Events
The derivation of variational models describing the epitaxial growth of thin films in the framework of the theory of Stress-Driven Rearrangement Instabilities (SDRI) will be presented, and the state of the art of the mathematical results described. By working in the context of both continuum and molecular mechanics, not only free boundary problems, but also atomistic models will be considered, and the discrete-to-continuum passage rigorously investigated in the intent to also provide a microscopical justification of the theory. An overview of the mathematical results achieved through the years with various co-authors for the existence, regularity and evolution of the solutions will be presented.
"Learning" complex technological or physical processes usually requires fusing the information provided by observational data with knowledge about a "background model" often formulated in terms of a family of parameter dependent PDEs. Frequent forward simulations as well as related inverse tasks calls for reduced models that facilitate an efficient evaluation of the underlying parameter-to-solution map despite expected obstructions caused by the typical high-dimensionality of parameter domains. Physics informed Neural Networks (PINN) offer a promising framework to that effect. The core issue addressed in this talk is the prediction capability of such methods. Related specific questions concern, for instance, the choice of "model compliant" metrics, the choice of training risks that convey certifiable information about the achieved accuracy in such metrics, the role of a priori versus a posteriori error bounds, connections with Generative Adversarial Networks, as well as related implications on training strategies and network adaptation.
Contact: alfio.quarteroni@polimi.it
Wolfgang Dahmen is currently a chaired professor in mathematics (SmartState and Williams-Hedberg-Hedberg Chair) at the University of South Carolina in Columbia, South Carolina, USA. His research interests are in Approximation Theory, Numerical, Applied and Harmonic Analysis as well as interdisciplinary applications. A central thematic thread is the development and analysis of adaptive and nonlinear solution concepts in a variety of contexts such as image and data analysis, machine learning, the numerical solution of singular integral and partial differential equations, and model reduction.
Wolfgang Dahmen received his PhD from RWTH Aachen in 1976 and his Habilitation from the University of Bonn. After an IBM Postdoctoral Fellowship at the IBM Research Center in Yorktown Heights, NY, he took (associate and full) professor positions at the University of Bielefeld and the Free University of Berlin before joining RWTH Aachen in 1992. In 2017 he became chaired professor at the University of South Carolina. Among his awards and honors are the Gottfried-Wilhelm-Leibniz Award, Keck Future Award of the US Academies (together with P. Binev, T. Vogt), election to the German National Academy of Sciences, Leopoldina, Robert-Piloti-Prize of the Technical University of Darmstadt, and SIAM Fellow. He has served on numerous Scientific Advisory boards such as the CRM in Barcelona and the Isaac Newton Institute in Cambridge, UK. From 2014 to 2017 he was the Chair of the Board of Directors of the Society Foundations of Computational Mathematics.
We study local asymptotics of solutions to fractional elliptic equations at boundary points, under some outer homogeneous Dirichlet boundary condition. Our analysis is based on a blow-up procedure which involves some Almgren type monotonicity formulæ and provides a classification of all possible homogeneity degrees of limiting entire profiles. As a consequence, we establish a strong unique continuation principle from boundary points. This is a joint work with A. De Luca and V. Felli.
Wolfgang Dahmen is currently a chaired professor in mathematics (SmartState and Williams-Hedberg-Hedberg Chair) at the University of South Carolina in Columbia, South Carolina, USA. His research interests are in Approximation Theory, Numerical, Applied and Harmonic Analysis as well as interdisciplinary applications. A central thematic thread is the development and analysis of adaptive and nonlinear solution concepts in a variety of contexts such as image and data analysis, machine learning, the numerical solution of singular integral and partial differential equations, and model reduction.
Wolfgang Dahmen received his PhD from RWTH Aachen in 1976 and his Habilitation from the University of Bonn. After an IBM Postdoctoral Fellowship at the IBM Research Center in Yorktown Heights, NY, he took (associate and full) professor positions at the University of Bielefeld and the Free University of Berlin before joining RWTH Aachen in 1992. In 2017 he became chaired professor at the University of South Carolina. Among his awards and honors are the Gottfried-Wilhelm-Leibniz Award, Keck Future Award of the US Academies (together with P. Binev, T. Vogt), election to the German National Academy of Sciences, Leopoldina, Robert-Piloti-Prize of the Technical University of Darmstadt, and SIAM Fellow. He has served on numerous Scientific Advisory boards such as the CRM in Barcelona and the Isaac Newton Institute in Cambridge, UK. From 2014 to 2017 he was the Chair of the Board of Directors of the Society Foundations of Computational Mathematics.
