Center for Computing Research (CCR)
Center for Computing Research
Ignacio Tomas
Ignacio Tomas Computational Mathematics Email: itomas@sandia.gov Phone: 505/8442667 Mailing address: Sandia National Laboratories P.O. Box 5800, MS 1320 Albuquerque, NM 871851320 
My main interests are PDEs and their respective discretization using methods that preserve essential “structural stability properties”, in particular: energy estimates, entropy inequalities, invariant sets, maximum/minimum principles, asymptotic limits, and pointwise positivity properties of the original PDE. One of my core interest lies in PDEs that preserve two or more of such notions of stability, in particular, when energystability is either nonapplicable or merely a complementary notion of stability. The vast majority of PDE models having actual physical/technical application fall in such category.
Influenced by my work within a research group with strong focus on multifluid plasma models, I am particularly interested in Eulerlike systems such as EulerMaxwell and EulerPoisson. Euler systems (in the broadest sense possible) are pervasive in mathematical physics: from selfgravitation, relativistic particle beams, to quantum mechanics. For instance the SchrodingerPoisson system can be rewritten as the QuantumEulerPoisson system using Madelung's transform, leading to an entire set of applications that goes well beyond traditional continuum physics and shockhydrodynamics. Systems such as EulerMaxwell and EulerPoisson do not have an evident (or explicit) variational structure that we can exploit numerically: the operator in space satisfies neither positivity, nor coercivity, nor Garding inequality, nor infsup inequality, nor related dissipative property. Boundedness innorm cannot be expected to hold true in such context and L2minimizers of the PDEresidual are unlikely to characterize the right notion of solution. This lack of an explicit variational structure is, to large extent, an early indicator that Galerkin techniques are unlikely to yield an outcome of long lasting value. This forces me to think about numerical schemes in terms of PDE properties rather than fabricated notions of stability.
The departure from the traditional energylike framework (Galerkin methods) is by no means a novel idea. Such change of priorities has already taken place in the PDE literature many decades ago with the introduction of entropysolutions and viscositysolutions in various settings. In this regard, Ansgar Jungel's book "Entropy Methods for Diffusive Partial Differential Equations" is particularly interesting, showing that in general we have consider a lot more than just energystability, even for PDEs of dissipative/parabolic nature. The challenge lies in the development of finite dimensional counterparts of such properties that can effectively be materialized as computer code.
Education/Background
Visiting Assistant Professor, 20152018
Department of Mathematics,
Texas A&M University, College Station.
Advisor/s: JeanLuc Guermond, Bojan Popov.
Ph.D. in Applied Mathematics, 2015.
University of Maryland College Park (UMCP).
Thesis: Ferrofluids: modeling, numerical analysis, and scientific computation.
Advisor: Prof. Ricardo H. Nochetto.
M.Sc. in Applied Mathematics, 2012.
University of Maryland College Park (UMCP).
Advisor: Prof. Ricardo H. Nochetto.
Mechanical Engineer, 2005.
Universidad Nacional de Mar del Plata (UNMdP), Argentina.
Selected Publications & Presentations
2020 

2019 

2018 
