·
16 Novembre 2021 ore 12 (aula Tricerri al Dini)

**Dott****. Andrea Farahat, Johann Radon Institute for Computational and
Applied Mathematics, Austrian Academy of Sciences, Austria**

A framework for the construction of a
C1-smooth isogeometric spline spaces over particular G1 multi-patch surfaces, called analysis-suitable
G1 (AS-G1) surfaces [1, 2], will be presented. The class of AS-G1 multi-patch
geometries is of importance since it includes exactly those G1-smooth
multi-patch geometries which allow the design of C1-smooth isogeometric
spline spaces with optimal approximation properties. The method extends the
construction [3] for AS-G1 planar multi-patch parametrizations to the AS-G1
multi-patch surface case. We also generate for the C1-smooth isogeometric spline space a local basis which is used to
solve, by a standard Galerkin approach, the
biharmonic equation –a particular fourth order partial differential
equation – over several AS-G1multi-patch surfaces. The obtained numerical
results exhibit optimal convergence orders in the L2, H1 and H2 norm, and
demonstrate the potential of our C1-smooth isogeometric
spline functions for solving fourth order partial differential equations over multipatch surfaces.

Joint work
with:

Mario Kapl (Carinthia University of Applied Sciences, Austria),

Bert Jüttler (Johannes Kepler University Linz, Austria),

Thomas Takacs
(Johannes Kepler University Linz, Austria).

References

[1] Annabelle
Collin and Giancarlo Sangalli and Thomas Takacs.
Analysis suitable G1 multi-patch parametrizations for C1 isogeometric
spaces. Comput. Aided Geom. Design, 47, 93-113, 2016.

[2] Mario Kapl and Giancarlo Sangalli and
Thomas Takacs. Construction of analysis-suitable G1
planar multi-patch parameterizations. Comput. Aided
Geom. Design, 97, 41-55, 2018.

[3] Mario Kapl and Giancarlo Sangalli and
Thomas Takacs. An isogeometric C1 subspace on
unstructured multi-patch planar domains. Comput.
Aided Geom. Design, 69, 55-75, 2019.

23 Ottobre 2020 ore 11.30
(aula Tricerri al Dini)

**Dott. Tadej Kanduč,
University of Ljubljana,
Slovenia
Titolo: **Regularization of
singular integrals for potential problems in 3D IgA-BEM

In this talk a novel numerical
integration scheme for the governing singular integrals that appear in Boundary
element method (BEM) is presented. By adopting the isoparametric
approach, curved geometries that are represented by mapped rectangles or
triangles from the parametric domain, are considered. The analytical
singularity extraction can be performed either as an operation of subtraction
or division, each having some advantages.

A particular series expansion of a singular kernel about a source point is
investigated. The series in the intrinsic coordinates consists of functions of
a type R^p x^q y^r, where R is a square root of a quadratic bivariate
homogeneous polynomial, corresponding to the first fundamental form of a smooth
surface, and p,q,r are
integers. By taking more terms in the series expansion, we can increase the
smoothness of the regularized kernel at the source point. Analytical formulae
for such surface integrals have closed form expressions.

Some numerical tests that demonstrate the use of the novel integration scheme
are provided.

- 24 Febbraio-20 Marzo 2020 (Dini):

Prof. **Oleg Davydov,
University of Giessen, Germany**

**PhD course, Local Approximation and Numerical Differentiation by
Polynomial and Kernel Methods**

- 28-29 Novembre 2019 (aula Tricerri
al Dini):

**Giornata di lavoro
del progetto GNCS 2019
"Metodi di approssimazione locale con applicazioni all'analisi isogeometrica e alle equazioni integrali di contorno"**

**4 Ottobre 2019 ore 11.30**(aula 108 Plesso Didattico Morgagni):

**Prof. ****Philippe Toint,**** ****University of Namur,
Belgium**

**Titolo**: Complexity of nonconvex
optimization

We present a review of results on the worst-case
complexity of minimization algorithms for nonconvex problems using potentially
high-degree models.

Global complexity bound are presented that are valid for any model's degree and
any order of optimality, thereby generalizing known results for first- and
second-order methods. An adaptive regularization algorithm using derivatives up
to degree p will produce an epsilon-approximate q-th
order minimizer in at most O( epsilon^( -(p+1)/(p−q+1)
) evaluations. We will also extend these results to the case of inexact
objective function and derivatives with an application to subsampling
algorithms for machine learning.

