vol. 51, Birkh\"auser Verlag, Basel, Boston, Stuttgart, 1979, 211--248.

 


        Prof. Nira Dyn,  Univ. Tel Aviv, Israel

        Titolo:   Reconstruction of 3D objects from their 2D cross-⁠sections by a subdivision schemes for sets.

        The first part of this talk consists of a short review on subdivision schemes for curves, and presents two important schemes, which are used later in the reconstruction task. The second part consists of few facts about sets and
          about  the approximation of set-⁠valued functions from their samples.  We conclude the talk with our adaptation of the 4-⁠point subdivision scheme to sets, which is based on the "measure average" designed for the reconstruction
          task, and give few examples of reconstructions.






Prof. Marco Donatelli,  Univ. dell'Insubria

Titolo:  Fast nonstationary preconditioned iterative methods for ill-posed problems, with application to image deblurring.

We introduce a new iterative scheme for solving linear ill-posed problems, similar to nonstationary iterated Tikhonov regularization, but with an approximation of the underlying operator to be used for the Tikhonov equations. For image deblurring problems such an approximation can be a discrete deconvolution that operates entirely in the Fourier domain. We provide a theoretical analysis of the new scheme, using regularization parameters that are chosen by a certain adaptive strategy.  
Some extensions with projection into a convex set and the use of regularization operators are discussed. Moreover, lest-square methods are usually employed as inner step in iterative methods for nonlinear models. In this framework we discuss the combination of our iterative scheme with the linearized Bregman splitting for image deblurring.  
The talk presents joint works with M. Hanke, D. Bianchi, A. Buccini, Y. Cai, and T. Z. Huang.



Prof. Bert JŘttler, Johannes Kepler University, Linz, Austria
Titolo:  Isogeometric Analysis with Geometrically Continuous Functions

Abstract: We study the linear space of C^s-smooth isogeometric functions defined on a multi-patch domain. We show that the construction of these functions is closely related to the concept of geometric continuity of surfaces, which has originated in geometric design. More precisely, the C^{s}-smoothness of isogeometric functions is found to be equivalent to geometric smoothness of the same order (G^s-smoothness) of their graph surfaces. This motivates us to call them C^s-smooth geometrically continuous isogeometric functions.
We present a general framework to construct a basis and explore potential applications in isogeometric analysis. The space of C^1-smooth geometrically continuous isogeometric functions on bilinearly parameterized two-patch domains is analyzed in more detail. Numerical experiments with bicubic and biquartic functions for performing L^2 approximation and for solving Poisson's equation and the biharmonic equation on two-patch geometries are presented and indicate optimal rates of convergence.
The talk presents joint work with K. Birner, F. Buchegger, M. Kapl, and V. Vitrih.






     


Prof. J.I.Montijano, Universidad de Zaragoza, Spain
Titolo:  Numerical methods for slow energy-varying problems (part 1: theoretical results; part 2: discussion)

Abstract: In this talk the numerical integration of  perturbations of Hamiltonian systems, as for example slowly dissipative Hamiltonian systems, is considered. The aim is to find numerical methods that are able to reproduce appropriately the evolution of the energy. On one side,  algorithms based on a combination of standard numerical integration methods and certain projection techniques  are proposed. On the other hand, conditions under which energy-preserving methods reproduce that desirable evolution are analysed. Some numerical experiments to confirm the theory and show a good qualitative and quantitative performance of the considered methods are presented.





Prof. Jacek Gondzio, University of Edinburgh, Great Britain, joint work with  K. Fountoulakis  
Titolo:  Preconditioners for higher order methods in big data optimization

Abstract: We address efficient preconditioning techniques for the second-order methods applied to solve various sparse approximation problems arising in big data optimization. The preconditioners cleverly exploit special features of such problems and cluster the spectrum of eigenvalues around one. The inexact Newton Conjugate Gradient method excels in these conditions. Numerical results of solving L1-regularization problems of unprecedented sizes will be presented.




Dr. Dominik Mokris, Univ. di Linz, Austria
Titolo:  Completeness of THB-splines

Abstract: We begin with an industrially motivated example of least squares  fitting, where THB-splines (particular set of piecewise polynomials  allowing for local refinement) outperform tensor-product B-splines.  Properly motivated, we then turn to theoretical properties of  THB-splines and similar constructions. In particular, we will  investigate their completeness, i.e., whether and when they generate the full space of piecewise polynomials on a given hierarchical mesh with prescribed smoothness.


