2024/08/15 by Vincenzo Mottola, Antonio Corbo Esposito, Mottola, Vincenzo +9 · 2 citations
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #Analysis of PDEs (math.AP) #FOS: Electrical engineering #FOS: Mathematics #Numerical methods in inverse problems #Optimization and Variational Analysis #Signal Processing (eess.SP) #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.2408.08040
openalex publication_date 2024/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Monotonocity Principle (MP), stating a monotonic relationship between a material property and a proper corresponding boundary operator, is attracting great interest in the field of inverse problems, because of its fundamental role in developing real time imaging methods. Moreover, under quite general assumptions, a MP for elliptic PDEs with nonlinear coefficients has been established. This MP provided the basis for introducing a new imaging method to deal with the inverse obstacle problem, in the presence of nonlinear anomalies. This constitutes a relevant novelty because there is a general lack of quantitative and physic based imaging method, when nonlinearities are present. The introduction of a MP based imaging method poses a set of fundamental questions regarding the performance of the method in the presence of noise. The main contribution of this work is focused on theoretical aspects and consists in proving that (i) the imaging method is stable and robust with respect to the noise, (ii) the reconstruction approaches monotonically to a well-defined limit, as the noise level approaches to zero, and that (iii) the limit contains the unknown set and is contained in the outer boundary of the unknown set. Results (i) and (ii) come directly from the Monotonicity Principle, while results (iii) requires to prove the so-called Converse of the Monotonicity Principle, a theoretical sults of fundamental relevance to evaluate the ideal (noise-free) performances of the imaging method. The results are provided in a quite general setting for Calderòn problem, and proved for three wide classes where the nonlinearity of the anomaly can be either bounded from infinity and zero, or bounded from zero only, or bounded by infinity only. These classes of constitutive relationships cover the wide majority of cases encountered in applications.