SEMINARI DI ANALISI NUMERICA

         Prossimo seminario riportato in rosso



        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.







          Prof. Oleg Davydov, University of Giessen, Germany
        PhD course, Local Approximation and Numerical Differentiation by Polynomial and Kernel Methods

  



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





       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.





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.



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


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 dierence 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.






        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].




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.



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.







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.






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.



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.


 




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.


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.



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).






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.



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.





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.


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.
 





 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.



        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.





        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, 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.




        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.





        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.