DREAMS Workshop, Futuro in Ricerca 2013 - Analisi Numerica: “Design of Reliable, Exact, and Application-oriented techniques for geometric Modeling and Numerical Simulation (DREAMS)” organizzato da Carlotta Giannelli (INdAM c/o University of Florence) e Hendrik Speleers (University of Rome “Tor Vergata”)

per i dettagli (speakers e time schedule) si veda al seguente link:

DREAMS_Workshop





Prof. Francesca Mazzia, Univ. di Bari
Titolo:  Conditioning and mesh selection for differential problems

Abstract: Boundary value problems for ordinary differential equations (BVODES) occur in many practical situations and they are generally much harder to solve than initial value problems. Traditionally, codes for BVODES did not take into account the conditioning of the problem and it was generally assumed that the problem being solved was well conditioned so that small local errors gave rise to correspondingly small global errors. Recently a new generation of codes which take account of conditioning has been developed. The lecture will describe a fundamental approach to defining sequences of meshes so that the continuous and discrete problems have the same conditioning. This is done by developing a monitor function which depends both on local accuracy and conditioning. This technique to compute the conditioning parameters and to define the mesh, which is ideally suited to BVODES, has been implemented in the codes: TOM, TWPBVPC, TWPBVPLC and ACDCC. The main difference is in how they choose the different values of a few heuristic parameters. The new codes can be considerably more efficient than other codes, due to the fact that they pay attention to the conditioning of the problem[2, 3, 5, 6]. A similar mesh selection strategy has been extended for the solution of Initial Value Problems and implemented in the codes DOPRI5 and CASHKARP [1, 4, 5].

[1] Mazzia F, Nagy AM (in stampa). A new mesh selection strategy with stiffness detection for explicit Runge-Kutta methods. APPLIED MATHEMATICS AND COMPUTATION, ISSN: 0096-3003, doi: http://dx.doi.org/10.1016/j.amc.2014.03.065.
[2] Mazzia F, Cash JR, Soetart K (2014). Solving boundary value problems in the open source software R: package bvpSolve. OPUSCULA MATHEMATICA. ROCZNIK AKADEMIA GRNICZO-HUTNICZA IM. STANIS?AWA STASZICA, vol. 34, p. 387-403, ISSN: 1232-9274.
[3] J. R. Cash JR, Hollevoet D, Mazzia F, Nagy AM (2013). Algorithm 927: The MATLAB code bvptwp.m for the numerical solution of Two Point BVPs.. ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE, vol. 39, 15, ISSN:           0098-3500, doi: http://dl.acm.org/citation.cfm?doid=2427023.2427032.
[4] Mazzia F, Cash JR, Soetaert K (2012). A Test Set for stiff Initial Value Problem Solvers in the open source software R: Package deTestSet. JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, vol. 236, p. 4119-4131, ISSN: 0377-0427, doi: 10.1016/j.cam.2012.03.014.
[5] Soetaert K, Cash JR, Mazzia F (2012). Solving Differential Equations in R . BERLIN HEIDELBERG:Springer-Verlag, ISBN: 978-3-642-28069-6, doi: 10.1007/978-3-642-28070-2.
[6] Cash JR, F. Mazzia F (2009) Conditioning and Hybrid Mesh Selection Algorithms for Two- Point Boundary Value Problems, Scalable Computing: Practice and Experience, vol. 10 (4), pp. 347-361.
 



Prof. Luigi Brugnano, Univ. di Firenze
Titolo:  Recent advances in the numerical solution of Hamiltonian problems

Abstract: The numerical solution of conservative problems is an active field of investigation dealing with the geometrical properties of the discrete vector field induced by numerical methods. The final goal is to reproduce, in the discrete setting, a number of geometrical properties shared by the original continuous problem. Because of this reason, it has become customary to refer to this field of investigation as geometric integration. In particular, we shall deal with the numerical solution of Hamiltonian problems, which are encountered in many real-life applications, ranging from the nano-scale of molecular dynamics to the macro-scale of celestial mechanics. Such problems are characterized by the conservation of the associated Hamiltonian function. Often, the Hamiltonian is also called the energy, since for isolated mechanical systems it has the physical meaning of total energy. Consequently, energy conservation is an important feature in the correct simulation of such problems. In this talk we review the main facts about the recently introduced family of energy-conserving Runge-Kutta-type methods named Hamiltonian Boundary Value Methods (HBVMs), and sketch their application to both Hamiltonian ODE and PDE problems.


