Events
How bad can singularities of a curve of degree d in projective n-space be? The study of this question is very classical. Even for plane curves, all possible configurations of singularities are only known in low degrees. In higher dimensions, much less is known. In the eighties, Piene and Eisenbud-Harris studied flags of osculating spaces attached to linear series of curves. In this talk, we introduce a gadget (called multifiltration) obtained by combining those flags. We use it to give upper bounds on the arithmetic genus of projective curves in some ranges (reproving a result due to Castelnuovo). We classify all configurations of singularities that can arise when any smooth curve is projected from a linear space of dimension at most two. With these techniques, one can describe the Schubert cycles giving rise to those projections. This is joint work with J. Buczynski and N. Ilten.
In this talk we present a series of new results in octonionic monogenic function theory. We introduce generalizations of the Weierstrass p and zeta function associated with eight-dimensional lattices that have an octonionic multiplication and explain some connections to some possible relations and applications to Class Field Theory.
Furthermore, we also give some explicit applications of these kind of functions to the study of Bergman and Hardy spaces in the octonionic cases. Octonionic monogenic generalizations of the cotangent and the cosecant can be obtained as subseries of the octonionic Weierstrass p-functions. These functions turn out to be the building blocks for the reproducing octonionic Bergman and Szegö kernel of strip domains in R^8.
Summarizing, these new functions seem to play a key role in octonionic function theories and their applications to number theory and function spaces.
In an inverse problem, one seeks to infer unknown parameters or parameter fields from measurements or observations of the state of a natural or engineered system. Such problems are fundamental to many fields of science and engineering: often available models possess unknown or uncertain input parameters that must be inferred from experimental or observational data. The Bayesian framework for inverse problems accounts for uncertainty in the inferred parameters stemming from uncertainties in the observational data, the model, and any prior knowledge. Bayesian inverse problems (BIPs) governed by large-scale complex models in high parameter dimensions (such as nonlinear PDEs with uncertain infinite dimensional parameter fields) quickly become prohibitive, since the forward model must be solved numerous times---as many as millions---to characterize the uncertainty in the parameters.
Efficient evaluation of the parameter-to-observable (p2o) map, definedby solution of the forward model, is the key to making BIPs tractable. Surrogate approximations of p2o maps have the potential to greatly accelerate BIP, provided that the p2o map can be accurately approximated using (far) fewer forward model solves than would be required for solving the BIP using the full p2o map. Unfortunately, constructing such surrogates presents significant challenges when the parameter dimension is high and the forward model is expensive. Deep neural networks (DNNs) have emerged as leading contenders for overcoming these challenges. We demonstrate that black box application of DNNs for problems with infinite dimensional parameter fields leads to poor results, particularly in the common situation when training data are limited due to the expense of the model. However, by constructing a network architecture that is adapted to the geometry
and intrinsic low-dimensionality of the p2o map as revealed through adjoint PDEs, one can construct a "parsimonious" DNN surrogate with superior approximation properties with only limited training data.
Application to an inverse problem in Antarctic ice sheet flow is discussed.
This work is joint with Tom O'Leary-Roseberry, Peng Chen, Umberto Villa, and Nick Alger.
Contact: luca.formaggia@polimi.it
Dr. Omar Ghattas is a Professor of Geological Sciences and Mechanical Engineering at the University of Texas at Austin. He is also the Director of the Center for Computational Geosciences and Optimization in the Oden Institute for Computational Engineering and Sciences and holds the John A. and Katherine G. Jackson Chair in Computational Geosciences. He is a member of the faculty in the Computational Science, Engineering, and Mathematics (CSEM) interdisciplinary PhD program in the Oden Institute, and holds courtesy appointments in Computer Science and Biomedical Engineering. He has general research interests in forward and inverse modeling, optimization, and uncertainty quantification of large-scale complex mechanical, geological, and biological systems. With collaborators, he received the ACM Gordon Bell Prize in 2003 (for Special Achievement) and again in 2015 (for Scalability), and was a finalist for the 2008, 2010, and 2012 Bell Prizes. He received the 2019 SIAM Computational Science & Engineering Best Paper Prize, and the 2019 SIAM Geosciences Career Prize. He is a Fellow of the Society for Industrial and Applied
Mathematics (SIAM).