**26 Giugno 2019 ore 14.30**(aula Tricerri al Dini)

**Prof. Kevin Burrage, University of Oxford and Queensland University of
Technology, Brisbane, Australia**

**Titolo****: ** Image based modelling and simulation: Perlin
Noise generation of physiologically realistic patterns of fibrosis

Fibrosis, the pathological excess of
fibroblast activity, is a significant health issue that hinders the function of
many organs in the body, in some cases fatally.
However, the severity of fibrosis-derived conditions depends on both the
positioning of fibrotic affliction, and the microscopic patterning of
fibroblast-deposited matrix proteins within affected regions. Variability in an
individual's manifestation of a type of fibrosis is an important factor in
explaining differences in symptoms, optimum treatment and prognosis, but a need
for ex vivo procedures and a lack of experimental control over conflating
factors has meant this variability remains poorly understood.

In this work, we present a computational methodology, based on Perlin
noise fields, Fast Fourier Transforms and SMC ABC parameter estimation, for the
generation of patterns of fibrosis microstructure. We demonstrate the technique
using histological images of four types of cardiac fibrosis. Our generator and
automated tuning method prove flexible enough to capture each of these very
distinct patterns, allowing for rapid generation of new realisations
for high-throughput computational studies. We also demonstrate via simulation,
using the generated fibrotic patterns, the importance of micro-scale
variability by showing significant differences in electrophysiological impact
even within a single class of fibrosis, hence quantifying arrhythmic risk. The
key novel impact of our methodology is, through data enhancement and image based simulation, to remove limitations posed by the
availability of ex-vivo data whilst being sophisticated enough to produce
physiologically realistic patterns that match the data available and then to
use image-based simulation to quantify arrhythmic risk.

**26 Giugno 2019 ore 15.30**(aula Tricerri al Dini)

**Prof. Pamela Burrage, Queensland University of Technology, Brisbane,
Australia**

**Titolo****: ** Integrated Approaches for Stochastic Chemical
Kinetics

In this talk I discuss how we can simulate stochastic
chemical kinetics when there is a memory component. This can occur when there
is spatial crowding within a cell or part of a cell, which acts to constrain
the motion of the molecules which then in turn changes the dynamics of the
chemistry. The counterpart of the Law of Mass Action in this setting is through
replacing the first derivative in the ODE description of the Law of Mass Action
by a time-fractional derivative, where the time-fractional index is between 0
and 1. There has been much discussion in the literature, some of it
wrong, as to how we model and simulate stochastic chemical kinetics in
the setting of a spatially-constrained domain – this is sometimes called
anomalous diffusion kinetics. In this presentation, I discuss some of these
issues and then present two (equivalent) ways of simulating fractional
stochastic chemical kinetics. The key here is to either replace the exponential
waiting time used in Gillespie's SSA by Mittag-Leffler
waiting times (MacNamara et al. [2]), which have
longer tails than in the exponential case. The other
approach is to use some theory developed by Jahnke and Huisinga
[1] who are able to write down the underlying
probability density function for any set of mono-molecular chemical reactions
(under the standard Law of Mass Action) as a convolution of either binomial
probability density functions or binomial and Poisson probability density
functions). We can then extend the Jahnke and Huisinga
formulation through the concept of iterated Brownian Motion paths to produce
exact simulations of the underlying fractional stochastic chemical process. We
demonstrate the equivalence of these two approaches through simulations and also by computing the probability density function of
the underlying fractional stochastic process, as described by the fractional
chemical master equation whose solution is the Mittag-Leffler
matrix function. This is computed based on a clever algorithm for computing
matrix functions by Cauchy contours (Weideman and Trefethen
[3]).

This is joint work with Manuel Barrio (University of Vallodolid,
Spain), Kevin Burrage (QUT), Andre Leier (University of Alabama), Shev MacNamara
(University of Technology Sydney) and T. Marquez-Lago (University of Alabama).

[1] T. Jahnke and W. Huisinga, 2007, Solving the
chemical master equation for

monomolecular reaction systems analytically, J. Math. Biology 54, 1,
1—26.

[2] S. MacNamara, B. Henry and W. McLean, 2017,
Fractional Euler limits and

their applications, SIAM J. Appl. Math. 77, 2, 447—469.

