14 Settembre, 2017 11:30

MOX Colloquia

Large-scale simulations with unfitted finite element techniques

Santiago Badia, Universitat Politècnica de Catalunya, Barcelona

Aula Consiglio VII Piano - Edificio 14, Dipartimento di Matematica POLITECNICO DI MILANO

abstract

abstract

The use of unfitted finite element methods (FEMs) is an appealing approach for different reasons. They are interesting in coupled problems or to avoid the generation of body-fitted meshes. One of the bottlenecks of the simulation pipeline is the body-fitted mesh generation step and the unstructured mesh partition. The use of unfitted methods on background octree Cartesian meshes avoids the need to define body-fitted meshes, and can exploit efficient and scalable space-filling curve algorithms. In turn, such schemes complicate the numerical integration, imposition of Dirichlet boundary conditions, and the linear solver phase. The condition number of the resulting linear system does depend on the characteristic size of the cut elements, the so-called small cut cell problem.
In this work, we will present an parallel unfitted framework that relies on adaptive octree background meshes and space-filling curve partitioners. In order to solve the small cut cell problem, we will pursue two different lines. The first one is a re-definition of the finite element spaces that solves this issue, leading to condition number bounds as the ones for body-fitted schemes without any kind of perturbation/stabilization of the Galerkin formulation. Another approach will be to define appropriate iterative linear solvers based on domain decomposition preconditioning that are robust with respect to the small cut cell problem.
We will briefly present the linear solver machinery within out framework, based on highly scalable multilevel domain decomposition methods for the simulation of large scale finite element problems. Our motivation here is not only to show the excellent scalability results of the preconditioners resulting from the framework, but also to present its customization to challenging applications, e.g., space-time parallelism and heterogeneous problems. Finally, its application to challenging engineering problems, e.g., the simulation of superconductor devices or additive manufacturing will be considered.
Contatto: christian.vergara@polimi.it

contact:christian.vergara@polimi.it

Santiago Badia is full profesor of computational engineering and science at the Universitat Politecnica de Catalunya and leader of the Large Scale Scientific Computing Dept. at CIMNE.

12 Settembre, 2017 15:00

Sezione di Probabilità Quantistica

The theory of quantum statistical comparison with some applications to quantum information science

Francesco Buscemi, Nagoya University

Aula Seminari III piano

abstract

abstract

In mathematical statistics a central role is played by the notion of statistical models (or, equivalently, statistical experiments). These objects, which can be imagined as families of probability distributions, are often used to describe the statistician's state of knowledge about an unknown parameter. It is then natural to compare statistical models on the basis of their "information content." This kind of problems led in the 1950s to the formulation of a whole new theory, sometimes named "statistical comparison theory," which soon developed into a very deep field with many applications, ranging from mathematical statistics, to physics and economics. In this lecture I will present an overview of the basic ideas of statistical comparison, some possible generalizations to the quantum setting, and a few applications in quantum thermodynamics, entanglement theory, quantum measurement theory, and the theory of open quantum systems.

Santiago Badia is full profesor of computational engineering and science at the Universitat Politecnica de Catalunya and leader of the Large Scale Scientific Computing Dept. at CIMNE.

4 Settembre, 2017 14:30

Sezione di Fisica Matematica

Modelling cell movement: from tissue organisation to disease

Kevin Painter, Heriot-Watt University, Edinburgh, UK

Aula Seminari III piano

abstract

abstract

Cells move and position themselves according to various environmental cues, ranging from short range adhesive and repulsive interactions with other cells to long range guided responses due to chemicals.Cells move and position themselves according to various environmental cues, ranging from short range adhesive and repulsive interactions with other cells to long range guided responses due to chemicals. In this talk I will describe the modelling of these fundamental processes from both the individual and continuous scale, and discuss their application in areas ranging from developmental biology to cancer invasion. Along the way I will discuss the numerous modelling, numerical and analytical challenges posed. In this talk I will describe the modelling of these fundamental processes from both the individual and continuous scale, and discuss their application in areas ranging from developmental biology to cancer invasion. Along the way I will discuss the numerous modelling, numerical and analytical challenges posed.

Santiago Badia is full profesor of computational engineering and science at the Universitat Politecnica de Catalunya and leader of the Large Scale Scientific Computing Dept. at CIMNE.

20 Luglio, 2017 14:00

MOX Colloquia

Multivariate Splines and Their Applications

Ming-Jun Lai, Dept. of Math. University of Georgia, Athens, GA, USA

Aula Consiglio, VII piano - Dipartimento di Matematica, Politecnico di Milano - Edificio 14

abstract

abstract

Multivariate splines are piecewise polynomial or rational functions over a collection of polygons. We shall explain some approximation properties of these splines and construction of locally supported basis functions. Then I will explain how to use them for numerical solution of linear and nonlinear partial differential equations and data fitting for statistical analysis. Our approach is based on barycentric coordinates (BB form) and generalized barycentric coordinates (GBC). The smoothness conditions, boundary conditions or interpolatory conditions will be set up as linear constraints when we minimize energy functional associated with the PDE to be solved or data to be fitted. Our approach enables us to use higher order polynomials and the smoothness higher than continuity easily. Several numerical examples of PDE and data fitting will be shown.

