Phase III: Probabilistic representations for boundary value problems

Period: 01.01.2022-31.08.2022.

Activity. The research conducted during the third (last) phase of this project was devoted to the study of boundary value problems in connection with probability representations for their solutions. The results are oriented in two directions: On the one hand we use probabilistic solutions for non-homogeneous Poisson problems with Holder continuous data to obtain numerical solvers that approximate (at a prescribed error) the exact solution with high probability, whilst the computational complexity of such approximation scales remarkably good with respect to the dimension of the space, the given error and the desired confidence. Most importantly, the curse of dimensionality is overcome. As a byproduct, we show that ReLU deep neural networks can also be used to solve such problems at a given error, and their sizes scale at most polynomially with respect to the dimension as well as the prescribed the error.

On the other hand, we continued our work from the previous phase of this project concerning the inverse Cauchy problem in anisotropic environments. More precisely, in contrast to the previous work, this time we tackle the inverse problem under the context of discontinuous (L^2 in fact) unknown boundary data as well ass irregular diffusion coefficients. We clarified the right notions of solutions to the corresponding direct problems and we showed that they are well-posed. Then, we derived numerical algorithms for which we showed theoretical and numerical convergence properties.

Detailed report of the entire project Phase I, II, and III (in Romanian):