[3] J.A.C. Weideman and L.N. Trefethen, 2007,
Parabolic and hyperbolic contours

for computing the Bromwich integral, Math. Comp.
76, 1341—13

**25 Giugno 2019 ore 14.30**(aula Tricerri al Dini)

**Prof. Marjeta Knez,
IMFM and Faculty of Mathematics
and Physics, University of Ljubljana, Slovenia, visiting at University of Siena**

**Titolo: **
Rigid body motion interpolation using Pythagorean-hodograph curves

Polynomial Pythagorean-hodograph (PH) curves in space,
which are characterized by the property that the unit tangent is rational, have
many important features for practical applications. One of them is that these
curves can be equipped with rational orthonormal frames called Euler{Rodriques (ER) frames, where
the

rst frame vector coincides with the unit tangent. The
second important property is that the arc{length
function is a polynomial. Joining these two properties we can construct motions
of a rigid{body that interpolate some given positions
and have a prescribed length of the center trajectory.

In the talk, the interpolation of motion data, i.e., interpolation of data
points and rotations at the points, with G1 continuous PH curves is presented,
where the rotational part of the motion is determined by the ER frame. In
addition, the length of the center trajectory is prescribed. It is shown how to
construct the interpolants using cubic PH biarc curves
and quintic PH curves. In both cases the solutions exist for any data and any
length greater than the di

erence between the interpolation points. Moreover,
the interpolants depend on some free parameters, that can be chosen so that the
center trajectory is of a nice shape and the total rotation of the ER frame
vectors, that lie in a normal plane, about the tangent is minimized. The
derived theoretical results are illustrated with numerical examples. Possible
extensions to PH curves of a higher degree and higher order of smoothness are
suggested.

**11 Giugno 2019 ore 11.30**(aula 108 al plesso didattico Morgagni):

**
****Prof.** **Luca
Dieci**,** Georgia Institute of Technology, Atlanta,
USA **

**Titolo**: Piecewise Smooth
Differential Systems and Their Discretization

We consider solution of piecewise smooth (PWS) systems of differential
equations of Filippov type. We review some of
the theoretical and numerical concerns and then look more specifically at
planar periodic orbits under discretization.

[Based on joint works
with: Timo Eirola, Cinzia Elia, Luciano Lopez].

**27 Marzo 2019 ore 15.30**(aula Tricerri al Dini)

**Prof. Oleg Davydov, University of Giessen, Germany**

**Titolo****: ** Meshless Finite Difference Methods

Meshless finite difference methods for partial
differential equations apply the methodology of the Finite Difference Method in
the grid-free setting by using numerical differentiation formulas on scattered
nodes. These formulas can be obtained by requiring polynomial consistency or
via optimal recovery of differential operators with the help of kernel (radial
basis) interpolation. Since no mesh has to be imposed
on the nodes, they can be freely distributed following the exact geometry of
the model and/or the features of the solution. After introducing the method I will introduce recent research results obtained
jointly with Dang Thi Oanh,
Hoang Xuan Phu, Robert Schaback,
Andriy Sokolov, Ngo Manh Tuong,
and Stefan Turek.

**25 Marzo 2019 ore 11.30**(aula 204 Plesso Didattico Morgagni)

**Prof. Jacek Gondzio, University of Edinburgh**

**Titolo****: ** An efficient primal-dual interior point method
for large-scale truss layout optimization problems.

Truss layout optimization problems are often
formulated by using a ground structure approach where a set of nodes is
distributed in the design domain and all the possible interconnecting bars are
generated. The main goal is then to determine the optimal cross-sectional areas
of these bars and obtain the minimum weight structure that is
able to sustain a given set of applied loads. However, such
consideration of the full connectivity of the nodes results in a large number of bars making the optimization problems
computationally challenging for solution techniques. We solve the problems
using a primal-dual interior point method where

we employ several novel techniques to deal with the large size of the problems.
The first step is to use a column generation procedure where the optimal
solution of the large original problem is obtained by solving a sequence of
smaller restricted master problems. However, after performing a few column
generation iterations, the size of the restricted master problems grows and the problems still challenge standard interior
point solvers. Therefore, we additionally exploit the algebraic structure of
the problems and reduce the normal equations originating from the interior
point algorithm to much smaller linear equation systems. We apply the
preconditioned conjugate gradient method to solve these reduced linear systems.
A special purpose preconditioner based on the mathematical properties of the
problem is designed. The efficiency and robustness of the method is supported
with several numerical experiments. This
is a joint work with Alemseged
Weldeyesus.

