Join me congratulating Distinguished Professor Terry Loring and Assistant Professor Jacob Schroder who have recently received NSF support for their proposals:
- Loring's "Numerical Methods in Noncommutative Matrix Analysis" is supported by the Analysis and Computational Mathematics programs. The project will develop algorithms that are useful for modern physics and quantum information. In particular, the methods developed will be important for modeling topological lasers which are applicable to photonic chips and quantum circuitry. These applications pose challenges to the computational science and in particular to the linear algebra algorithms so they can work with joint measurement in multivariable setting and incommensurate observables recognizing the theoretical limits that exist. The project will consider a variety of multivariable linear algebra algorithms, mainly those filling an immediate need in computational quantum physics and quantum information, but also those that can increase the speed and accuracy of computer shape analysis. The project will involve students and provide training in interdisciplinary projects.
This project will develop numerical methods for collections of matrices and operators arising in quantum physics and image analysis. This includes algorithms that work with finite-dimensional approximations to infinite-dimensional systems, leading to better computer modeling of quasicrystals, amorphous systems and periodic systems with defects. Methods and algorithms to be developed will be useful in the study of topological insulators, including periodically-driven systems. The project will study various forms of spectra, including variations of the local density of states, a standard tool in many areas of physics and chemistry. The anticipated work on K-theory is expected to produce new and better tools that can be used by theoretical physicists in numerical studies. At the core of these methods is the study of joint approximate eigenvectors.
- Schroder's "Collaborative Research: Parallel Space-Time Solvers for Systems of Partial Differential Equations" is supported by the Computational Mathematics program. Computer simulations and the mathematical methods supporting these are central to the modern study of engineering, biology, chemistry, physics, and other fields. Many simulations are computationally costly and require the large resources of modern supercomputers. New mathematical methods are urgently needed to efficiently utilize next generation supercomputers with millions to billions of processors. This project will develop new parallel-in-time algebraic multigrid methods for complex physical systems specifically designed for next generation computers. These new methods will add a new dimension of parallel scalability (time) and promise dramatically faster simulations in many important application areas, such as the gas and fluid dynamics problems considered (e.g., with relevance to wind turbines and viscoelastic flow). Graduate students will be involved and trained, and open source code will be developed.
This project will develop fast, parallel, and flexible space-time solvers for systems of partial differential equations (PDEs). The project will focus on algebraic multigrid (AMG) within block preconditioning traditionally appropriate for large adaptively refined spatial systems. These techniques will be extended to general space-time systems with a flexible approach that allows for adaptive space-time refinement. This adaptivity helps to accurately resolve lower dimensional features such as shocks at a fraction of the cost and storage of uniform refinement. Furthermore, the project will produce new practical AMG theory for non-SPD (symmetric positive definite) problems as well as solvers for adaptively refined space-time discretizations for a variety of parabolic and hyperbolic PDEs including the Euler and Navier-Stokes equations and Cahn-Hilliard system. The project will design, analyze, and tune parallel AMG solvers that are robust, efficient, and fast over a wide range of PDEs and parameters and will contribute to the widely used packages MFEM and hypre. The solvers will be developed and tested for applications in wind turbines, as well the high Weissenberg number problem in viscoelastic flows.