Work Description

My main goal is solving nature's equations. In most branches of physics, chemistry and engineering, the fundamental laws of the investigated system are well established, stating the relations that always hold between its microscopic parts. Yet, to derive from these laws the macroscopic behavior of the system at given surrounding conditions is usually a formidable computational task, even with modern supercomputers. The inherent inefficiency of existing computational methods is the major bottleneck in many fields of study. It bars scientists from the theoretical derivation of, for example, the mass of the photon and other properties of elementary particles. It defeats attempts to compute the structure and interactions of chemical compounds, needed to understand and design materials, proteins, drugs, etc. This inefficiency also hampers calulations in fluid dynamics, medical imaging, radar analysis, astrophysics, weather prediction, oil prospecting, lubrication theory, acoustics, image processing, and so on. New mathematical methods to eliminate the inefficiency and to drastically reduce the complexity of all these computational tasks are being developed, based on hierarchical approaches to the organization of space and time. See details in the 2000 survey and the systematic upscaling paper.

Current Projects:

  1. Top-efficiency multigrid methods for steady-state fluid dynamics at all Mach and Reynolds numbers, and other non-elliptic stationary PDE systems.

  2. Multilevel approaches to time-dependent partial-differential equations, emphasizing applications to oceanic and atmospheric flows.

  3. Grid adaptation techniques for bounded and unbounded domains, exploiting multigrid structures and resulting in a one-shot solver-adaptor.

  4. Direct multigrid solvers for inverse problems, including system identification (e.g., impedance tomography) and data assimilation (in atmospheric simulations).

  5. Optimal control: Feedback control via very fast updating of open-loop solutions, based on their multiscale representations.

  6. Top-efficiency multilevel algorithms for highly indefinite (e.g., standing wave) problems, with ray (geometrical-optics) equations at the limit of very coarse grids.

  7. Multigrid solvers for the Dirac equations arising in quantum field theory.

  8. Compact multiresolution representation of the inverse matrix of a discretized differential operator; fast updating of the inverse matrix and of the value of the determinant upon changing an arbitrary term in the matrix itself; with application to the QCD fermionic interaction.

  9. Collective multiscale representation and fast calculation of many eigenfunctions of a differential operator. Fast expansion in terms of the eigenfunctions of a general differential operator.

  10. Multiscale Monte-Carlo algorithms for eliminating both the critical slowing down and the volume factor in increasingly advanced models of statistical physics, including non-equilibrium models.

  11. Electronic structures: One-shot (no self-consistency iterations) highly accurate multigrid solver for all-electron (no pseudo potentials) Kohn-Sham equations.

  12. Multigrid Monte-Carlo approaches for solving the high-dimensional (several-particle) Schrodinger equation by real-time path integrals.

  13. Introducing multiscale computations to many-particle (macromolecule or many-small-molecule) calculations, including fast evaluation of forces, fast convergence to local and global ground states, fast Monte Carlo simulations and large time steps, with applications to molecular mechanics.

  14. Multigrid methods for integral transforms and integro-differential equations, on adaptable grids, with applications to tribology.

  15. Multiscale methods for the fast evaluation and inversion of the Radon transform and other line-integral transforms; applications to medical tomography (CT, MRI, PET and SPECT) and airplane and satellite radar reconstruction.

  16. Multiscale algorithms for early vision tasks such as surface reconstruction, edge and fiber detection, segmentation, and meaningful picture coarsening.

  17. Rigorous quantitative theory for predicting the performance of multigrid solvers.

  18. Generalizing Algebraic Multigrid (AMG) fast solvers for systems of linear equations: relaxation-based measurement of algebraic distance between variable and least-square derivation of interpolation; multiple interpolation for highly indefinite systems.

  19. AMG-inspired methods for graph problems: multiscale distances in a graph: fast graph ordering, clustering and partitioning.

  20. Systematic Upscaling (SU): Accurate derivation of increasingly larger-scale processing of systems, starting from their fundamental fine-scale equations.

* * Prof. Brandt's home page * * Faculty of Mathematics home page