Numerical Methods and Applications (GMNA)

Physics-Informed Neural Networks (PINN) for Partial Differential Equations

Physics-Informed Neural Networks (PINNs) are a scientific machine learning technique used to solve problems involving Partial Differential Equations (PDEs). Essentially, PINNs approximate PDE solutions by training a neural network to minimize a loss function. This novel mesh-free methodology has emerged as a powerful technique to solve a wide range of PDE problems, including fractional equations, integral differential equations, and stochastic PDEs. In this project, we propose the study/development of some aspects of this approach.

Compressed sensing for imaging

Although the role of Mathematics is not much appreciated in the new imaging technologies used in modern medicine, Mathematics is at the heart of these technologies that provide detailed anatomical information in all different types of tissue. Examples include X-ray Computed Tomography (CT), Positron Emission Tomography (PET), Magnetic Resonance Imaging (MRI), as well as new photo and thermoacoustic imaging techniques. On the other hand, medical imaging data have an associated cost and, hence, it is desirable to minimize the amount of data needed for imaging. For this reason, the new ideas developed for Compressive Sensing, that is able to produce high quality images with less data, has attracted much interest in the biomedical community. In this project, we propose to explore these ideas that can help reduce the cost of imaging and, at the same time, can improve the quality of the produced images. 

Inverse problems

While direct problems consist in calculating the effects of known causes, an inverse problem is the process of calculating from a set of observations the causal factors that produced them. There are interesting inverse problems in medical imaging, geophysics, astronomy, machine learning, etc... each one with different characteristics. Also, there is a wide spectrum of mathematical techniques that are used to solve them. This project focuses on the analyses, study and development of the existing techniques.

Probabilistic algorithms for high performance computing (HPC)

Realistic mathematical applications are so computationally costly that they can only be tackled by supercomputers with hundreds or thousands of concurrent processors. Taking full advantage of the parallelisation scope of those machines requires highly scalable algorithms. In our group, a novel paradigm based on stochastic calculus, which is free from "Amdahl's curse", is currently being developed. This topic combines differential equations with stochastic numerics and parallel programming, and offers the chance of learning CUDA for programming NVidia GPUs.

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RBF methods for nonlinear elliptic problems

Many problems in applied maths, such as wave scattering or fluid-structure interaction, can be modeled by nonlinear elliptic equations. The RBF meshless method, owing to its unique advantages---such as ease of coding and geometric flexibility---furnishes a particularly suitable approach, and very hot, to solving those applications. 

Monte Carlo methods for financial mathematics

Monte Carlo methods have become the keystone of mathfin, ranging from relatively simple, yet high-dimensional problems such a Black-Scholes European option, to options over complex swing contracts that call for sophisticated stochastic optimal control. Besides option valuation, statistical calibration of stochastic models is an equally interesting area, and even quantifying the cost of the uncertainty borne by the calibrated parameters. Even the mathematical algorithms themselves are often of interest to the industry, since in this context time (to solution) is literally money. 

Semi-Lagrangian methods for transport equations with applications to geophysical flows

The semi-Lagrangian (SL) method is a widely used numerical technique for solving the transport equation in fluid mechanics. It is an unconditionally stable scheme which exhibits very good phase speeds and little numerical dispersion. The practical result of these theoretical properties is that it allows stable integrations with long timesteps, at CFL numbers much larger than unity. Furthermore, SL schemes can be coupled with semi-implicit (SI) time discretizations, combining the virtues of the Lagrangian and Eulerian approach in a single scheme: although a Lagrangian approach is used for advection, the transported fields are remapped at every timestep to the model grid. The undesirable side-effects of large grid deformation of the purely Lagrangian approach is therefore limited and physical processes can be accurately computed. In this project, we propose to explore the use of SL schemes for simple tracer transport with applications to geophysical flows.

Div-free and curl-free vector field approximation 

The problem of approximating vector fields from scattered samples arises in many scientific applications, including, for example, fluid dynamics, electromagnetics, and computer graphics. Often these vector fields have certain differential invariant properties related to an underlying physical principle. For example, in incompressible fluid dynamics the velocity of the fluid is divergence-free (div-free) as a consequence of the conservation of mass. Similarly, in electromagnetics the electric field is curl-free in the absence of a time varying magnetic field as a consequence of the conservation of energy. To enforce these differential invariants on the approximant, one cannot approximate the individual components of the field separately, but must combine them in a particular way. Various approaches exist for this purpose, such as Helmholtz-Hodge decompositions or tailored vector basis. In this project we aim to explore these ideas and analyze their use in different physical applications.