I'm interested in mathematical, numerical and computational treatment of Partial Differential Equations (PDE) that arise from various applications, PDE constrained optimization and recently in uncertainty quantification. My favorite numerical  scheme for discretizing PDEs, and the one I'm mostly experienced with, is the Galerkin finite element method. I've been also working on mesh-free methods, in particular on Smoothed Particle Hydrodynamics and Generalized Moving Least Squares.  I'm interested in the design and analysis of numerical methods and the mathematical analysis (existence/uniqueness results) of PDEs. I am the lead of the Trilinos Discretization product.

Recently I have been working on Scientific Machine Learning, with the goal of combining learning techniques with well estabilished numerical methods.

• Ice sheet dynamics: we develop a finite element multi-physics implementation (FELIX) of equations governing ice dynamics, including nonlinear Stokes-like equations for ice-flow and enthalpy equation for ice temperature . We also solve PDE constrained optimization to invert for unknown/uncertain parameters by minimizing the mismatch with observed data and climate forcings. Code is written in Albany applications and relies on several Trilinos packages.
• Meshless discretization: We aim at analysing and developing consistent and compatible meshless discretizations using generalized moving least square methods. We mainly target elliptic and transport equations and data-transfer problems.
• Intrepid2: We develop a finite element library that provides performant and portable high-order finite element discretizations spanning all the De Rham complex. The code is a refactoring and enhancement of the library Intrepid.
• Physics Informed Learning Machines: We investigate the role of deep learning in modeling (or enhancing current models for) scientific problems. Specifically I'm targeting ice sheet modeling. We are also aim at better understadning the approximation properties of neural networks.