Seminars

Speaker Affiliation: Ramapo College, USA.

In recent years due to the huge amount of time series data available in real time, there has been a constant interest by researchers and practitioners to develop models to describe and enhance the understanding of these data sets. The development of efficient models to correctly quantify and predict the sample paths of these kinds of time series is essential since it helps prevent losses or maximize profits in the field of financial modeling. Therefore understanding the behavior of stock prices will help investors and practitioners make informed financial decisions. In this talk, we will discuss a stochastic model arising on the superposition of Ornstein Uhlenbeck processes to describe the paths of a time series data. We will also estimate parameters that are useful for making inferences and predicting these types of events. We will test our approach and present numerical examples by using real financial stock market data.

Speaker affiliation: Western Norway University of Applied Science, Norway.

In this talk, we present an overview on applications of mathematics in relation to solving problems in the financial and oil industries. The concept of the Curse of Dimensionality will be used to explain why machine learning techniques are playing a key role for developing novel numerical techniques that overcome the curse of dimensionality while solving high dimensional problems.

Speaker affiliation: University of the Littoral Opal Coast, Calais, France

In this talk, we consider large-scale continuous-time differential Lyapunov and differential Riccati equations having low rank right-hand sides. These equations appear in many problems such in control theory for finite horison or in model reduction for large scale time-dependent dynamical systems. (DRE) and (DLE) are generally solved by Backward Differentiation Formula (BDF) (or Rosenbrock) methods leading to large scale algebraic Lyapunov or Riccati equation which has to be solved for each timestep. However, these techniques are not effective for large problems because, at each timestep, one has to solve large scale algebraic Lyapunov or Riccati equations which could be very expensive. Here, we propose new approaches based on projection on small subspaces. For differential Lyapunov equations, we construct approximate solutions from the exponential expression of the exact solution using Krylov subspace methods to approximate exponential of a matrix times a block of vectors. For differential Riccati equations, we project the problem onto a small block Krylov or extended block Krylov subspace and then and obtain a low-dimentional differential algebraic Riccati equation. The latter matrix differential problem is solved by Backward Differentiation Formula (BDF) method and the obtained solution is used to reconstruct an approximate solution of the original problem. We give some theoretical results and simple expressions of the residual norms allowing the implementation of a stop test in order to limit the dimension of the projection spaces. Uppers bounds for the norm of the errors are also given. The proposed numerical experiments show the effectiveness of our approaches.

Speaker affiliation: Norwegian University of Science and Technology, Norway.

Many advanced engineering problems require the numerical solution of multidomain, multidimension, multiphysics and multimaterial problems with interfaces. When the interface geometry is highly complex or evolving in time, the generation of conforming meshes may become prohibitively expensive, thereby severely limiting the scope of conventional discretization methods.

For instance, the simulation of blood flow dynamics in vessel geometries requires a series of highly non-trivial steps to generate a high quality,full 3D finite element mesh from biomedical image data. Similar challenging and computationally costly preprocessing steps are required to transform geological image data into conforming domain discretizations which respect

complex structures such as faults and large scale networks of fractures. Even if an initial mesh is provided, the geometry of the model domain might change substantially in the course of the simulation, as in, e.g., fluid-structure interaction and free surface flow problems, rendering even recent algorithms for moving meshes infeasible. Similar challenges arise in more elaborated optimization problems, e.g. when the shape of the problem domain is subject to the optimization process and the optimization procedure must solve a series of forward problems for different geometric configuration.

In this talk, we focus on recent unfitted finite element technologies as one possible remedy. The main idea is to design discretization methods which allow for flexible representations of complex or rapidly changing geometries by decomposing the computational domain into several, possibly overlapping domains. Alternatively, complex geometries only described by some surface representation can easily be embedded into a structured background mesh. In the first part of this talk, we briefly review how finite element schemes on cut and composite meshes can be designed by either using a Nitsche-type imposition of interface and boundary conditions or, alternatively, a partition of unity approach. Some theoretical and implementational challenges and their rectifications are highlighted. In the second part we demonstrate how unfitted finite element techniques can be employed to address various challenges from mesh generation to fluid-structure interaction problems, solving PDE systems on embedded manifolds of arbitrary co-dimension and PDE systems posed on and coupled through domains of different topological dimensionality.