**24 Gennaio 2019 ore 11.00**(aula 120 Plesso Didattico Morgagni)

**Prof. Alberto Bemporad,
IMT Lucca**

**Titolo: ** Embedded
Quadratic Programming: Algorithms
and Industrial Applications

A large variety of industrial systems rely on the solution of a convex quadratic programming (QP) problems. Applications include machine learning,
automated trading, signal processing, and control systems just to mention a
few. The QP solver is often required to run at high rates in an embedded
computing platform with limited CPU and memory resources. As
a consequence, the code solving the QP problem must require a low memory
footprint and be fast, robust in executing arithmetic operations under limited
machine precision, certifiable for worst-case execution time, and simple enough
for software certification. In my talk I will propose several approaches to
solve convex QPs that address such requirements. I will also show examples of
real application of QP in the automotive industry related to model predictive
control (MPC), one of the most successful techniques to control multivariable
systems in which the various actuators are decided automatically by a QP solver
in real-time.

**21 Novembre 2018 ore 11.00**(aula Tricerri al Dini)

**Dr. Philipp Morgenstern, Leibniz Universität of Hannover, Germany**

**Titolo: ** Mesh refinement for T-splines in 2D and nD

We
introduce mesh refinement algorithms for the Adaptive Isogeometric Method using bivariate and multivariate T-splines.
We address the boundedness of mesh
overlays, linear independence of the T-splines, nestedness
of the T-spline spaces, and linear complexity of the refinement procedure.

In order to justify the proposed methods and
theoretical results in this thesis, numerical experiments underline their
practical relevance, showing that they are not outperformed by currently
prevalent refinement strategies. As an outlook to future work, we outline an
approach for the handling of zero knot intervals and multiple lines in the interior
of the domain, which are used in CAD applications for controlling the
continuity of the spline functions, and we finally sketch basic ideas for the
local refinement of two-dimensional meshes that do not have tensor-product
structure.

**7 Novembre 2018 ore 11.00**(aula Tricerri al Dini)

**Dr. Michael Barton, Basque Center for Applied Mathematics (BCAM), Bilbao, Spain**

**Titolo: ** Gaussian quadrature rules for univariate splines and their applications to tensor-product isogeometric analysis

Univariate Gaussian quadrature rules for spline spaces
that are frequently used in Galerkin discretizations to build mass and stiffness matrices will
be discussed [2]. Their computation is based on the homotopy
continuation concept [1] that transforms Gaussian
quadrature rules from the so called source space to the target space. Starting
with the classical Gaussian quadrature for polynomials, which is an optimal
rule for a discontinuous odd-degree space, and building the source space as a
union of such discontinuous elements, we derive rules for
target spline spaces with higher continuity across the elements. We
demonstrate the concept by computing numerically Gaussian rules for spline
spaces of various degrees, particularly those with non-uniform knot vectors and
non-uniform knot multiplicities. We also discuss
convergence of the spline rules over finite domains to their asymptotic counterparts, that is, the analogues of the half-point
rule of Hughes et al. [4], that are exact and Gaussian over the infinite domain.
Finally, the application of spline Gaussian rules in the context of isogeometric analysis on
subdivision surfaces will be discussed [3], showing the advantages and
limitations of the tensor product Gaussian rules.

[1] M. Barton and V. M. Calo. Gaussian
quadrature for splines via homotopy continuation:
rules for C^2 cubic splines. Journal of Computational and Applied Mathematics,
296:709--723, 2016.

[2] M. Barton and V. M. Calo. Optimal quadrature rules for odd-degree spline
spaces and their application to
tensor-product-based isogeometric analysis. Computer
Methods in Applied Mechanics and Engineering, 305:217--240, 2016.

[3] P. Barendrecht and M. Barton and J. Kosinka. Efficient quadrature rules for subdivision
surfaces in isogeometric analysis. Computer
Methods in Applied Mechanics and Engineering}, 340:1--23,
2018.