Ming-Jun Lai is professor at University of Georgia, Georgia, USA. Is main scientific interest are multivariate splines. He is the author of the book, Spline Functions over Triangulations (joint with Larry Schumaker), Cambridge University Press, 2007. He has authored more than 100 publications in the fields of Numerical Analysis, Numerical Solution of Partial Differential Equations, Approximation Theory, Compressed Sensing, Computational Optimization, Computer Aided Geometric Design, Mathematical Image Analysis, Multivariate Splines, Wavelet and Frame Analysis.

14 Luglio, 2017 11:00

Sezione di Analisi

Doubly nonlocal Cahn-Hilliard equations

Ciprian G. Gal, Florida International University, Miami, USA

Aula seminari 3° piano

abstract

abstract

We consider a doubly nonlocal nonlinear parabolic equation
which describes phase-segregation of a two-component material in a
bounded domain. This model is a more general version than the recent
nonlocal Cahn–Hilliard equation proposed by Giacomin and Lebowitz,
such that it reduces to the latter under certain conditions. It turns
out that there are four possible cases of double interaction in which
the nonlocality is reflected, we shall discuss some of them.

Ming-Jun Lai is professor at University of Georgia, Georgia, USA. Is main scientific interest are multivariate splines. He is the author of the book, Spline Functions over Triangulations (joint with Larry Schumaker), Cambridge University Press, 2007. He has authored more than 100 publications in the fields of Numerical Analysis, Numerical Solution of Partial Differential Equations, Approximation Theory, Compressed Sensing, Computational Optimization, Computer Aided Geometric Design, Mathematical Image Analysis, Multivariate Splines, Wavelet and Frame Analysis.

12 Luglio, 2017 11:30

MOX Seminar

A Three-Field Formulation for Poroelasticity

Ricardo Ruiz Baier, Mathematical Institute, University of Oxford

MOX, Aula Saleri, Dipartimento di Matematica, VI piano

abstract

abstract

In this talk we present a stable and convergent conforming finite element method for the discretisation of the linear poroelasticity equations in a new formulation, where the volumetric contributions to the total stress are merged into an additional unknown. The resulting saddle point formulation can be analysed by means of a Fredholm alternative, after realizing that the problem is a compact perturbation of a Stokes-like invertible system. A generic Galerkin scheme is constructed, whose solvability properties follow closely those from the continuous variational form, and more importantly, given that specific finite dimensional spaces are chosen adequately, it is stable even in the incompressible limit.
Contatto: luca.formaggia@polimi.it

Ming-Jun Lai is professor at University of Georgia, Georgia, USA. Is main scientific interest are multivariate splines. He is the author of the book, Spline Functions over Triangulations (joint with Larry Schumaker), Cambridge University Press, 2007. He has authored more than 100 publications in the fields of Numerical Analysis, Numerical Solution of Partial Differential Equations, Approximation Theory, Compressed Sensing, Computational Optimization, Computer Aided Geometric Design, Mathematical Image Analysis, Multivariate Splines, Wavelet and Frame Analysis.

11 Luglio, 2017 14:00 in punto

Sezione di Probabilità Quantistica

Quantum Markov Chains and Associated Open Quantum Random Walks

Ameur Dhahri, Chungbuk National University

Aula Seminari III piano

abstract

abstract

We establish the connection between Open Quantum Random Walks and the Quantum Markov Chains. In particular, we construct tow kinds of Quantum Markov Chains associated with Open Quantum Random Walks. Finally, we study the recurrence, transience and the property of accessibility associated to these Quantum Markov Chains.

Ming-Jun Lai is professor at University of Georgia, Georgia, USA. Is main scientific interest are multivariate splines. He is the author of the book, Spline Functions over Triangulations (joint with Larry Schumaker), Cambridge University Press, 2007. He has authored more than 100 publications in the fields of Numerical Analysis, Numerical Solution of Partial Differential Equations, Approximation Theory, Compressed Sensing, Computational Optimization, Computer Aided Geometric Design, Mathematical Image Analysis, Multivariate Splines, Wavelet and Frame Analysis.

11 Luglio, 2017 15:00 in punto

Sezione di Probabilità Quantistica

Glauber Dynamics on the Cycles: Spectral Distribution of the Generator

Hyun Jae Yoo, Hankyong National University

Aula Seminari III piano

abstract

abstract

We consider Glauber dynamics on finite cycles. By introducing a vacuum state we consider an algebraic probability space for the generator of the dynamics. We obtain a quantum decomposition of the generator and construct an interacting Fock space. As a result we obtain a distribution of the generator in the vacuum state.

We also discuss the monotonicity of the moments of spectral measure as the couplings increase.