In an inverse problem, one seeks to infer unknown parameters or parameter fields from measurements or observations of the state of a natural or engineered system. Such problems are fundamental to many fields of science and engineering: often available models possess unknown or uncertain input parameters that must be inferred from experimental or observational data. The Bayesian framework for inverse problems accounts for uncertainty in the inferred parameters stemming from uncertainties in the observational data, the model, and any prior knowledge. Bayesian inverse problems (BIPs) governed by large-scale complex models in high parameter dimensions (such as nonlinear PDEs with uncertain infinite dimensional parameter fields) quickly become prohibitive, since the forward model must be solved numerous times---as many as millions---to characterize the uncertainty in the parameters.
Efficient evaluation of the parameter-to-observable (p2o) map, definedby solution of the forward model, is the key to making BIPs tractable. Surrogate approximations of p2o maps have the potential to greatly accelerate BIP, provided that the p2o map can be accurately approximated using (far) fewer forward model solves than would be required for solving the BIP using the full p2o map. Unfortunately, constructing such surrogates presents significant challenges when the parameter dimension is high and the forward model is expensive. Deep neural networks (DNNs) have emerged as leading contenders for overcoming these challenges. We demonstrate that black box application of DNNs for problems with infinite dimensional parameter fields leads to poor results, particularly in the common situation when training data are limited due to the expense of the model. However, by constructing a network architecture that is adapted to the geometry
and intrinsic low-dimensionality of the p2o map as revealed through adjoint PDEs, one can construct a "parsimonious" DNN surrogate with superior approximation properties with only limited training data.
Application to an inverse problem in Antarctic ice sheet flow is discussed.
This work is joint with Tom O'Leary-Roseberry, Peng Chen, Umberto Villa, and Nick Alger.
Contact: luca.formaggia@polimi.it
Dr. Omar Ghattas is a Professor of Geological Sciences and Mechanical Engineering at the University of Texas at Austin. He is also the Director of the Center for Computational Geosciences and Optimization in the Oden Institute for Computational Engineering and Sciences and holds the John A. and Katherine G. Jackson Chair in Computational Geosciences. He is a member of the faculty in the Computational Science, Engineering, and Mathematics (CSEM) interdisciplinary PhD program in the Oden Institute, and holds courtesy appointments in Computer Science and Biomedical Engineering. He has general research interests in forward and inverse modeling, optimization, and uncertainty quantification of large-scale complex mechanical, geological, and biological systems. With collaborators, he received the ACM Gordon Bell Prize in 2003 (for Special Achievement) and again in 2015 (for Scalability), and was a finalist for the 2008, 2010, and 2012 Bell Prizes. He received the 2019 SIAM Computational Science & Engineering Best Paper Prize, and the 2019 SIAM Geosciences Career Prize. He is a Fellow of the Society for Industrial and Applied
Mathematics (SIAM).
How can we understand the closed subschemes in a projective space? Hilbert schemes provide the geometric answer to this question. After surveying some features of these natural parameter spaces, we will classify the smooth Hilbert schemes. Time permitting, we will also describe the geometry of the nonsingular Hilbert schemes by interpreting them as suitable generalizations of partial flag varieties. This talk is based on joint work with Roy Skjelnes (KTH).
Dr. Omar Ghattas is a Professor of Geological Sciences and Mechanical Engineering at the University of Texas at Austin. He is also the Director of the Center for Computational Geosciences and Optimization in the Oden Institute for Computational Engineering and Sciences and holds the John A. and Katherine G. Jackson Chair in Computational Geosciences. He is a member of the faculty in the Computational Science, Engineering, and Mathematics (CSEM) interdisciplinary PhD program in the Oden Institute, and holds courtesy appointments in Computer Science and Biomedical Engineering. He has general research interests in forward and inverse modeling, optimization, and uncertainty quantification of large-scale complex mechanical, geological, and biological systems. With collaborators, he received the ACM Gordon Bell Prize in 2003 (for Special Achievement) and again in 2015 (for Scalability), and was a finalist for the 2008, 2010, and 2012 Bell Prizes. He received the 2019 SIAM Computational Science & Engineering Best Paper Prize, and the 2019 SIAM Geosciences Career Prize. He is a Fellow of the Society for Industrial and Applied
Mathematics (SIAM).