[4] T.J.R. Hughes and A. Reali and G. Sangalli. Efficient quadrature for NURBS-based isogeometric analysis. Computer Methods in Applied
Mechanics and Engineering, 199:301--313, 2010.

**2 Ottobre 2018 ore 14.30**(aula Tricerri al Dini)

**Dr. Mario Kapl, Johann
Radon Institute for Applied
and Computational Mathematics,
Austrian Academy of Sciences,
Linz, Austria**

**Titolo: ** C^1
isogeometric spaces over analysis-suitable G^1 multi-patch geometries

In this talk, we deal with a particular class of C^0
planar multi-patch spline parametrizations called analysis-suitable G^1 (AS-G^1)
multi-patch parametrizations (cf. [1]). This class of parametrizations has to satisfy specific geometric continuity constraints,
and is of importance since it allows to construct, on the multi-patch domain,
C^1 isogeometric spaces with optimal approximation
properties.

We first present a method which
approximates a given planar multi-patch parameterization by an AS-G^1
multi-patch geometry. The potential of this algorithm for modeling complex
planar multi-patch domains by AS-G^1 multi-patch parameterizations is
demonstrated on the basis of several examples. Then,
we describe a basis construction for a specific C^1 isogeometric
spline space W over a given AS-G^1 multi-patch parametrization. The considered
space W is a simpler subspace of the entire C^1 isogeometric
space maintaining the optimal approximation properties. The construction of the
basis functions is easy and works uniformly for all
multi-patch configurations. In addition, the resulting functions possess a

simple explicit representation and a local
support. Finally, some numerical experiments are performed, which exhibit the
optimal approximation order of the space W and demonstrate the applicability of
our approach for isogeometric analysis.

Joint work with: Giancarlo Sangalli (Dipartimento di Matematica "F. Casorati",
Universita' degli Studi di Pavia, Italy)

and Thomas Takacs (Institute of Applied Geometry, Johannes Kepler University
Linz, Austria).

[1] A. Collin, G. Sangalli, and T. Takacs.
Analysis-suitable G1 multi-patch parametrizations for C1 isogeometric
spaces. Computer Aided Geometric
Design, 47:93-113, 2016.

**5 Settembre 2018 ore 11.00**(aula Tricerri al Dini)

**Dr. Gregor Gantner, TU
Wien, Austria**

**Titolo: ** Adaptive isogeometric
methods with optimal convergence rates

The CAD standard for spline representation in 2D or 3D
relies on tensor-product splines. To allow for adaptive refinement, several
extensions have emerged, e.g., analysis-suitable T-splines, hierarchical
splines, or LR-splines. All these concepts have been studied via numerical
experiments, but there exists only little literature concerning the thorough
analysis of adaptive isogeometric methods.

The work [1] investigates linear
convergence of the weighted-residual error estimator (or equivalently: energy
error plus data oscillations) of an isogeometric
finite element method (IGAFEM) with truncated hierarchical B-splines. Optimal
convergence was independently proved in [2, 3]. In [3], we employ hierarchical
B-splines and propose a refinement strategy to generate a sequence of refined
meshes and corresponding discrete solutions.

Usually, CAD provides only a
parametrization of the boundary $\partial\Omega$ instead of the domain $\Omega$
itself. The boundary element method, which we consider in the second part of
the talk, circumvents this difficulty by working only on the CAD provided
boundary mesh. In 2D, our adaptive algorithm steers the mesh-refinement and the
local smoothness of the ansatz functions. The corresponding a posteriori error
analysis has been investigated in [4, 5, 6]. Recently, we proved linear
convergence of the employed weighted-residual estimator at optimal algebraic
rate in [7, 8]. In 3D, we consider an adaptive IGABEM with hierarchical splines
and prove linear convergence of the estimator at optimal rate; see [8].

REFERENCES

[1] A. Buffa and C. Giannelli,
Adaptive isogeometric methods with hierarchical splines:
error estimator and convergence. Math. Mod. Meth. Appl. S., Vol. 26, 2016.

[2] A. Buffa and C. Giannelli,
Adaptive isogeometric methods with hierarchical
splines: Optimality and convergence rates. Math. Mod. Meth. in Appl. S., Vol.
27, 2017.

[3] G. Gantner, D. Haberlik, and Dirk Praetorius,
Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines. Math.
Mod. Meth. in Appl. S., Vol. 27, 2017.