Given a smooth path X, the signature of a path is the infinite sequence of the iterated integral of X with itself. If X is a stochastic process with an integration structure (e.g. semimartingale), the corresponding signature is the key-object to understand the "lack of continuity" between the solution of a SDE driven by X and the process itself. In this talk, we will review the main properties of the signature and introduce the new notion of "non-commutative signature", which we tailored to study a new class of rough/stochastic differential equations arising in the context of non-commutative probability. Joint paper with Nicolas Gilliers (Université de Toulouse) and Yannick Vargas (Postdam Universität).
Wolfgang Dahmen is currently a chaired professor in mathematics (SmartState and Williams-Hedberg-Hedberg Chair) at the University of South Carolina in Columbia, South Carolina, USA. His research interests are in Approximation Theory, Numerical, Applied and Harmonic Analysis as well as interdisciplinary applications. A central thematic thread is the development and analysis of adaptive and nonlinear solution concepts in a variety of contexts such as image and data analysis, machine learning, the numerical solution of singular integral and partial differential equations, and model reduction.
Wolfgang Dahmen received his PhD from RWTH Aachen in 1976 and his Habilitation from the University of Bonn. After an IBM Postdoctoral Fellowship at the IBM Research Center in Yorktown Heights, NY, he took (associate and full) professor positions at the University of Bielefeld and the Free University of Berlin before joining RWTH Aachen in 1992. In 2017 he became chaired professor at the University of South Carolina. Among his awards and honors are the Gottfried-Wilhelm-Leibniz Award, Keck Future Award of the US Academies (together with P. Binev, T. Vogt), election to the German National Academy of Sciences, Leopoldina, Robert-Piloti-Prize of the Technical University of Darmstadt, and SIAM Fellow. He has served on numerous Scientific Advisory boards such as the CRM in Barcelona and the Isaac Newton Institute in Cambridge, UK. From 2014 to 2017 he was the Chair of the Board of Directors of the Society Foundations of Computational Mathematics.
We consider a time discretization scheme of Euler type for the 2d stochastic Navier-Stokes equations on the torus.
We prove a mean square rate of convergence. This refines previous results established with a rate of convergence in probability only.
Using exponential moment estimates of the solution of the Navier-Stokes equations and a convergence of a localized scheme, we can prove strong convergence of fully implicit and semi-implicit time Euler discretization and also a splitting scheme. The speed of convergence depends on the diffusion coefficient and the viscosity parameter.
When the noise is additive, we are able to get strong convergence without localization.
Wolfgang Dahmen is currently a chaired professor in mathematics (SmartState and Williams-Hedberg-Hedberg Chair) at the University of South Carolina in Columbia, South Carolina, USA. His research interests are in Approximation Theory, Numerical, Applied and Harmonic Analysis as well as interdisciplinary applications. A central thematic thread is the development and analysis of adaptive and nonlinear solution concepts in a variety of contexts such as image and data analysis, machine learning, the numerical solution of singular integral and partial differential equations, and model reduction.
Wolfgang Dahmen received his PhD from RWTH Aachen in 1976 and his Habilitation from the University of Bonn. After an IBM Postdoctoral Fellowship at the IBM Research Center in Yorktown Heights, NY, he took (associate and full) professor positions at the University of Bielefeld and the Free University of Berlin before joining RWTH Aachen in 1992. In 2017 he became chaired professor at the University of South Carolina. Among his awards and honors are the Gottfried-Wilhelm-Leibniz Award, Keck Future Award of the US Academies (together with P. Binev, T. Vogt), election to the German National Academy of Sciences, Leopoldina, Robert-Piloti-Prize of the Technical University of Darmstadt, and SIAM Fellow. He has served on numerous Scientific Advisory boards such as the CRM in Barcelona and the Isaac Newton Institute in Cambridge, UK. From 2014 to 2017 he was the Chair of the Board of Directors of the Society Foundations of Computational Mathematics.