In particular, when the couplings are assumed to be uniform, as the cycle grows to an infinite chain, we show that the distribution (under suitable dilation and translation) converges to a Kesten distribution.

Ming-Jun Lai is professor at University of Georgia, Georgia, USA. Is main scientific interest are multivariate splines. He is the author of the book, Spline Functions over Triangulations (joint with Larry Schumaker), Cambridge University Press, 2007. He has authored more than 100 publications in the fields of Numerical Analysis, Numerical Solution of Partial Differential Equations, Approximation Theory, Compressed Sensing, Computational Optimization, Computer Aided Geometric Design, Mathematical Image Analysis, Multivariate Splines, Wavelet and Frame Analysis.

4 Luglio, 2017 14:00 in punto

Sezione di Finanza Quantitativa

Debt overhang and sovereign debt restructuring

Mattia Picarelli, Università di Roma 1

Aula Seminari Terzo Piano

abstract

abstract

Debt overhang is defined as a situation where a large amount of debt distorts the optimal investment decisions and discourages the government's efforts of the debtor country to undertake the necessary "adjustment policies". In this paper, I study some different strategies that can be used to solve a sovereign debt overhang problem. In particular, I consider two strategies based on a debt restructuring process, via haircut or rescheduling, and a third one based on conditional-additional lending. This strategy relies on the idea that the debtor country can get new lending from the existing creditors, in order to undertake investments that can affect the productivity shock distribution in a positive way (or reduce the probability of default). The aim is to study the consequences, deriving from the three strategies described, on the incentives to invest in a "troubled country". According to these consequences and under some specific conditions, I am able to build a ranking for these three strategies in order to see which is the most effective one. In particular, I find that if the change in investments due to the conditional-additional lending makes the probability of default low in this scenario, the conditional lending strategy will be the most effective one.

Ming-Jun Lai is professor at University of Georgia, Georgia, USA. Is main scientific interest are multivariate splines. He is the author of the book, Spline Functions over Triangulations (joint with Larry Schumaker), Cambridge University Press, 2007. He has authored more than 100 publications in the fields of Numerical Analysis, Numerical Solution of Partial Differential Equations, Approximation Theory, Compressed Sensing, Computational Optimization, Computer Aided Geometric Design, Mathematical Image Analysis, Multivariate Splines, Wavelet and Frame Analysis.

29 Giugno, 2017 14:00

MOX Colloquia

Dynamical low rank approximation of random time dependent PDEs

Fabio Nobile, Mathematics Institute, CSQI, Ecole Polytechnique Fédérale de Lausanne, Switzerland

Aula Consiglio, VII piano - Dipartimento di Matematica, Politecnico di Milano - Edificio 14

abstract

abstract

Partial differential equations with random coefficients and input data (random PDEs in short) arise in many applications in which the data of the PDE need to be described in terms of random variables/fields due either to a lack of knowledge of the system or to its inherent variability. The numerical approximation of statistics of the solution poses several challenges when the number of random parameters is large and/or the parameter-to-solution map is complex, and effective surrogate or reduced models are of great need in this context.
In this talk we consider time dependent PDEs with few random parameters and seek for an approximate solution in separable form that can be written at each time instant as a linear combination of linearly independent spatial functions multiplied by linearly independent random variables (low rank approximation) in the spirit of a truncated Karhunen-Loève expansion. Since the optimal deterministic and stochastic modes can significantly change over time, we consider here a dynamical approach where those modes are computed on the fly as solutions of suitable evolution equations. From a geometrical point of view, this corresponds to constraining the original dynamics to the manifold of fixed rank functions, i.e. functions that can be written in separable form with a fixed number of terms. Equivalently, the original equations are projected onto the tangent space to the manifold of fixed rank functions along the approximate trajectory, similarly to the Dirac-Frenkel variational principle in quantum mechanics.
We discuss the construction of the method as well as practical numerical aspects for several time dependent PDEs with random parameters, including the heat equation with a random diffusion coefficient; the incompressible Navier-Stokes equations with random Dirichlet boundary conditions; the wave equation with random wave speed. In the latter case, we propose a dynamical low rank approximation that preserves the symplectic structure of the equations.

Fabio Nobile is associate professor in Mathematics at the Ecole Polytechique Federale de Lausanne, Switzerland, since 2011. Previously he has been assistant professor in numerical analysis at Politecnico di Milano. He graduated in electronic engineering at Politecnico di Milano in 1998; PhD in Applied Mathematics at EPFL in 2001; 2 year post-doc position (ICES fellowship) at ICES, University of Texas at Austin. Invited speaker to several international conferences and workshops, including the 7th European Congress of Mathematics 7ECM in 2016. Co-Chair and main organizer of the SIAM UQ16 conference, EPFL. Awarded the second prize at the 12th Leslie Fox Prize in 2005. He is author of more than 60 publications on international journals. He is member of the editorial board of 5 international journals. His main research activity is in numerical approximation of PDEs with stochastic coefficients, sparse approximation techniques, numerical solution of fluid-structure interaction problems, simulation of the cardiovascular system.