In this second talk I will address in details some of the challenges and applications mentioned in the first talk. Specifically, I will describe two techniques to tackle the unresolved treatment of nonlocal interfaces in the simulation of heterogeneous material and media. The first technique is based on the minimization of an energy principle and yields a well-posed and physically consistent nonlocal interface problem; the second is based on a new fractional model for anomalous diffusion with increased variability.
Then, I will describe a technique for optimal image denoising using nonlocal operators as filters. The optimal imaging problem is formulated as a bilevel optimization problem where the control variables are the denoising parameters. Several numerical results on benchmark images illustrate the applicability and improved accuracy of our approach.
If time allows, I will also present two recently developed machine-learning techniques for nonlocal model identification. These techniques are physics-informed, data-driven, tools that allow us to reconstruct model parameters from sparse observations. I will also show one- and two-dimensional numerical tests that illustrate robustness and accuracy of our approaches
01/10/2020
BIO
Dr. Marta D'Elia obtained her bachelor and master degree in Mathematical Engineering at Politecnico of Milano under the supervision of Prof. Quarteroni (Dec 2007). She obtained her PhD in Applied Mathematics at Emory University under the supervision of Prof. Veneziani (Dec 2011). Here, she worked on optimal control and data assimilation in CFD for cardiovascular applications. She was a postdoctoral fellow from Jan 2012 to Jan 2014 at Florida State University where she worked with Prof. Gunzburger on optimization for nonlocal and fractional models. In February 2014 she joined Sandia National Laboratories, where she works now as a Member of the Technical Staff in the Computational Science and Analysis group at the California site. Her interests include nonlocal modeling and simulation, optimization and optimal control, and scientific machine learning.