In this talk, we are going to discuss the dynamics of a polaron, at large coupling. For initial data of Pekar product form, with a coherent phonon field and with the electron minimising the corresponding field energy, we provide a norm approximation of the evolution, valid up to times quadratic in the coupling constant. The approximation is given in terms of a Pekar product state, evolved through the Landau-Pekar equations and corrected by a Bogoliubov dynamics describing quantum fluctuations. I will explain the similarities with the study of the evolution of interacting bosons. This talk is based on joint work with Nikolai Leopold, David Mitrouskas, Simone Rademacher and Robert Seiringer.
Wolfgang Dahmen is currently a chaired professor in mathematics (SmartState and Williams-Hedberg-Hedberg Chair) at the University of South Carolina in Columbia, South Carolina, USA. His research interests are in Approximation Theory, Numerical, Applied and Harmonic Analysis as well as interdisciplinary applications. A central thematic thread is the development and analysis of adaptive and nonlinear solution concepts in a variety of contexts such as image and data analysis, machine learning, the numerical solution of singular integral and partial differential equations, and model reduction.
Wolfgang Dahmen received his PhD from RWTH Aachen in 1976 and his Habilitation from the University of Bonn. After an IBM Postdoctoral Fellowship at the IBM Research Center in Yorktown Heights, NY, he took (associate and full) professor positions at the University of Bielefeld and the Free University of Berlin before joining RWTH Aachen in 1992. In 2017 he became chaired professor at the University of South Carolina. Among his awards and honors are the Gottfried-Wilhelm-Leibniz Award, Keck Future Award of the US Academies (together with P. Binev, T. Vogt), election to the German National Academy of Sciences, Leopoldina, Robert-Piloti-Prize of the Technical University of Darmstadt, and SIAM Fellow. He has served on numerous Scientific Advisory boards such as the CRM in Barcelona and the Isaac Newton Institute in Cambridge, UK. From 2014 to 2017 he was the Chair of the Board of Directors of the Society Foundations of Computational Mathematics.
The talk is concerned with the convergence analysis of finite element methods for the approximate solution of a system of nonlinear elliptic partial differential equations that arise in models of chemically reacting viscous incompressible biological fluids. The shear-stress appearing in the model involves a power-law type nonlinearity, where, instead of being a fixed constant, the power law-exponent is a function of a spatially varying nonnegative concentration function, which, in turn, solves a nonlinear convection-diffusion equation. In order to prove the convergence of the sequence of finite element approximations to a solution of this coupled system of nonlinear PDEs, a uniform Hölder norm bound needs to be derived for the sequence of finite element approximations to the concentration in a setting, where the diffusion coefficient in the convection-diffusion equation satisfied by the concentration is merely an $L^\infty$ function. This necessitates the development of a finite element counterpart of the De Giorgi--Nash--Moser theory. Motivated by an early paper by Aguilera and Caffarelli (1986) in the simpler setting of Laplace's equation, we derive such uniform Hölder norm bounds on the sequence of continuous piecewise linear finite element approximations to the concentration. We then use results from the theory of variable-exponent Sobolev spaces equipped with a Luxembourg norm, Minty's method for monotone operators and an extension to variable-exponent Sobolev spaces of the finite element version of the Acerbi--Fusco Lipschitz-truncation method, originally developed in classical Sobolev spaces in collaboration with Lars Diening and Christian Kreuzer (SIAM J. Numer. Anal. 51(2): 984--1015 (2014)), to pass to the limit in the coupled system of nonlinear PDEs under consideration.
The talk is based on joint work with Seungchan Ko and Petra Pustejovska, and recent results obtained in collaboration with Lars Diening and Toni Scharle.
Endre Süli is Professor of Numerical Analysis in the Mathematical Institute at the University of Oxford. He is a Foreign Member of the Serbian Academy of Sciences and Arts (2009), Fellow of the European Academy of Sciences (2010), SIAM Fellow (2016), Member of the Academia Europaea (2020), and a Fellow of the Royal Society (2021). He received his doctorate at the University of Belgrade in March 1985 and has held a faculty position at the University of Oxford since October 1985. His research is concerned with the numerical analysis of nonlinear partial differential equation in continuum mechanics.
