Adversarial AI and the concentration of measure
Mauro Barni, University of Siena
AI and machine learning have revolutionized the way we process information with a dramatic impact on many aspects of our society and lives. Yet many aspects of AI are still poorly understood casting some shadows on the use of AI in security-critical applications. One of these aspects is the surprising vulnerability of AI models to adversarial examples, that is slightly perturbed data leading to completely wrong predictions. For instance, it turns out be pretty easy to slightly modify an image in such a way that the image perturbation is invisible, yet enough to induce unexplainable and catastrophic interpretation errors by AI networks. Several attempts to explain the apparently ubiquitous existence of adversarial examples have been made, some of which trace the existence of adversarial examples to the concentration of measure phenomenon. In this talk I will first provide some background on adversarial examples, then I will review some recent works linking adversarial examples to the concentration of measure and present some new results of my group in this field. I will conclude the talk with some perspectives on future research directions.
Models and methods for multifluid flow
in Lagrangian coordinates
Giovanni Russo, University of Catania
Metamaterials are composed by assembling a large number of small unit cells. At scales much larger than. the size of the cells, they appear as a homogeneous material, which can have electrical or mechanical properties quite different from ano standaard material. Multifluids, which can be considered a sort of fluid metamaterial , have attracted a lot of attention in recent years. In particular, stratified fluids composed by a sequence of alternate layers show interesting macroscopic properties, which may be quite different from those of the individual fluids which constitute the stratified system. Several models have been proposed to describe the stratified system as a single effective fluid. In the first part of the talk, we present two homogenized models of the stratified fluid system, which are based on isentropic approximation [1]. The behavior of the models is compared with a detailed solution of the multifluid system, computed by an explicit scheme for the Euler equations in Lagrangian coordinates. Lagrangian description seems ideal, since it automatically maintains sharp interfaces between the different fluids. However, the formulation poses a real challenge for explicit schemes if the two fluids have a large mass density ratio, as is the case, for example, of air and water, because of the corresponding large ratio in the Lagrangian speed of sound, which would require a very small time step. With this motivation, the second part of the talk is devoted to the derivation and testing of a fully implicit scheme for multifluid in Lagrangian coordinates [2]. The method exploits the special structure of the Lagrangian formulation of the Euler equations and is based on a simple and effective non conservative implicit predictor and a conservative corrector, which allows a very robust and efficient solution of the Euler equations, with no CFL stability restrictions and no need of complicated Riemann solvers. The method is second order accurate in space. High order accuracy in time is achieved by singly diagonally implicit Runge-Kutta schemes in time. Several numerical tests illustrate the efficiency and robustness of the method.
References
[1] Duyen T.M. Phan, Sergey L. Gavrilyuk, Giovanni Russo, Numerical validation of homogeneous multi-fluid models, Applied Mathematics and Computation, Volume 441, March 15, 2023.
[2] Simone Chiocchetti, Giovanni Russo, An efficient implicit scheme for the multimaterial Euler equations in Lagrangian coordinates, in preparation.
Isogeometric Boundary Element Methods
Maria Lucia Sampoli, University of Siena
In the numerical solution of many differential problems arising in the applications, a possible approach is to reformulate the problem by integral equations defined on the boundary of the given domain, giving rise to the so called Boundary Element Methods (BEMs). These methods have two main advantages, the dimension reduction of the computational domain and the simplicity for treating external problems. As a major drawback, the resulting integrals can be singular and therefore robust and accurate quadrature formulas are necessary for their numerical computation.
On the other side, the new Isogeometric analysis approach (IgA) establishes a strict relation between the geometry of the problem domain and the approximate solution representation, giving surprising computational advantages. In the IgA setting a new formulation of BEMs has been studied, where the discretization spaces are splines spaces represented in B-spline form. In order to take all the possible benefits from using B-splines instead of Lagrangian basis, an important issue is the development of specific new quadrature formulas for efficiently implementing the assembly phase of the method. In addition, as the resulting linear systems are large and dense, requiring huge memory and high computational cost, another important issue is an efficient approximation of such matrices.
In this talk the problem of constructing appropriate and accurate quadrature rules, tailored on B-splines, for Boundary Integral Equations is addressed. The first results on the application of a new fast IGA-BEM technique will be also presented. Such technique, based on the hierarchical matrices (H-matrices), allows to approximate the fully-populated IgA-BEM matrices by data-sparse ones.
The research is part of a collaboration with A. Aimi, L. Desiderio, G. A. D’Inverno, T. Kanduč, A. Falini, and A. Sestini.
Explicit error estimates for spline approximation in isogeometric analysis
Hendrik Speleers, University of Roma "Tor Vergata"
Isogeometric analysis is a well established paradigm to improve interoperability between geometric modeling and numerical simulation. It is commonly based on B-splines and shows important advantages over classical finite element analysis. In particular, the inherent high smoothness of B-splines leads to a higher accuracy per degree of freedom. This has been numerically observed in a wide range of applications, and recently a mathematical explanation has been given thanks to error estimates in spline spaces that are explicit, not only in the mesh size, but also in the polynomial degree and the smoothness.
In this talk we review some recent results on explicit error estimates for approximation by splines. These estimates are sharp or very close to sharp in several interesting cases and allow us to explain that certain spline subspaces are able to approximate eigenvalue problems without spurious outliers.
Device accelerated solvers with PETSc
Stefano Zampini, KAUST
In his talk, Dr. Stefano Zampini will review recent developments towards exploiting Graphical Processing Units (GPUs) from the Portable and Extensible Toolkit for Scientific Computing (PETSc). Current status and future perspective will be discussed. Examples include multi GPU distributed sparse matrix-matrix multiplications, device accelerated linear solvers, and applications related with the CFD open-source OpenFOAM and machine-learning training.