I am a computational mathematician at Sandia National Laboratories. My research interests include numerical analysis, applied mathematics and computational science, with particular focus on

• Compatible and alternative discretizations of PDEs, including mimetic, DG, and stabilized methods.
• Finite element methods of least-squares type
• Finite element methods for PDE constrained optimization
• Multiscale methods: atomistic-to-continuum coupling, domain bridging

Current Research projects

My current research is in the following areas:

Compatible discretizations

Discretization is a model reduction process which converts infinite dimensional mathematical models into finite dimensional algebraic equations that can be solved on a computer. Consequently, discretization is accompanied by inevitable losses of information which can adversely affect the predictiveness of the discrete models. Compatible and mimetic discretizations control these losses directly by using discrete fields and operators which inherit the key structural properties of their continuum counterparts. As a result, compatible discretizations transform partial differential equations into discrete algebraic problems that mimic fundamental properties of the continuum equations. I work on a common framework for mimetic discretizations using algebraic topology to guide the analysis. This framework provides the basis for the development of the Intrepid package.

Optimization-based modeling

This research is outgrowth of previous work on compatible discretizations and extends it in an important complementary direction. Compatible and regularized methods excel in controlling "structural" information losses responsible for the stability and well-posedness of the discrete equations. However, they encounter considerable difficulties in at least two cases: multi-physics models which combine constituent components with fundamentally different mathematical properties, and loss of "qualitative" properties such as maximum principles, monotonicity and local bounds preservation.

The main objective of this research is to develop a formal approach which uses optimization ideas to control externally information losses which are difficult (or impractical) to manage directly in the discretization process. Ultimately, the goal is to use optimization and control ideas in order to improve predictiveness of computational models, increase robustness and accuracy of solvers, and enable efficient reuse of code.

Software tools for compatible discretizations

Intrepid is a library of interoperable tools for compatible discretizations of Partial Differential Equations (PDEs). Design of Intrepid is motivated by the common framework for compatible discretizations. A key feature of the design philosophy is separation of cell topology from the reconstruction process. As a result, Intrepid can be easily extened to include new cell shapes and/or function spaces. Development of Intrepid started in 2007 and the package was publicly released in 2009 with Trilinos 10.0. Intrepid is part of the Discretization Capability Area (DCA) in Trilinos. The goal of DCA is to provide, over time, a collection of libraries and interfaces that enable rapid development of application codes for applications that require numerical solution of Partial Differential Equations (PDE).

Previous Research projects

Atomistic-to-continuum coupling

Atomistic-to-Continuum (AtC) coupling is a critical component in computational materials science and other applications of interest to the DOE Office of Science. Our goal is to understand and quantify mathematically the limits in AtC coupling and the resulting impact on multiscale simulations.

Multiscale methods

Part of this research deals with application of Variational Multiscale Analysis (VMS)to develop new finite element methods for PDEs. Another research direction is domain bridging where we look at novel Lagrange-multiplier formulations for mesh tying

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