Dipartimento di Matematica

Abstract

There is increasing interest in personalized prediction of disease progression. Prediction models are statistical models based on patient and disease
characteristics which are used to inform treatment decisions, to provide personalized risk estimates for a patient. Many available prediction models are limited to predictions from a specific baseline time like diagnosis or shortly before treatment is initiated. It is at this time that the most important decisions on primary treatment are made. It is well known that available prognostic models are important tools for physicians to guide treatment decisions at diagnosis. However, once primary treatment has been initiated, the prognosis of the patient will change over the course of time, as a result of the effect of treatment, like treatment toxicity, clinical events such as disease recurrence that may have occurred, or simply, because of the fact that the patient is still alive. This implies that prediction models need to be updated by using new information about a specific patient that has become available since baseline. Prediction models that incorporate this dynamic aspect are called dynamic prediction models. In the first part of the talk the methodology for dynamic prediction will be discussed. The dynamic aspect of dynamic prediction use information on events and/or measurements up to the present, in order to update the prediction. It will be shown how dynamic predictions may be obtained using the concept of landmarking. In the second part of the talk a dynamic prediction model of survival for patients with high-grade extremity soft tissue sarcoma, will be presented. The model provides updated 5-year survival probabilities from different prediction time points during follow-up. Baseline covariates as well as time-dependent covariates, such as status of local recurrence and distant metastases, are included in the model. This dynamic prediction model which updates survival probabilities over time can be used to make better individualized treatment decisions based on a dynamic assessment of a patient's prognosis.
- van Praag VM , Rueten-Budde A, PERSARC group, van de Sande MA, Fiocco M. A prediction model for treatment decisions in high-grade extremity soft-tissue sarcomas. European Journal of cancer 2017, Volume 83, Pages 313{323
- Rueten-Budde A, van Praag VM , PERSARC group, van de Sande MA, Fiocco M. Dynamic Prediction of Overall Survival for Patients with High-Grade Extremity Soft Tissue Sarcoma. Surgical Oncology Volume 27, Issue 4, December 2018, Pages 695-701.
- Hans van Houwelingen & Hein Putter (2011). Dynamic Prediction in Clinical Survival Analysis. Chapman & Hall. - H. C. van Houwelingen (2007). Dynamic Prediction by Landmarking in Event History Analysis. Scand. J. Stat. 34: 70{85.

Contact: francesca.ieva@polimi.it

Abstract

We are interested in thermal convection as described by the Rayleigh-B ?enard convection model. In this model the Navier-Stokes equations for the (divergence-free) velocity u with no-slip bound- ary conditions is coupled to an advection-diffusion equation for the temperature T with inhomo- geneous Dirichlet boundary conditions. The problem of understanding the (average) upward- heat-transport properties is of great interest for the applications and challenging for the rigorous analysis. We show how the PDE theory (in particular, regularity analysis) can contribute to the understanding of the scaling regimes for the heat transport. After reviewing the theory of Constantin& Doering ’99 we will present some recent results and discuss new challenges.

Abstract

It is a well-known fact in the theory of several complex variables that a function
is holomorphic if and only if it is holomorphic in each variable separately. This
result goes back to Hartogs. It is natural to consider a boundary version of Hartogs’ theorem. The general problem is to take a boundary function and ask if holomorphic extensions on some families of complex curves are enough to guarantee an extension which is holomorphic in all variables simultaneously.
We will talk about the known results on the subject and show some new results obtained in collaboration with M. Fassina and S. Pinton for the spiecial case of the unit ball in C^n.

Abstract

: Real-life applications often require the optimization of nonlinear functions with several unknowns or parameters - where the function is the result of highly expensive and complex model simulations involving noisy data (such as climate or financial models, chemical experiments), or the output of a black-box or legacy code, that prevent the numerical analyst from looking inside to find out or calculate problem information such as derivatives. Thus classical optimization algorithms, that use derivatives (steepest descent, Newton's methods) often fail or are entirely inapplicable in this context. Efficient derivative-free optimization algorithms have been developed in the last 15 years in response to these imperative practical requirements. As even approximate derivatives may be unavailable, these methods must explore the landscape differently and more creatively. In state of the art techniques, clouds of points are generated judiciously and sporadically updated to capture local geometries as inexpensively as possible; local function models around these points are built using techniques from approximation theory and carefully optimised over a local neighbourhood (a trust region) to give a better solution estimate.
In this talk, I will describe our implementations and improvements to state-of-the-art methods. In the context of the ubiquitous data fitting/least-squares applications, we have developed a simplified approach that is as efficient as state of the art in terms of budget use, while achieving better scalability. Furthermore, we substantially improved the robustness of derivative-free methods in the presence of noisy evaluations. Theoretical guarantees of these methods will also be provided. Finally, despite derivative-free optimisation methods being able to only provably find local optima, we illustrate that, due to their construction and applicability, these methods can offer a practical alternative to global optimisation solvers, with improved scalability. This work is joint with Lindon Roberts (Oxford), Katya Scheinberg (Lehigh), Jan Fiala (NAG Ltd) and Benjamin Marteau (NAG Ltd).

