Mini-Courses

This short course presents some guidelines for writing a research paper.
We will start discussing how to organize and present research ideas.
We then analyze good and bad practices in benchmarking and results presentation with practical and interactive examples.
We discuss some common mistakes that might impact the results' meaningfulness and interpretability.
Last, we conclude by discussing how to guarantee the reproducibility of research results.

Physics Informed Neural network for Advanced modeling (PINA) is an open- source Python library capable of solving differential equations using artificial intelligence models. It is built on top of PyTorch with PyTorchLightning as backend and enables users to define their own problems and create models to easily compute differential equation solutions using Physics-based Neural Networks and Neural Operators (NOs).

The modular structure of PINA allows it to be tailored to specific user needs, providing the freedom to choose the most suitable learning techniques for their specific problem domain. Additionally, by exploiting the capabilities of the Lightning package, PINA can adapt to various hardware setups, including GPUs and TPUs. This adaptability makes PINA an excellent choice for implementing these methodologies in production and industrial pipelines, where computational efficiency and scalability are crucial.

Simulating computer models is of paramount importance in today's world. From climate modelling and weather forecasting to medical imaging and applications, mathematical models have revolutionised what can be achieved with computational budget at hand. Typically, these are subject to uncertainty arising from unknown model parameters which have to be estimated from (scarce) data, numerical errors arising from numerical approximations or round-off errors and incomplete knowledge of the system. Tracing the propagation of the aforementioned uncertainty through the system is crucial, but it frequently increases the computational complexity of the deterministic algorithms associated with the original problem. Hence, this mini-course is concerned with an efficient method for tackling this issue, namely the multilevel Monte Carlo method.

In many numerical applications, we often encounter phenomena for which we only have measurements at a set of equally spaced points. When using standard polynomial interpolation to approximate such phenomena, the results can be highly inaccurate due to the Runge phenomenon. Several techniques have been introduced to mitigate this issue, for example, the mock-Chebyshev subset interpolation and the constrained mock-Chebyshev least squares approximation. The excellent accuracy achieved by these approximations has led to their widespread use in various applications. Motivated by the success of these techniques in the classical polynomial interpolation, we aim to extend the mock-Chebyshev subset interpolation and the constrained mock-Chebyshev least squares approximation to the case of interpolation on segments. Specifically, we present three detailed generalizations of these methods in this context. The interpolation on segments is a mathematical technique used to approximate a function f over a specific interval I = [a, b]. It offers a distinct approach compared to classical polynomial interpolation. While the classical polynomial interpolation relies solely on function evaluations at specific points, the interpolation on segments leverages information about the integral of the function f over a set of subintervals of the interval I. This difference is crucial because the interpolation on segments only requires the function to be essentially bounded, a less restrictive condition than the continuity required for classical polynomial interpolation. We demonstrate that two of these three new methods achieve optimal growth rates for the Lebesgue constant of the corresponding Vandermonde matrix. Specifically, one method boasts logarithmic growth, while another exhibits growth between logarithmic and square-root. Finally, we compare the performance of these new approximation techniques through various numerical experiments.

• F. Dell’Accio, A. Guessab, and F. Nudo. “A new quadratic and cubic polynomial en- richment of the Crouzeix-Raviart finite element”. In: arXiv preprint arXiv:2403.05844 (2024).

• F. Nudo. “A general quadratic enrichment of the Crouzeix–Raviart finite element”. In: arXiv preprint arXiv:2403.11915 (2024).

• F. Nudo. “Two one-parameter families of nonconforming enrichments of the Crouzeix– Raviart finite element”. In: Applied Numerical Mathematics (2024).