Numerical Methods and Applications (NMA)

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.

Simulating non-local diffusion processes on graphs

Graphs are a important mathematical tool to characterize complex systems, such as social networks, transportation systems, or chemical reactions. One common use is to study how something spreads, or diffuses across the network over time, which can be described mathematically and solved efficiently with existing methods.

However, not all processes are purely local. Some involve long-range effects, where distant parts of the network influence each other. To capture this, it has been proposed fractional diffusion models, which extend the usual mathematical tools to allow more flexible types of spreading. These models have been applied to both simple and complex networks and can reveal behaviors, like faster-than-normal spreading, that standard models miss. This projects tackles the computational challenges of fractional dynamics, with an emphasis on efficiently solving the fractional diffusion equation on graphs. While classical method based on Krylov methods provide strong performance, probabilistic methods based on Monte Carlo stand out as a particularly promising alternative, especially due to their computational advantages as easily scalable algorithms suited for HPC for solving large networks.
 

Neuromorphic computing for solving differential equations

Neuromorphic computing is an emerging paradigm that designs hardware inspired by the brain’s architecture. Instead of sequential instructions as in traditional CPUs, or dense matrix operations as in GPUs, neuromorphic processors rely on spiking neural networks (SNNs), where information is transmitted through discrete electrical spikes.
These large networks of spiking neurons can carry out computations in parallel, making them well-suited for tasks that benefit from scalability and energy efficiency. Although much of the recent focus has been on machine learning, these systems can support a much wider range of applications. In this project, we use the event-driven and parallel structure of neuromorphic hardware to solve differential equations with a stochastic approach. The random walk nature of this stochastic approach is carried out directly within a spiking neural network by taking advantage of the natural stochastic behavior of neurons.

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 powerful and widely-used numerical technique for solving transport equations, particularly in fluid mechanics. Known for its unconditional stability, this method offers excellent phase speed accuracy with minimal numerical dispersion, making it ideal for stable integrations even with large timesteps and Courant-Friedrichs-Lewy (CFL) numbers greater than one.

One of the key strengths of SL schemes is their ability to be coupled with semi-implicit (SI) time discretization methods, effectively merging the advantages of both Lagrangian and Eulerian approaches. In SL schemes, advection is handled in a Lagrangian manner, while the transported fields are remapped onto a fixed grid at each timestep. This approach minimizes the grid distortion commonly associated with purely Lagrangian methods, enabling more accurate computations of physical processes over time.

In this project, we will explore the application of SL schemes for tracer transport, with applications to geophysical flows.

Mimetic Approximations for Divergence-Free and Curl-Free Vector Field Approximation

Approximating vector fields from scattered samples is a critical problem across numerous scientific disciplines, including fluid dynamics, electromagnetics, and computer graphics. Often, these vector fields must satisfy certain differential invariants that arise from fundamental physical principles. For example, in incompressible fluid dynamics, the fluid’s velocity field must be divergence-free (div-free) due to mass conservation. Similarly, in electromagnetics, the electric field is curl-free in the absence of a time-varying magnetic field, reflecting energy conservation.

To accurately enforce these differential properties in approximations, one cannot simply approximate the field components independently; instead, the components must be combined in a manner that respects the underlying mathematical structure. Approaches like Helmholtz-Hodge decomposition and custom vector basis functions have been developed to address this challenge.

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In this project, we will explore these techniques for vector field approximation, focusing on their ability to preserve physical invariants. The goal is to analyze their effectiveness in different scientific and engineering applications, assessing their utility in real-world scenarios.

Numerical Study of the Incompressible Stokes Problem 

This project investigates numerical solutions for the incompressible Stokes equations, which are essential for modeling slow-moving, viscous fluid flows at low Reynolds numbers. These equations are foundational in simulating fluid behavior in scenarios such as highly viscous environments and microfluidics.

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The focus of the project will be on developing and analyzing numerical methods to achieve stable and accurate solutions, with particular emphasis on fluid dynamics applications. By assessing and refining existing computational approaches, the goal is to enhance the simulation of incompressible flows, offering improvements relevant to both engineering and scientific applications.