DEEP LEARNING AND MODEL REDUCTION - IS ACCURACY CONTROL FEASIBLE?
speaker: Wolfgang Dahmen
According to a general perception, modern machine learning methodologies, in particular, deep learning (DL) are exerting a transformative impact on societal interactions, especially in \Big Data" scenarios.
REDUCED ORDER MODELLING IN COMPUTATIONAL FLUID DYNAMICS: STATE OF THE ART, CHALLENGES AND PERSPECTIVES
speaker: Gialuigi Rozza
We do provide the state of the art of Reduced Order Methods (ROM) for parametric Partial Differential Equations (PDEs), and we focus on some perspectives in their current trends and developments, with a special interest in parametric problems arising in offline-online Computational Fluid Dynamics (CFD).
SIMULATION OF PDES ON GEOMETRIES DEFINED VIA BOOLEAN OPERATIONS
speaker: Annalisa Buffa
We consider the scenario of a computational domain described as a combination of primitives and spline/NURBS boundary representations which are combined via boolean operations, such as intersections and unions. We aim to develop numerical methods that are robustly able to tackle the simulation of PDEs over such \unstructured" geometric representation.
The Virtual Element Method (VEM) is an extension of the finite element method to polytopic meshes. Test and trial spaces are made up of functions that are solutions to local problems related to the PDE to be approximated.
In numerical linear algebra it has long been customary to distin guish between dense and sparse matrices, the latter class being characterized by the fact that, for an n x n matrix, only O(n) of its entries are nonzero. As is well known, significant differences exist, in terms of solution algorithms, their complexity and implementation aspects, between dense and sparse matrix computations.
THE ICDD METHOD TO MODEL THE FILTRATION OF FLUID IN POROUS MEDIA
speaker: Paola Gervasio
The talk focuses on the validation of the Interface Control Domain Decomposition (ICDD) method in the context of the Stokes-Darcy problem to model the filtration of fluids in porous media.
ON THE NUMERICS OF NON-NEWTONIAN FLUIDS: VIRTUAL ELEMENTS AND NEURAL NETWORKS
speaker: Marco Verani
In the first part of the talk, we will present recent results on the numerical approximation of non-newtonian fluids on polygonal meshes obtained in collaboration with Paola F. Antonietti, Michele Botti (Politecnico di Milano), Lourenco Beirão da Veiga (Università di Milano Bicocca), Giuseppe Vacca (Università di Bari).
The office was miserable, hot in summer and cold in winter, with a variable sealing ceiling. The company, however, was excellent, and the small window overlooking the three towers of Pavia allowed us to dream without distracting us from what we were most passionate about at that time: spectral methods. Then, our professional lives diverged. Friendship, on the other hand, has stood the test of time, unchangeable.
QUANTITATIVE STUDY OF THE STABILIZATION PARAMETER IN THE VIRTUAL ELEMENT METHOD
speaker: Alessandro Russo
The choice of stabilization term is a critical component of the Virtual Element Method. However, the theory of VEM provides only asymptotic guidance for selecting the stabilization term, which ensures convergence as the mesh size approaches zero, but does not provide a unique prescription for its exact form.
Matrices are an important building block in the numerical treatment of partial differential equations. Depending on the adopted discretization methodology, matrices with special structural or spectral properties arise.
After a quick reminder of classical Virtual Element approximations on polygonal and polyhedral decompositions we present a couple of new ideas for the construction of VEM-approximations on domains with curved boundary, both in two and three dimensions.
WAN DISCRETIZATION OF PDES: BEST APPROXIMATION, STABILIZATION AND BOUNDARY CONDITIONS
speaker: Silvia Bertoluzza
We present a theoretical analysis of the Weak Adversarial Networks (WAN) method, recently introduced in [1, 2], where its was proposed as a method for approximating the solution of partial differential equations in high dimensions and tested in the framework of inverse problems.
APPROXIMATION THEORY AND CLAUDIO HELP: A CERTIFIED WAVELET-BASED PHYSICS-INFORMED NEURAL NETWORK
speaker: Karsten Urban
Physics Informed Neural Networks (PINNs) have frequently been used for the numerical approximation of Partial Differential Equations (PDEs). In this talk, we present recent work on loss functions for PINNs along with a computable upper bound of the error, which is particularly relevant for model reduction of Parameterized PDEs (PPDEs).
THE WEAK BOUNDARY BETWEEN THEORETICAL AND APPLIED MATHEMATICS IN SPACE SCIENCE
speaker: Alessandra Celletti
Very often, mathematical theories have been deeply motivated by questions arising in Celestial Mechanics and Astrodynamics. Among the others, perturbative methods, KAM theory, Nekhoroshev theorem.
A LAGRANGE MULTIPLIER METHOD FOR FLUID-STRUCTURE INTERACTIONS
speaker: Lucia Gastaldi
In this talk I present some recent advances on the discretization of fluid-structure interaction problems based on the Lagrange multiplier formulation presented in [1].
VIRTUAL ELEMENT METHODS FOR ELLIPTIC PDES IN NON-DIVERGENCE FORM
speaker: Ricardo H. Nochetto
Elliptic PDEs in non-divergence form with (discontinuous) Cordes coefficients admit a Miranda-Talenti estimate on convex domains that yields well posedness in H2.