Applied Mathematics Research
I design and analyse numerical methods for multi-physics, multi-rate partial differential equations. In particular, I focus on Adjoint based A posteriori error estimation to quantify the error in a quantity of a interest in a numerical solution. My research is interdisplinary and I enjoy working with mathematicians, engineers and computer scientists. I actively work with Dr. Eric Cyr and Dr. John Shadid from Sandia National Labs and many of students do their internships at the national labs.
My recent interests include:
- Parallel-in-time integration
- Uncertainty quantification
- Stochastically perturbed domains
- Reduced order modeling
- Least-Squares finite element methods
- Poisson-Boltzmann and Poisson-Nernst-Planck equations
For more information, see my personal webpage.
Emeritus Professor
I'm an Applied Mathematician actively involved in Dynamical Systems, Probabilistic Methods, Stochastic Processes, Perturbation Theory, and Scientific Computing. I have worked on both fundamental and formal aspects of averaging methods in both deterministic and stochastic dynamical systems, but always with an eye for proofs of asymptotic validity. Our main current focus is on problems defined by mean field Vlasov, Vlasov Fokker-Planck and Vlasov-Maxwell systems. The applications I have worked on are related to Beam Dynamics in Modern Particle Accelerators and Particle Channeling in Crystals .
For example, we are investigating coherent radiation and space charge in the context of charged particle bunches evolving under the mean field Vlasov-Maxwell dynamics. This involves both analytic and numerical (e.g., meshless) approaches and the development of fast numerical algorithms in a high performance computing environment. In addition, we are beginning a study of the probabilistic Large Deviation Theory as a tool to study large N particle dynamics and the above mean field approximations.
My first area of research was classical general relativity, and in particular (quasilocal) notions of gravitational energy and momentum. Over the last 15 years, I've migrated towards solution of PDE via spectral-element methods, and have written a large code for solving the Einstein equations reduced by helical symmetry. This work remains unfinished, as I'm struggling with issues of boundary conditions and gauge. In short, the code solves the helically reduced Einstein equations in the harmonic (or de Donder) gauge. However, the boundary conditions I'm currently using violate this gauge, as do the computed solutions (so they are not, strictly speaking, solutions to the Einstein equations). Possibly through the use of spectral penalty methods, I hope to implement better boundary conditions soon.
The primary aim of my research program is to develop and analyze mathematical models and computational methods for problems at the intersection of fluid dynamics, biology, and transport in multi-phase media. Through mathematical models, my work elucidates the physical processes necessary for biological function. In my opinion, a model is only as good as the assumptions underpinning it and must respect the physics that impact the system of interest. Non-Newtonian rheology, immersed elastic structures, and electro-diffusion all play critical roles in the biological systems I study. A modeling approach that incorporates these non-linearities (or other complex effects) often results in mathematical equations which have not been previously studied. Analyzing such models requires the use of new or novel mathematical techniques. Consequently, my work typically requires the solution and analysis of partial differential equations via PDE theory, clever application of perturbation analysis, or the development and use of numerical methods.
At small scales, biological function is a consequence of the laws of physics, and the laws of physics are written in the language of mathematics. Individual cells control the transport of dissolved species in coordinated ways, respond to diffuse chemical signals, store chemical energy, and later expend it to perform mechanical work. Larger collections of cells work together to manipulate the distribution of ions within the body, secrete enzymes and other proteins, or generate the forces and deformations necessary to drive larger scale transport through organs. All of these processes are carried out in heterogeneous and dynamic environments. Mathematical modeling (particularly when used in collaboration with the work of experimentalists) is an essential tool that allows one to address important and complex questions about the physical principles underlying the function of living systems, often providing insights that are simply not accessible through experimental techniques. In my experience, such an approach leads to mathematical problems, analyses, and techniques that are interesting in their own right.
My areas of reseach are Applied Analysis and Scientific Computing. Specific problems that I have worked on lately include: the stability of shock and spiral waves; the modeling of ocean circulations for climate studies; the transition from smooth to rough attractors and Lyapunov-type numbers.
I am broadly interested in partial differential equations and their numerical approximation. Most of the applications I have worked on are related to fluids, but a recent interest is a nonlinear Black-Scholes equation in financial mathematics. The Black-Scholes equation is used extensively for option pricing; it becomes nonlinear if one assumes dependence of the volatility on the asset price.
My current research is a blend of operator theory, C*-algebras, K-theory, numerical algorithms, mostly aimed at better computer modeling of of topological phases such as topological metals, higher order topological insulators. Some of my research is in collaboration with the Nanostructure Physics department at the Center for Integrated Nanotechnologies, Sandia National Laboratories.
More information on my research is listed on the research page of my webpage.
My research interests include a wide range of topics in applied mathematics, nonlinear waves and theoretical physics. Among them are laser fusion and laser-plasma interaction; dynamics of fluids with free surface, Kelvin-Helmholtz instability and nonlinear interactions of surface waves; theory of the wave collapse, singularity formation and its application to plasma physics, hydrodynamics, biology and nonlinear optics; bacterial aggregation, chemotaxis, cell-cell interactions; collapse of bacterial colonies, stochastic Potts model of biological cell; pattern formation in photorefractive crystals and other nonlinear optical media; high-bit-rate optical communication; dispersion-managed optical fiber systems; soliton propagation in optical systems; high performance parallel simulations of optical fiber systems; Bose-Einstein condensation of ultracold dipolar gases.
My general research is in predictive computational science, a discipline concerned with assessing the predictability of mathematical and computational models of physical systems. It involves and bridges several fields and procedures such as computational mathematics, uncertainty quantification, and verification and validation. In the past few years, I have been working on the analysis and computation of partial differential equations with uncertain parameters described by stochastic and fuzzy functions. I work on both forward and inverse problems with applications to materials science, earthquake engineering, and high-frequency waves. I am currently developing new mathematical and computation models for the prediction of physical systems which are subject to hybrid random and non-random uncertainties.
I work in the area of computational fluid dynamics. In particular, I have studied vortex dynamics in separated flows, interfacial flows, both in inviscid flow and in Stokes flow, evolution of internal waves in flows with density gradients, and numerical methods to compute the fluid motion accurately.
For more information, please see my homepage https://math.unm.edu/~schroder/index.html
