- 12 Gennaio 2018 ore 11.30 (aula 215 al Plesso Didattico Morgagni)

Prof. Rida T. Farouki, Univ. of California,

Titolo: The Bernstein polynomials: a centennial retrospective

Titolo: The Bernstein polynomials: a centennial retrospective

The Bernstein polynomial basis was introduced by Sergei Natanovich Bernstein in 1912 to provide a constructive proof of the Weierstrass approximation theorem. However, the leisurely convergence rate of Bernstein polynomial approximations to continuous functions caused them to languish in obscurity for more than half a century, pending the advent of digital computers. With the desire to exploit computers for geometric design applications, the Bernstein form began to enjoy widespread acceptance as a versatile means of intuitively constructing and manipulating geometric shapes, spurring further development of the basic theory, simple and efficient recursive algorithms, recognition of its excellent numerical stability properties, and an increasing diversification of its repertoire of applications. This talk surveys the historical evolution of the Bernstein form, and current state of theory, algorithms, and applications associated with this remarkable representation of polynomials over finite domains.

- 29 Novembre 2017 ore 11.30 (aula 230 Plesso Didattico Morgagni): two successive seminars:

1) Dr. Svenja Huning, Graz University of Technology, Institute of Geometry, Austria

Titolo: Adaption of linear subdivision schemes to Riemannian geometry

Titolo: Adaption of linear subdivision schemes to Riemannian geometry

Linear subdivision schemes produce limit curves by refining discrete data. These algorithms are based on linear rules. The convergence of those refinement rules and the smoothness of their limit curves are well-studied.

In this talk, we introduce different methods to adapt linear refinement rules to Riemannian geometry. We present methods using intrinsic properties of the manifold (e.g. log-exp-analogue, Riemannian center of mass) as well as the projection analogue which is based on an extrinsic property. In particular, we discuss the advantages and disadvantages of different procedures.

2) Dr. Sergio Lopez Urena, Dept. of Mathematics, Faculty of Mathematics, Valencia, Spain

Titolo: Combining multiresolution representation with optimization techniques. Applications in yacht designing.

Titolo: Combining multiresolution representation with optimization techniques. Applications in yacht designing.

In some optimization problems, the number of parameters is very large, but the values of the parameters themselves describe a smooth function. For instance, some yacht design optimization problems consist in modifying the shape of some pieces of the yacht to improve the navigation. In such optimizations, we are looking for the best location of points, which describes the optimal shape.

This talk is about a multilevel strategy on the parameter space, which main idea relies on the Harten’s multiresolution representation of data. The strategy requires to solve an optimization problem at each level, taking as initial guess the solution of the previous level.

We study this technique in some academic examples, showing its performance. A real yacht optimization problem was solved with this multilevel strategy, leading to an ’optimal’ shape for a keel, that minimizes the drag of the yacht.

[1] A. Harten, Multiresolution Representation of Data: A General Framework, SIAM J. Numer. Anal., 33(3), pp. 1205-1256, 1996.

[2] I. H. Abbot, A. E. Von Doenhoff, Theory of Wing Sections. Dover Publication, 1959.

[3] F.Fossati,Aero-Hydrodynamics and the Performance of SailingYachts: The Science Behind Sailboats and Their Design, A&C Black, 2009.

- 7 Novembre 2017 ore 11.30 (aula Tricerri al Dini)

Dr. Jaka Speh, MTU Aero Engines AG, Munich, Germany

Titolo: Tutorial of fitting tools in G+SMO

Titolo: Tutorial of fitting tools in G+SMO

We will present a fitting algorithm using G+Smo software library. The algorithm constructs a surface which approximates given points.

We will first present a theoretical background of the fitting and then we will show how to do the fitting in G+Smo.

- 24 Ottobre 2017 ore 14.30 (aula 212 Plesso didattico Morgagni):

Titolo: High-order optimality in nonlinear optimization: necessary conditions and a conceptual approach of evaluation complexity

We
consider recent progress in the use of high-order models to derive
worst-case evaluation complexity of algorithms for solving nonlinear
optimization problems, both constrained and unconstrained. This include
a discussion of what is meant by high-order critical points and the
intrinsic limitations that arise in connection with this concept.

We give some new results which improve existing evaluation complexity bounds and finally consider the difficulties of of further progress in this domain.

We give some new results which improve existing evaluation complexity bounds and finally consider the difficulties of of further progress in this domain.

- 12 Maggio 2017 ore 10.30 (aula 121 al plesso didattico Morgagni):

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.

- 1 Marzo 2017 ore 12 (aula Tricerri al Dini):

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.

- 21 Febbraio 2017 ore 15 (aula Tricerri al Dini):

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. J¨uttler, 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. J¨uttler, D. Großmann 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.

- 25 Gennaio 2017 ore 11 (aula Tricerri al Dini):

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.

- 13 Maggio 2016 ore 11.30 (aula 118 Plesso didattico Morgagni):

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.

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.

- 17 Marzo 2016 ore 11.30 (aula 212 del Plesso didattico Morgagni):

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.

Key words: spectral projected gradient, constrained stochastic problems, sample average approximation, variable sample size.

- 4 Dicembre 2015 ore 14.30 (aula Tricerri del Dip. di Matematica e Informatica):

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.

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.

- 18 Settembre 2015 ore 14.00 (aula 214 del plesso didattico Morgagni):

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.

- 15 Aprile 2015 ore 11.00 (aula Tricerri del Dip. di Matematica e Informatica):

- 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.

- 8 Aprile 2015 ore 11 (aula Tricerri del Dip. di Matematica e Informatica):

- Prof. Bert Jüttler, 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.

- 30 Marzo 2015 ore 14.30 (aula Tricerri del Dip. di Matematica e Informatica):

- 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.

- 20 Marzo 2015 ore 14 (aula 223 al Plesso didattico Morgagni):

- 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.

- 13 Marzo 2015 ore 11.30 (aula Tricerri del Dip. di Matematica e Informatica):

- 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.

- 19-20 Febbraio 2015 (aula Tricerri del Dip. di Matematica e Informatica):

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

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

DREAMS_Workshop

- 20 Gennaio 2015 ore 11.30 (aula Tricerri del Dip. di Matematica e Informatica):

- 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.

- 24 Ottobre 2014 ore 11.30 (aula Tricerri del Dip. di Matematica e Informatica):

- 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.

- 27 Giugno 2014 ore 12.15 (aula Tricerri del Dip. di Matematica e Informatica):

- 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, BÈzier 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. Mokriö, B. Jüttler 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.

- 5 Giugno 2014 ore 11 (aula Tricerri del Dip. di Matematica e Informatica) :

- Dott. Gianluca Frasca Caccia, Univ. di Firenze

- Titolo: Hamiltonian 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.

- 9 Maggio 2014 ore 11 (aula Tricerri del Dip. di Matematica e Informatica):

- 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.

- 11 Marzo 2014 ore 11 (aula 215 del Plesso didattico Morgagni):

- Dott. Laura Iapichino, Universität 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 più 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 complessità 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 complessità
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 sensitività, 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.

- 8 Gennaio 2014 ore 14.30 (aula Tricerri del Dip. di Matematica e Informatica):

- 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.