Shallow water models of geophysical flows must be adapted to geometric characteristics in the presence of a general bottom topography with non-negligible slopes and curvatures, such as a mountain landscape. In this work we derive an intrinsic shallow water model from the Navier-Stokes equations defined on a local reference frame anchored on the bottom surface. The resulting equations are characterized by non-autonomous flux functions and source terms embodying only the geometric information. We show that the proposed model is rotational invariant, admits a conserved energy, is well-balanced, and it is formally a second order approximation of the Navier-Stokes equations with respect to a geometry-based order parameter. We then derive a numerical discretization by means of a first order upwind Godunov Finite Volume scheme intrinsically defined on the bottom surface. We study convergence properties of the resulting scheme both theoretically and numerically. Next, we extend the approach to derive a second order Discontinuous Galerkin method and show its properties.
Simulations on several synthetic test cases are used to validate the theoretical results as well as more experimental properties of the solver. The results show the importance of taking into full consideration the bottom geometry even for relatively mild and slowly varying curvatures.
Lagrangian and Eulerian formulations for multi-agent optimal control problems Giulia Cavagnari, Politecnico di Milano mercoledì 26 febbraio 2020 alle ore 15:15, Aula seminari 3° piano
In this talk we present and compare two approaches to study deterministic optimal control problems for interacting multi-agent systems: Lagrangian and Eulerian.
Different research fields comes into play: optimal control and transport theory to set out the variational model and analyze the underlying principles, and a random variable approach to deal with the problem in its various Lagrangian formulations on a fixed probability space ? (space of parametrizations).
On one side, the state of the system is expressed by a random variable in L^2(?) and the nonlocal velocity field driving the dynamics depends on the realizations of a Borel measurable control map. The lack of existence of minimizers for a generic cost functional, even after relaxation, makes it fundamental to recover optimality through the study of the problem from an Eulerian viewpoint. Here the problem is set in the 2-Wasserstein space of probability measures. The relation with the previous approach lies on the identification of the time-dependent measure, solving a controlled continuity equation, as the law of the evolving random variable.
After stating the problem, we prove the equivalence between the Lagrangian and Eulerian value functions through an intermediate formulation based on the Superposition Principle by Ambrosio-Gigli-Savaré.
Finally we deal with stability and ?-convergence results for the corresponding problems involving a finite number of agents to the mean-field ones.
This is a joint work with Stefano Lisini (University of Pavia), Carlo Orrieri (University of Trento) and Giuseppe Savaré (University of Pavia).
Finite Element Methods at Realistic Complexities Wolfgang Bangerth, Department of Mathematics, Campus Delivery, Fort Collins, CO, USA NUMETH-HPC venerdì 28 febbraio 2020 alle ore 14:00, Aula Saleri VI piano
Solving realistic, applied problems with the most modern numerical methods introduces many levels of complexity. In particular, one has to think about not just a single method, but a whole collection of algorithms: a single code may utilize fully adaptive, unstructured meshes; nonlinear, globalized solvers; algebraic multigrid and block preconditioners; and do all this on
1,000 processors or more with realistic material models.
Codes at this level of complexity can no longer be written from scratch. However, over the past two decades, many high quality libraries have been developed that make writing advanced computational software simpler. In this talk, I will briefly introduce the deal.II finite element library
(http://www.dealii.org) whose development I co-lead and show how it has enabled us to develop the ASPECT code (http://aspect.geodynamics.org) for simulation of convection in the earth mantle. I will discuss some of the results obtained with this code and comment on the lessons learned from developing this massively parallel code for the solution of a complex problem.
Some global results for homogeneous Hormander sums of squares Stefano Biagi, Politecnico di Milano mercoledì 4 marzo 2020 alle ore 15:15, Aula seminari 3° piano
In this talk we present several global results concerning homogeneous Hormander sums of squares L.
These operators intervene in various contexts of interest, such as CR geometry, Lie group Theory, sub-Riemannian manifolds, Mathematical Finance, etc.