We will introduce and study some discrete dynamical models intrinsically related to "continuous" ones.
In particular, the following topics will be addressed.
A. Cellular automata and related dynamical systems. Fractals through celluar automata. The game of "Life".
B. Sierpinski gasket, laplacian on the Sierpinski gasket as a limit of ``discrete laplacians''. Spectrum of the laplacian and
harmonic functions on the Sierpinsky gasket.
Dr. Omar Ghattas is a Professor of Geological Sciences and Mechanical Engineering at the University of Texas at Austin. He is also the Director of the Center for Computational Geosciences and Optimization in the Oden Institute for Computational Engineering and Sciences and holds the John A. and Katherine G. Jackson Chair in Computational Geosciences. He is a member of the faculty in the Computational Science, Engineering, and Mathematics (CSEM) interdisciplinary PhD program in the Oden Institute, and holds courtesy appointments in Computer Science and Biomedical Engineering. He has general research interests in forward and inverse modeling, optimization, and uncertainty quantification of large-scale complex mechanical, geological, and biological systems. With collaborators, he received the ACM Gordon Bell Prize in 2003 (for Special Achievement) and again in 2015 (for Scalability), and was a finalist for the 2008, 2010, and 2012 Bell Prizes. He received the 2019 SIAM Computational Science & Engineering Best Paper Prize, and the 2019 SIAM Geosciences Career Prize. He is a Fellow of the Society for Industrial and Applied
Mathematics (SIAM).
The development of reduced order models for complex applications, offering the promise for rapid and accurate evaluation of the output of complex models under parameterized variation, remains a very active research area. Applications are found in problems which require many evaluations, sampled over a potentially large parameter space, such as in optimization, control, uncertainty quantification, and in applications where a near real-time response is needed. However, many challenges remain unresolved to secure the flexibility, robustness, and efficiency needed for general large-scale applications, in particular for nonlinear and/or time-dependent problems.
After giving a brief general introduction to projection based reduced order models, we discuss the use of artificial feedforward neural networks to enable the development of fast and accurate nonintrusive models for complex problems. We demonstrate that this approach offers substantial flexibility and robustness for general nonlinear problems and enables the development of fast reduced order models for complex applications.
In the second part of the talk, we discuss how to use residual based neural networks in which knowledge of the governing equations is built into the network and show that this has advantages both for training and for the overall accuracy of the model.
Time permitting, we finally discuss the use of reduced order models in the context of prediction, i.e. to estimate solutions in regions of the parameter beyond that of the initial training. With an emphasis on the Mori-Zwansig formulation for time-dependent problems, we discuss how to accurately account for the effect of the unresolved and truncated scales on the long term dynamics and show that accounting for these through a memory term significantly improves the predictive accuracy of the reduced order model.
Contatto: alfio.quarteroni@polimi.it
After receiving his PhD in 1995 from the Technical University of Denmark, Professor Hesthaven joined Brown University, USA where he became Professor of Applied Mathematics in 2005. In 2013 he joined EPFL as Chair of Computational Mathematics and Simulation Science and since 2017 as Dean of the School of Basic Sciences. His research interests focus on the development, analysis, and application of high-order accurate methods for the solution of complex time-dependent problems, often requiring high-performance computing. A particular focus of his research has been on the development of computational methods for problems of linear and non-linear wave problems and the development of reduced basis methods, recently with an emphasis on combining traditional methods with machine learning and neural networks with broad applications, including structural health monitoring.
He has received several awards for both his research and his teaching, and has published 4 monographs and more than 160 research papers. He is on the editorial board of 8 journals and serves as Editor-in-Chief of SIAM J. Scientific Computing.
Homepage - www.epfl.ch/labs/mcss/
Motivated by questions from computational biology, we tackle the problem of a combinatorial classification of finite metric spaces by means of a new polyhedral invariant introduced by Vershik in 2010: the metric space's ``fundamental polytopes''. These originate from the theory of optimal transport (where they are often named after Wasserstein or Kantorovich-Rubinstein) and have recently found applications in a host of different contexts, from algebraic statistics to tropical geometry to the theory of reaction networks. Nevertheless, the most basic questions on their structure remain to date unanswered.
