# Talks

## Forthcoming

• NUMERICAL METHODS FOR THE ROBUST DESIGN AND OPTIMIZATION OF A JET ENGINE LOW PRESSURE TURBINE ROTOR
Giulia Antinori (TU München) (TBA)

The design of a low pressure turbine rotor is usually deterministic. The variability of the input parameters due to ambient condition or to the presence of engine to engine variations and manufacturing tolerances, are typically not considered in the design. This leads to the fact that the generated design can be unreliable, not robust and cause in-service problems.
To avoid this behavior a probabilistic analysis, which includes a sensitivity analysis followed by an uncertainty analysis, is performed. This seminar illustrates some of the numerical methods, like HDMR (High Dimensional Model Representation) decomposition and Monte Carlo simulations, that can be used to run such analysis.
The aim of the sensitivity analysis is to gain a better understanding of the coupled flow-thermomechanical system robustness, to identify the important variables and to reduce the number of design parameters which will be used in the optimization. The uncertainty analysis using probability distributions derived from the manufacturing process, allows to predict the influence of the input uncertainties on the life duration of the rotor. Finally, the ultimate goal is to achieve a robust optimization, so that the optimal design is insensitive to statistical variation of the initial conditions. To do that classical optimization methods, like Newton methods or genetic algorithms, are coupled with orthogonal polynomial decomposition through which robustness measures are efficiently obtained.

## Past

• SOLVING PDEs ON SURFACES WITH RADIAL BASIS FUNCTIONS: FROM GLOBAL TO LOCAL METHODS
Prof. Grady B. Wright, Department of Mathematics, Boise State University - USA (Apr 10, 2014)

Radial basis function (RBF) methods are becoming increasingly popular for numerically solving partial differential equations (PDEs) because they are geometrically flexible, algorithmically accessible, and can be highly accurate. There have been many successful applications of these techniques to various types of PDEs defined on planar regions in two and higher dimensions, and to PDEs defined on the surface of a sphere. Originally, these methods were based on global approximations and their computational cost was quite high. Recent efforts have focused on reducing the computational cost by using "local" techniques, such as RBF generated finite differences (RBF-FD).
In this talk, we first describe our recent work on developing a new, high-order, global RBF method for numerically solving PDEs on relatively general surfaces, with a specific focus on reaction-diffusion equations.
The method is quite flexible, only requiring a set of scattered'' nodes on the surface and the corresponding normal vectors to the surface at these nodes. We next present a new scalable local method based on the RBF-FD approach with this same flexibility. This is the first application of the RBF-FD method to general surfaces. We conclude with applications of these methods to some biologically relevant problems.
This talk represents joint work with Ed Fuselier (High Point University), Aaron Fogelson, Mike Kirby, and Varun Shankar (all at the University of Utah).
• ON THE CONSTRAINED MOCK-CHEBYSHEV LEAST-SQUARES
Mariarosa Mazza, Universita' dell'Insubria (Nov 28, 2013)

The algebraic polynomial interpolation on uniformly distributed nodes is affected by the Runge phenomenon, also when the function to be interpolated is analytic. Among all techniques that have been proposed to defeat this phenomenon, there is the mock-Chebyshev interpolation which is an interpolation made on a subset of the given nodes which elements mimic as well as possible the Chebyshev-Lobatto points. In this work we use the simultaneous approximation theory to combine the previous technique with a polynomial regression in order to increase the accuracy of the approximation of an analytic function. We give indications on how to select the degree of the simultaneous regression so as to obtain polynomial approximant good in the uniform norm and provide a sufficient condition to improve, in that norm, the accuracy of the mock-Chebyshev interpolation with a simultaneous regression. Numerical results are provided.
• DUAL-MODALITY IMAGING
Davide Poggiali (Nov 21, 2013)
• NEWTON BASIS: ADAPTIVE CALCULATION ALGORITHM AND APPLICATIONS
Silvia Guglielmo (Nov 21, 2013)
• KINETICS OF THE TRACER IN PET
Davide Poggiali (Nov 8, 2013)
• CIRCULANT MATRICES AND APPLICATIONS
Maria Angela Narduzzo (Oct 29, 2013)

Circulant matrices have a strong presence in various parts of modern (and classical) mathematics. We present the most important properties of Toeplitz and circulant matrices. We have tried to make the topic accessible to a wide audience by supplying rather full details in most of the arguments. We focus on the application of this kind of matrices to obtain a faster implementation of a matrix-vector product. Once we have reached this goal, we present a concrete use of this fast algorithm in the resolution of a linear system through the Conjugate Gradient Method. The seminar ends with an application of our result to the Image Reconstruction from Radon Data in the setting of CT.
• POSITRON EMISSION TOMOGRAPHY: AN INTRODUCTION
Davide Poggiali (Oct 17, 2013)
• SEQUENCES AND RECONTRUCTION IN MRI
Davide Poggiali (Sep 27, 2013)
• PHYSICAL BASIS OF MAGNETIC RESONANCE IMAGING
Davide Poggiali (Sep 19, 2013)
• AN ALTERNATIVE RADON TRANSFORM FOR THE CORRECTION OF PARTIAL VOLUME EFFECT
Davide Poggiali (Jun 28, 2013)

