We are pleased to announce that our colleagues Dr. Mihai Nechita and Rareș Răhăian are participating in the 11th Regional French-Romanian Summer School on Applied Mathematics, held in Sinaia, Romania, 8–16 July 2026.
As part of the scientific program, Dr. Mihai Nechita will deliver a mini-course entitled “Numerical Analysis for Ill-Posed PDEs: Stability, Finite Elements and Neural Networks”.
The course will present modern numerical methods for solving ill-posed partial differential equations (PDEs). While classical numerical analysis typically focuses on well-posed PDEs, many practical applications involve incomplete or inaccessible boundary data, making the underlying problems ill-posed and requiring regularization techniques. Beginning with conditional stability estimates for elliptic PDEs arising in unique continuation and Cauchy problems, the mini-course will introduce a framework for designing stabilized finite element methods that provide discrete regularization tailored to the problem and achieve proven quasi-optimal convergence rates. It will also explore neural network-based approaches, discussing how physics-informed neural networks (PINNs) can be used to approximate solutions and how variationally consistent loss functions can provide a posteriori error estimates.
We congratulate Mihai on the invitation to present this mini-course and wish both Mihai and Rareș a successful and inspiring summer school.