Dott. Cesare Bracco, Univ. di Torino
Titolo:  T-spline spaces
Abstract: The concept of T-spline, first introduced in [9] brought significant advancements to the application of spline spaces both to modelling techniques and to methods for the numerical solution of differential problems (in particular in isogeometric analysis, see, e.g., [1]): the possibility to perform local refinements allowed the introduction of new and improved techniques in several areas.
The spaces spanned by the T-splines are widely studied by several authors (see, e.g., [6,2]). In this seminar, we will focus on some particularly relevant issues related to these spaces. The need to construct linearly independent T-splines led to the introduction of analysis-suitable (dual-compatible, equivalently) T-meshes (see [6,2]), where T-meshes guaranteeing the linear independence for the associated T-splines are characterized from the topological point of view. In the seminar, the possibility to consider larger classes of T-meshes guaranteeing the linear independence will be discussed. The same issues can be also considered and studied in the non-polynomial case, that is, in the case of Generalized T-splines (see, e.g., [7,3]). Recently, some authors also suggested the possibility to use subspaces of T-splines (see [5]) and hierarchical versions of T-spline spaces (see [4], and [8] for classical hierarchical spline spaces).

[1] Y. Bazilevs, V.M. Calo, J.A. Cottrell, J.A. Evans, T.J.R. Hughes, S. Lipton, M.A. Scott and T.W. Sederberg, Isogeometric analysis using T-splines, Comput. Methods Appl. Mech. Engrg. 199 (2010), 229-263.
[2] L. Beirao da Veiga, A. Buffa, G. Sangalli and R. Vazquez, Analysis-suitable T-splines of arbitrary degree: definition and properties, Math. Mod. Meth. Appl. Sci. 23 (2013), 1979-2003.
[3] C. Bracco, D. Berdisnky, D. Cho, M. Oh and T. Kim, Trigonometric Generalized T-splines, Comput. Methods Appl. Mech. Engrg. 268 (2014), 540-556.
[4] E.J. Evans, M.A. Scott, X. Li and D.C. Thomas, Hierarchical analysis-suitable T-splines: Formulation, B╚zier extraction, and application as an adaptive basis for isogeometric analysis, available on arXiv (2014).
[5] H. Kang, F. Chen and J. Deng, Modified T-splines, Computer Aided Geometric Design 30 (2013), 827-843.
[6] X. Li and M.A. Scott, Analysis-suitable T-splines: Characterization, refineability, and approximation, Math. Models Methods Appl. Sci. 24 (2014), 1141-1164.
[7] C. Manni, F. Pelosi and M.L. Sampoli, Generalized B-splines as a tool in isogeometric analysis, Comput. Methods Appl. Mech. Engrg. 200 (2011), 867-881.
[8] D. Mokri÷, B. JŘttler and C. Giannelli, On the completeness of hierarchical tensor-product B-splines, Journal of Computational and Applied Mathematics 271 (2014), 53-70.
[9] T.W. Sederberg, J. Zheng, A. Bakenov and A. Nasri, T-splines and T-nurccs, ACM Trans. Graph. 22 3 (2003), 477-484.

Dott. Gianluca Frasca Caccia, Univ. di Firenze
TitoloHamiltonian Boundary Value Methods (HBVMs) and their efficient implementation
Abstract: One of the main features when dealing with Hamiltonian systems is the conservation of energy. In this talk I will expose the main fact about a family of conservative Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs) for the efficient numerical integration of these problems. These methods yield exact conservation for polynomial energy of arbitrarily high degree and an at least "practical" conservation for non polynomial energy. We will also discuss about the efficient implementation of HBVMs by means of two different procedures: the "blended" implementation and a new iterative procedure based on a particular triangular splitting of the corresponding Butcher's matrix. The linear convergence analysis of these two procedures exhibits excellent properties that make these procedures more efficient than a classical fixed point iteration for stiff problems. A few numerical tests confirming all the theoretical findings will be shown.