• A SOUND MODEL FOR MUSICAL SIGNALS
Matteo Briani (Jun 21, 2013)

Sinusoidal modeling is one of the most popular techniques in digital audio signal processing. This model is useful in a variety of applications like morphing, time and pitch scaling, compression, source separation or music analysis.
In order to fully understand the model we first review the main tools in the Fourier-related analysis, both with linear and non-linear frequency resolution. Then we introduce the Prony-related spectrum analysis techniques treating both noiseless and noisy audio signal. Before discussing the sinusoidal model few psychoacoustic notions are given to the partecipants. The seminar ends with a comparison between the sinusoidal model implemented both through Fourier and Prony-related techniques.
• RESOLUTION AND RECONSTRUCTION ISSUES IN CT AND SPECT
Davide Poggiali (Mar 6, 2013)

CT and SPECT reconstruction and resolution issues from a mathematician's point of view. A description of the most common algorithms to reconstruct the patient's slice from projection data and of some methods to find and check the resolution of the machine in use.
(slides of the seminar)
• SPHERICAL POLYNOMIALS AND SPHERICAL HARMONICS
Mariano Gentile (Jan 16, 2013)

In this talk we will introduce the space of spherical polynomials $\mathbb{P}_n(\mathbb{S}^d)$, determining an orthogonal basis for this space over the $d$-dimensional sphere $\mathbb{S}^d$. In particular, we will consider the basic properties of spherical harmonics, focusing the discussion on their construction and that they provide an orthogonal basis for $\mathbb{P}_n(\mathbb{S}^d)$.
• SMOOTH TRANSFORMATIONS AND SMALL PERTURBATIONS OF WEAKLY ADMISSIBLE POLYNOMIAL MESHES
Federico Piazzon (Mar, 2012)

As an introduction we give a quick survey on (Weakly) Admissible Meshes, say (W)AM, for polynomial approximation as they were defined by J.P. Calvi and N. Levenberg in 2008. Then we present two new result from the joint work with M. Vianello. First we show that any given compact $K\subset \mathbb R^d\text{ or }\mathbb C^d$ preserving a Markov Inequality and being the smooth image of a compact set $Q$ where we are able to build a WAM admits a WAM. Second we show that the property of being a WAM is stable under small perturbations of the domain or the WAM itself, actually this last statement implies the first one. (extended abstract)
• A NEW STABLE BASIS FOR RBF APPROXIMATION
Gabriele Santin (Feb, 2012)

It's well know that Radial Basis Function interpolants suffers of bad conditioning if the simple basis of translates is used. A recent work of M.Pazouki and R.Schaback gives a quite general way to build stable, orthonormal bases for the native space based on a factorization of the kernel matrix $A_{\Phi}$. Starting from that setting we describe a particular $N_{\Phi}(\Omega)$-orthonormal, $l_{2,w}$-orthogonal basis that arises from a weighted singular value decomposition of $A_{\Phi}$. This basis is related to a basis of the space which arises from an eigen-decomposition of a compact and self adjoint operator. We give convergence estimates and stability bound for the interpolation and the discrete least-squares approximation based on this basis, which involves the eigenvalues of such an operator.
• A RADIAL BASIS FUNCTION NETWORK
Giulia Antinori (Feb, 2012)

A method based on a Radial Basis Function Network (RBFN) to solve a system of ODEs, which models diabetes and insulin therapy, is studied. The network is firstly tested on the approximation of a function and its derivatives. Then, the solution of a differential equation is approximated. Finally, by using the theory of radial basis functions in the vectorial case, the solution of a system of differential equations is approximated. A deep analysis on the choice of the kernels is also performed by studying kernels localization properties and the relation between a good approximation and the properties of the native space of the kernel.
• KERNEL-BASED ALGEBRAIC RECONSTRUCTION OF MEDICAL IMAGES
Amos Sironi (Jan, 2012)

We present a novel kernel-based algebraic reconstruction method for medical image reconstruction from scattered Radon data. Our reconstruction relies on generalized Hermite-Birkhoff interpolation by positive definite kernel functions in combination with a suitable regularization of the Radon transform. This leads to a very flexible reconstruction method for medical images. Good performance is supported by numerical examples and comparisons with classical Fourier-based methods relying on the filtered back projection formula.