[4] G. Gantner,
Adaptive isogeometric BEM, Master's thesis, TU Wien,
2014.

[5] Michael Feischl, Gregor Gantner,
and Dirk Praetorius. Reliable and efficient a
posteriori error estimation for adaptive IGA boundary element methods for
weakly-singular integral equations. Comput. Methods
Appl. Mech. Engrg., Vol. 290, 2015.

[6] Michael Feischl,
Gregor Gantner, Alexander Haberl, and Dirk Praetorius. Adaptive 2D IGA boundary element methods. Eng.
Anal. Bound. Elem., Vol. 62, 2016.

[7] M. Feischl, G. Gantner, A. Haberl, and D. Praetorius,
Optimal convergence for adaptive IGA boundary element methods for
weakly-singular integral equations. Numer. Math.,
Vol. 136, 2017

[8] G. Gantner,
Optimal adaptivity for splines in finite and boundary element methods, PhD
thesis, TU Wien, 2017.

**8 Giugno 2018 ore 10.30**(aula 204 al Plesso Didattico Morgagni)

**Prof. Haomin Zhou, Georgia Institute of Technology of Atlanta,
USA**

**Titolo****: ** Method of Evolving Junctions (MEJ) and Its
Application in Robotic Path Planning

We design a new stochastic differential
equation (SDE) based algorithm that can efficiently compute the solutions of a
class of infinite dimensional optimal control problems with constraints on both
state and control variables. The main ideas include two parts. 1) Use junctions
to separate paths into segments on which no constraint changes from active to
in-active, or vice versa. In this way, we transfer the original infinite
dimensional optimal control problems into finite dimensional optimizations. 2)
Employ the intermittent diffusion (ID), a SDE based global optimization
strategy, to compute the solutions efficiently. It can find the global optimal
solution in our numerical experiments. We illustrate the performance of this
algorithm by several shortest path problems, the frogger problem and
generalized Nash equilibrium examples.This
is joint work with Shui-Nee Chow (Math, Georgia Tech), Magnus Egerstedt (ECE, Georgia Tech). Wuchen
Li (Math, UCLA), Jun Lu (Shunfeng) and Haoyan Zhai (Math, Georgia Tech).

**31 Maggio 2018 ore 11.30**(aula Tricerri al Dini)

**Prof. Xinyuan Wu, Normal University,
Nanjing, China**

**Titolo: ** Structure-preserving
algorithms for highly oscillatory differential equations

In the last few decades, the structure-preserving
numerical simulation for nonlinear oscillators has received a great deal of
attention. This talk begins with ERKN integrators for a system of
multi-frequency highly oscillatory second-order differential equations and ends
with the applica- tions to
KG equations based on the operator-variation-of-constants formula for nonlinear
wave equations.

**24 Maggio 2018 ore 10.00**(aula 202 al Plesso Didattico Morgagni)

**Prof. Demetrio Labate**, Univ. of Houston, USA

**Titolo****: ** Sparsity-based computed tomography and
region-of-interest tomographic reconstruction

Computed tomography is a non-invasive scanning method that is widely employed in medical and industrial
imaging to reconstruct the unknown interior structure of an object from a
collection of projection images. The mathematical problem of recovering
an unknown density function from its linear
projections is a classical ill-posed problem, and many methods have been
proposed and applied in the literature. This talk will be divided into
two parts.

The first part will discuss classical and
more advanced methods of regularized tomographic
reconstruction. In particular, we show how a wavelet-vaguelette decomposition of the Radon operator can
take advantage of sparse multiscale representations to obtain
regularized reconstruction outperforming more conventional regularization
methods. In the second part of the talk, we consider region-of-interest (ROI) tomographic reconstruction - a particularly

challenging mathematical and computational
problem. Using an appropriate sparsity prior based on the theory of compressed sensing, we derive performance guarantees for ROI
tomographic reconstruction by establishing error bounds for stable
recovery. We show numerical tests from
experimental data to compare sparsity-based and state-of-the-art
reconstruction methods.

**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

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

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.

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

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):

Prof. **Philippe Toint,** University of Namur, Belgio

**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.

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

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.

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

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.

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

Dott. **Cesare
Bracco**, Università 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. 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):

Dott. **Gianmarco
Gurioli**, Università 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.

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

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.

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

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.