In this talk I will begin by defining the fundamental polytopes of finite metric spaces and sketching the motivation for our work. I will then show how matroid theory allows to describe the combinatorial structure of the fundamental polytopes associated to tree-like metric spaces. I will also discuss some partial results for the case of a special type of phylogenetic networks and, time permitting, I will also present some lines of current research.
After receiving his PhD in 1995 from the Technical University of Denmark, Professor Hesthaven joined Brown University, USA where he became Professor of Applied Mathematics in 2005. In 2013 he joined EPFL as Chair of Computational Mathematics and Simulation Science and since 2017 as Dean of the School of Basic Sciences. His research interests focus on the development, analysis, and application of high-order accurate methods for the solution of complex time-dependent problems, often requiring high-performance computing. A particular focus of his research has been on the development of computational methods for problems of linear and non-linear wave problems and the development of reduced basis methods, recently with an emphasis on combining traditional methods with machine learning and neural networks with broad applications, including structural health monitoring.
He has received several awards for both his research and his teaching, and has published 4 monographs and more than 160 research papers. He is on the editorial board of 8 journals and serves as Editor-in-Chief of SIAM J. Scientific Computing.
Homepage - www.epfl.ch/labs/mcss/
The talk is devoted to a rather old topic in Complex Analysis - representation of univalent (= holomorphic and injective) functions of a complex variable via integrals of Loewner's differential equation and its analogues, with applications to problems in conformal mapping. After introducing Loewner's classical method and important results obtained with its help, the main focus will be made on recent developments in the topic for the last 20 years, including applications to univalent functions with quasiconformal extensions and conformal mappings with prescribed boundary fixed points.
After receiving his PhD in 1995 from the Technical University of Denmark, Professor Hesthaven joined Brown University, USA where he became Professor of Applied Mathematics in 2005. In 2013 he joined EPFL as Chair of Computational Mathematics and Simulation Science and since 2017 as Dean of the School of Basic Sciences. His research interests focus on the development, analysis, and application of high-order accurate methods for the solution of complex time-dependent problems, often requiring high-performance computing. A particular focus of his research has been on the development of computational methods for problems of linear and non-linear wave problems and the development of reduced basis methods, recently with an emphasis on combining traditional methods with machine learning and neural networks with broad applications, including structural health monitoring.
He has received several awards for both his research and his teaching, and has published 4 monographs and more than 160 research papers. He is on the editorial board of 8 journals and serves as Editor-in-Chief of SIAM J. Scientific Computing.
Homepage - www.epfl.ch/labs/mcss/
Machine learning and biophysical modelling are very complementary approaches. The recent progress in computing power and available data makes it possible to develop accurate data-driven approaches for healthcare, while biophysical models offer a principled way to represent anatomy and physiology. In this talk, I will present research where we combine both methodologies in order to leverage their strengths. Different clinical applications in computational cardiology will be presented.
This seminar is organized within the ERC-2016-ADG Research project iHEART - An Integrated Heart Model forthe simulation of the cardiac function, that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 740132).
Contact: alfio.quarteroni@polimi.it
Maxime Sermesant is a researcher at Inria, the French research institute on informatics and mathematics, chair of « AI & Biophysics » at 3IA Côte d’Azur AI Institute, and head of « Multimodal Data Science » at IHU Liryc, Bordeaux. His research interests include biomedical image processing, organ modelling and machine learning. His main focus has been the application of patient-specific models of the heart to cardiac pathologies. He received his Diploma in General Engineering from Ecole Centrale Paris, France in 1999, his MSc from Ecole Normale Superieure de Cachan, France in 1999, and his PhD in Control, Signal and Image Processing from the University of Nice – Sophia Antipolis, France in 2003. From June 2003 to December 2005, he was a Research Fellow with the Cardiac MR Research Group, Guy’s Hospital, King’s College London, UK.
Mathematical Seminars
Politecnico di Milano
- Analisi
- Cultura Matematica
- Seminari FDS
- Geometry and Algebra
- Probabilità e Statistica Matematica
- Probabilità Quantistica