Speaker affiliation: University of Lille, France

In numerical analysis, Cholesky is well known for his direct method for solving a system of linear equations with a symmetric positive definite matrix. In this talk, I will first describe the life of Cholesky. Then I will review his works, and, in particular, his method for linear systems. I will explain why Cholesky was interested in this problem, how his unpublished paper on it has been discovered and I will analyze it. Other historical remarks will conclude the talk.

Speaker affiliation: University of L'Aquila, Italy.

In this talk we present recent advances in the numerical preservation of the qualitative behaviour characterizing the underlying dynamics of various evolutive problems. This approach, well known in the literature as geometric numerical integration, is here applied to both deterministic and stochastic problems, with a rigorous theoretical investigation matched with a proper experimental one confirming the effectiveness of the introduced methodologies.

We first address deterministic and stochastic Hamiltonian problems, numerically handled in order to obtain a long-term energy conservation. Specifically, concerning stochastic Hamiltonian problems, a structure-preserving framework aims to retain the known long-term properties of the expected Hamiltonian. Specifically, we study the behaviour of stochastic Runge-Kutta methods arising as stochastic perturbation of symplectic Runge-Kutta methods. The analysis is provided through ɛ-expansions of the solutions (where ɛ is the amplitude of the stochastic fluctuation) and shows the presence of secular terms destroying the long-term preservation of the expected Hamiltonian. Then, an energy-preserving scheme is developed and analyzed.

We finally consider the nonlinear stability properties of stochastic θ-methods with respect to nonlinear test problems such that the mean-square deviation between two solutions exponentially decays, i.e., a mean-square contractive behaviour is visible along the stochastic dynamics. We aim to make the same property visible also along the numerical discretization via stochastic θ-methods: this issue is translated into sharp stepsize restrictions depending on some parameters of the problem, accurately estimated. A selection of numerical tests confirming the effectiveness of the analysis and its sharpness is also provided.

Speaker affiliation: Temple University, USA.

Asynchronous methods refer to parallel iterative procedures where each process performs its task without waiting for other processes to be completed, i.e., with whatever information it has locally available and with no synchronizations with other processes. For the numerical solution of a general linear partial differential equation on a domain, Schwarz iterative methods use a decomposition of the domain into two or more (possibly overlapping) subdomains. In essence one is introducing new artificial boundary conditions on the interfaces between these subdomains. In the classical formulation, these artificial boundary conditions are of Dirichlet type. Given an initial approximation, the method progresses by solving for the PDE restricted to each subdomain using as boundary data on the artificial interfaces the values of the solution on the neighboring subdomain from the previous step. This procedure is inherently parallel, since the (approximate) solutions on each subdomain can be performed by a different processor. In the case of optimized Schwarz, the boundary conditions on the artificial interfaces are of Robin or mixed type. In this way one can optimize the Robin parameter(s) and obtain a very fast method.

Instead of using this method as a preconditioner, we use it as a solver, thus avoiding the pitfall of synchronization required by the inner products. In this talk, an asynchronous version of the optimized Schwarz method is presented for the solution of differential equations on a parallel computational environment. A coarse grid correction is added and one obtains a scalable method. Several theorems show convergence for particular situations.

Numerical results are presented on large three-dimensional problems illustrating the efficiency of the proposed asynchronous parallel implementation of the method. The main application shown is the calculation of the gravitational potential in the area around the Chicxulub crater, in Yucatan, where an asteroid is believed to have landed 66 million years ago contributing to the extinction of the dinosaurs.

Speaker affiliation: University of Nottingham & Imperial College London.

Hamilton-Jacobi PDEs are a central object in optimal control and differential games, enabling the computation of controls in feedback form. High-dimensional HJ PDEs naturally arise in the feedback synthesis for high-dimensional control systems, and their numerical solution must be sought outside the framework provided by standard grid-based discretizations. In this talk, I will discuss two novel computational methods for the approximation of high-dimensional HJ PDEs. In the first part of the talk, I will present a numerical method based on tensor decompositions. Such a compressed representation of the value function has a complexity that scales linearly with respect to the dimension of the control system, allowing the solution of control problems with over 100 states. In the second part of the talk, I will discuss the construction of a class of causality-free, data-driven methods which circumvent the numerical solution of the HJ PDE. I will address the generation of a synthetic dataset based on the use of representation formulas (such as Lax-Hopf or Pontryagin’s Maximum Principle), which is then fed into a high-dimensional sparse polynomial/ANN model for training. The use of representation formulas providing gradient information is fundamental to increase the data efficiency of the method. I will present applications in nonlinear dynamics, control of PDEs, and agent-based models.