- 23 Ottobre 2020 ore 11.30 (aula Tricerri al Dini)

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.

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):

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"

"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):

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.

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

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.

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

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

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

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.

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.

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

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

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.

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.

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

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

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.

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

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

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.

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

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].

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

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

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

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.

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

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.