Dott. Carlotta Giannelli, INdAM c/o Univ. di Firenze
Titolo:  Adaptive techniques for isogeometric analysis
Abstract: The isogeometric approach to approximate partial differential equations promotes the use of a common representation model for the description of the geometry and the basis for analysis. Exact geometries can then be introduced in the simulation setting by considering suitable spline models. Since the spline standard in commercial Computer Aided Design (CAD) systems relies on tensor-product B-spline structures, an adaptive isogeometric method necessarily requires to exploit alternative paradigms which allow highly localized refinement procedures. The talk will address recent results concerning hierarchical spline constructions and their effective use for the development of adaptive techniques in geometric design and isogeometric methods.

Dott. Laura Iapichino, Universitńt Konstanz, Germany
Titolo:  Metodo a basi ridotte per la soluzione di problemi di ottimizzazione  multi-obiettivo
Abstract: In questo seminario viene descritto un modello di ordine ridotto per la risoluzione numerica di problemi di ottimizzazione multiobiettivo governati da equazioni alle derivate parziali parametrizzate. Numerose applicazioni di interesse industriale e ingegneristico sono caratterizzate dalla presenza contemporanea di pi¨ obiettivi, ovvero funzioni a valori reali da massimizzare e/o minimizzare, tipicamente in contrasto tra loro.  In generale, per questi problemi non esiste un'unica soluzione ottimale, ma  un insieme (possibilmente infinito) di soluzioni ottimali secondo Pareto [1]. A causa del numero elevato di soluzioni ottimali richieste e della discretizzazione numerica delle equazioni alle derivate parziali, questi problemi sono caratterizzati da una notevole complessitÓ computazionale. Per ridurre quest'ultima, il problema viene riformulato in forma parametrizzata e viene utilizzata un'approssimazione a basi ridotte [3]  della soluzione combinando opportunamente un insieme di soluzioni (o basi) precedentemente calcolate (tramite una tecnica di discretizzazione tradizionale, come il metodo degli elementi finiti). Il metodo alle basi ridotte applicato in questo contesto, rappresenta una tecnica di ordine ridotto capace di ridurre considerevolmente la complessitÓ computazionale e i tempi di risoluzione necessari per il calcolo di ogni soluzione ottimale, garantendo un adeguato livello di accuratezza tenendo conto di una rigorosa analisi a posteriori dell'errore [2]. Infine, un'efficiente analisi di sensitivitÓ, permette di ridurre il numero di soluzioni ottimali da calcolare al fine di definire un insieme sufficientemente adeguato di soluzioni di Pareto e quindi ridurre ulteriormente i tempi richiesti per un'accurata ed affidabile risoluzione dei problemi multiobiettivo presentati.
[1] C. Hillermeier. Nonlinear multiobjectiveoptimization. A generalized homotopy approach. Birkhaeuser Verlag, Basel, 2001.
[2] F. Negri, G. Rozza, A. Manzoni and A. Quarteroni, Reduced basis method for parametrized elliptic optimal control problems, SIAM Journal on Scientific Computing, vol. 35, num. 5, p. A2316--A2340, 2013.
[3]G. Rozza, D.B.P. Huynh, and A.T. Patera. Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Engrg., 15:229--275, 2008.


Dott. Alessandra Aimi, Univ. di Parma
Titolo: Energetic BEM-FEM coupling for the numerical solution of wave propagation problems in unbounded multi-domains
Abstract: The talk will be focused on the numerical solution of wave propagation problems defined in unbounded multi-domains. The approximation is operated by a suitable coupling of boundary and finite element methods, directly written in space-time domain, used as local discretization techniques and both considered in an energetic framework. Fundamentals which allow the adopted boundary integral reformulation of the differential wave propagation problem will be shortly recalled; some details on quadrature schemes developed for the numerical evaluation of matrix elements in the linear system of the final time-marching procedure will be explained; emphasis will be given to the stability analysis of the proposed energetic approach. At last, several numerical results on wave propagation model problems will be presented and discussed.