After a brief introduction on general sub-elliptic operators (of which our L’s are a particular case), we properly introduce the class of the homogeneous Hormander sums of squares and we discuss some global `qualitative' aspects regarding these operators: global lifting on Carnot groups; existence/global estimates for the associated fundamental solution and heat kernel; maximum principles on unbounded domains.
The results presented in this talk are contained in several papers in collaboration with A. Bonfiglioli, M. Bramanti and E. Lanconelli.
Modelling and Simulation of Magnetoquasistatic and Electroquasistatic Fields Herbert De Gersem, Institute for Accelerator Science and Electromagnetic Fields (TEMF), TU Darmstadt, Germany giovedì 12 marzo 2020 alle ore 14:00, Dip. di Matematica, Politecnico di Milano ed. 14 "La Nave", via Ponzio 31/P- Aula Consiglio, VII piano
Magnetoquasistatic field problems arise in, e.g., transformers, electric machines and superconducting magnets. Electroquasistatic field problems arise in high-voltage engineering and on printed circuit boards. Both are expressed by a nonlinear parabolic partial differential equation, which is typically discretised by finite elements on an unstructured tetrahedral mesh and time-stepped by implicit methods. The talk addresses contemporary challenges such as, e.g., the increasing level of detail, multiphysical phenomena, multirate behaviour and nontrivial materials. Recent developments in numerical methods dedicated to the quasistatic setting will be discussed, e.g., hybrid spatial discretisation schemes, field-circuit coupling, multirate time stepping and sparse surrogate modelling. The methods will be illustrated by examples from electrical engineering and accelerator science.
Design and Health Stefano Capolongo , Politecnico di Milano mercoledì 18 marzo 2020 alle ore 12:15, aula B21
Radial positive solutions for a class of Neumann problems without growth conditions Benedetta Noris, Politecnico di Milano mercoledì 25 marzo 2020 alle ore 15:15, Aula seminari 3° piano
We consider a class of semilinear and quasilinear partial differential equations on bounded radial domains, subject to homogeneous Neumann boundary conditions. The nonlinear term is possibly supercritical in the sense of Sobolev embeddings.
We investigate the existence of positive non-constant radial solutions: we prove existence and multiplicity, and examine the oscillatory behavior.
In case the nonlinearity is subcritical in the sense of Sobolev embeddings, we also prove a priori bounds of solutions.
The talk is based on papers in collaboration with several authors.
Homoclinic orbits to a center in a class of autonomous Hamiltonian systems, with an application in the theory of interfacial waves Boris Buffoni, Ecole Polytechnique Federale de Lausanne mercoledì 10 maggio 2000
Problemi di rilassamento per sistemi iperbolici P. Marcati, Università di l'Aquila lunedì 17 aprile 2000 alle ore 17:00, Dipartimento di Matematica - Università degli Studi di Milano - Via Saldini 50 - Milano - Sala di Rappresentanza
I The continuous coagulation-fragmentation model Philippe Laurencot, CNRS & Institut Elie Cartan, Università di Nancy mercoledì 12 aprile 2000
Topology, deformations and automorphsims of real algebraic surfaces V. Kharlamov, Université L. Pasteur, Strasbourg, France lunedì 10 aprile 2000 alle ore 17:00, Dipartimento di Matematica - Università degli Studi di Milano - Via Saldini 50 - Milano - Sala di Rappresentanza
Modelli dinamici per membrane biologiche http://smmm.unipv.it Epifanio Virga, Dip. di Matematica (Università di Pavia) mercoledì 5 aprile 2000
Semigruppi ed equazioni della Finanza Matematica Fabrizio Colombo, Dip. di Matematica del Politecnico di Milano mercoledì 29 marzo 2000
Contributions to the Mathematics of the nonstandard finite difference methods with applications to certain discrete schemes Jean Lubuma, Dept. of Mathematics and Statistics, Vista University, Pretoria (South Africa) mercoledì 22 marzo 2000
Calcolo variazionale e ottimizzazione rispetto ad una misura Ilaria Fragalà, Dip. Matematica Politecnico di Milano mercoledì 23 febbraio 2000