Goal of this talk is to present some recent progress in the theory of holomorphic function
spaces on homogenous Siegel domains of Type II. These domains are unbounded realisation of the homogeneous bounded domains and include the tube domains over homogeneous cones as a particular case.
We first describe the structure and geometrical properties of such domains. Second, we introduce some classes of holomorphic function spaces, including the weighted Bergman spaces, and describe their boundary behaviour. Finally, we concentrate on the boundedness of the Bergman projections, presenting some recent results and open problems. This talk is based on joint work with Mattia Calzi.
Endre Süli is Professor of Numerical Analysis in the Mathematical Institute at the University of Oxford. He is a Foreign Member of the Serbian Academy of Sciences and Arts (2009), Fellow of the European Academy of Sciences (2010), SIAM Fellow (2016), Member of the Academia Europaea (2020), and a Fellow of the Royal Society (2021). He received his doctorate at the University of Belgrade in March 1985 and has held a faculty position at the University of Oxford since October 1985. His research is concerned with the numerical analysis of nonlinear partial differential equation in continuum mechanics.
The notion of a holomorphic motion was introduced by Mané, Sad and Sullivan in the 1980's, motivated by the observation that Julia sets of rational maps often move holomorphically with holomorphic variations of the parameters. In the years that followed, the study of the behavior of various set-functions under holomorphic motions became an area of significant interest. For instance, holomorphic motions played a central role in the work of Astala on distortion of Hausdorff dimension and area under quasiconformal mappings.
In this talk, I will first review the basic notions and results related to analytic capacity and holomorphic motions, including the extended lambda lemma. I will then present some recent results on the behavior of analytic capacity under holomorphic motions. The proofs involve different notions such as conformal welding, quadratic Julia sets and harmonic measure. This is joint work with Tom Ransford and Wen-Hui Ai.
Endre Süli is Professor of Numerical Analysis in the Mathematical Institute at the University of Oxford. He is a Foreign Member of the Serbian Academy of Sciences and Arts (2009), Fellow of the European Academy of Sciences (2010), SIAM Fellow (2016), Member of the Academia Europaea (2020), and a Fellow of the Royal Society (2021). He received his doctorate at the University of Belgrade in March 1985 and has held a faculty position at the University of Oxford since October 1985. His research is concerned with the numerical analysis of nonlinear partial differential equation in continuum mechanics.
The minimum spanning tree (MST) problem is a combinatorial optimization problem with many applications, well beyond its historical introduction for network design. The study of its random instances on Euclidean models, e.g., on complete graphs obtained by sampling i.i.d. uniform points on a d-dimensional cube, is classical, with many limit results as the number of the points grows. In this talk, I will present two new results for its bipartite counterpart, i.e., with an additional colouring (red/blue) of the points and allowing connections only between different colours. First, we prove that the maximum vertex degree of the MST grows logarithmically, in contrast with the non-bipartite case, where a uniform bound holds, depending on d only -- a fact crucially used in many classical results. Despite this difference, we then argue that the cost of the MST, suitably normalized, converge a.s. to a limiting constant that can be represented as a series of integrals, thus extending a result of Avram and Bertsimas to the bipartite case and confirming a conjecture by Riva, Malatesta and Caracciolo. Joint work with M. Correddu, Università di Pisa.
Endre Süli is Professor of Numerical Analysis in the Mathematical Institute at the University of Oxford. He is a Foreign Member of the Serbian Academy of Sciences and Arts (2009), Fellow of the European Academy of Sciences (2010), SIAM Fellow (2016), Member of the Academia Europaea (2020), and a Fellow of the Royal Society (2021). He received his doctorate at the University of Belgrade in March 1985 and has held a faculty position at the University of Oxford since October 1985. His research is concerned with the numerical analysis of nonlinear partial differential equation in continuum mechanics.
Mathematical Seminars
Politecnico di Milano
- Analisi
- Cultura Matematica
- Seminari FDS
- Geometry and Algebra
- Probabilità e Statistica Matematica
- Probabilità Quantistica