Contact: alfio.quarteroni@polimi.it

Abstract

In "Esquisse d'un programme" Grothendieck argues that general topology, which was developed for the needs of analysis, should be replaced by a "tame topology" if one wants to study the topological properties of natural geometric forms.
Such a tame topology has been developed by model theorists under the name "o-minimal structures". The goal of this lecture will be to explain in simple topological terms the notion of o-minimal structure and its applications in algebraic geometry, in particular for studying periods of algebraic varieties.

Abstract

It is a longstanding problem to show how the irreversible behaviour of macroscopic systems can be reconciled with the time-reversal invariance of these same systems when considered from a microscopic point of view. A theorem by Lanford shows that, under certain conditions, the famous Boltzmann equation, describing the irreversible behaviour of a dilute gas, can be obtained from the time-reversal invariant Hamiltonian equations of motion for the hard spheres model. This raises the question whether and how Lanford’s theorem succeeds in deriving this remarkable emergence of irreversibility. Many authors (Cercigniani, Illner & Pulverenti, 1994; Lebowitz 1983, Spohn 1991) have expressed very different views on this question. In this talk, I will argue that the theorem actually does not imply irreversibility at all.

Abstract

Since the pioneering works of A. Einstein, M. von Smoluchowski, and P. Langevin, stochastic processes and the related Fokker-Planck equations have been used to model irregular motion in different situation. These equations are also considered to be adequate for modelling the motion of one pedestrian and of a crowd of people under certain circumstances. Therefore they can play a central role in the design of control strategies for different purposes as well as in the analysis and calibration of microscopic models of pedestrian interaction. The purpose of this talk is to review some mathematical results in this field, which includes different modelling issues concerning the optimal control of a single pedestrian subject to perturbation, the development of a new framework for pedestrians' avoidance dynamics based on the formulation of a Nash game, and a new approach to the analysis of superresolution 2D images of moving molecules on a cell membrane to obtain quantitative results on the organization of the membrane and on the interaction between the molecules.

[1] M. Annunziato, A. Borzì, A Fokker-Planck control framework for stochastic systems, EMS Surveys In Mathematical Sciences, 5 (2018), 65-98.
[2] M. Annunziato, A. Borzì, A Fokker-Planck control framework for multidimensional stochastic
processes, Journal of Computational and Applied Mathematics, 237 (2013), 487-507.
[3] S. Roy, A. Annunziato, A. Borzì, C. Klingenberg, A Fokker-Planck approach to control collective motion, Computational Optimization and Applications, 69 (2018), 423-459.
[4] S. Roy, A. Annunziato, A. Borzì, A Fokker-Planck feedback control-constrained approach for modelling crowd motion, Journal of Computational and Theoretical Transport, 45 (2016), 442-458.
[5] S. Roy, A. Borzì, A. Habbal, Pedestrian motion modelled by Fokker-Planck Nash games, Royal Society open science, 4, (2017) 170648-17.

Contact: marco.verani@polimi.it

Abstract

This talk outlines a novel numerical approach for accurate and computationally efficient integrations of PDEs governing all-scale atmospheric dynamics. Such PDEs are not easy to handle, due to a huge disparity of spatial and temporal scales as well as a wide range of propagation speeds of natural phenomena captured by the equations. Moreover, atmospheric dynamics constitutes only a small perturbation about dominant balances that result from the Earth gravity, rotation, composition of its atmosphere and the energy input by the solar radiation. Maintaining this mean equilibrium, while accurately resolving the perturbations, conditions the design of atmospheric models and subjects their numerical procedures to stringent stability, accuracy and efficiency requirements.
The novel Finite-Volume Module of the Integrated Forecasting System (IFS) at ECMWF (hereafter IFS-FVM) solves perturbation forms of the fully compressible Euler/Navier-Stokes equations under gravity and rotation using non-oscillatory forward-in-time semi-implicit time stepping and finite-volume spatial discretisation. The IFS-FVM complements the established semi-implicit semi-Lagrangian pseudo-spectral IFS (IFS-ST) with the all-scale deep-atmosphere formulation cast in a generalised height-based vertical coordinate, fully conservative and monotone advection, flexible horizontal meshing and a predominantly local communication footprint. Yet, both dynamical cores can share the same quasi-uniform horizontal grid with co-located arrangement of variables, geospherical longitude-latitude coordinates and physics parametrisations, thus facilitating their synergetic relation.
The focus of the talk is on the mathematical/numerical formulation of the IFS-FVM with the emphasis on the design of semi-implicit integrators and the associated elliptic Helmholtz problem. Relevant benchmark results and comparisons with corresponding IFS-ST results attest that IFS-FVM offers highly competitive solution quality and computational performance.

Contact: luca.bonaventura@polimi.it