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:

Topefficiency multigrid methods for steadystate fluid dynamics at all Mach and
Reynolds numbers, and other nonelliptic stationary PDE systems.

Multilevel approaches to timedependent partialdifferential equations, emphasizing
applications to oceanic and atmospheric flows.

Grid adaptation techniques for bounded and unbounded domains, exploiting multigrid
structures and resulting in a oneshot solveradaptor.

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

Optimal control: Feedback control via very fast updating of openloop solutions, based
on their multiscale representations.

Topefficiency multilevel algorithms for highly indefinite (e.g., standing wave)
problems, with ray (geometricaloptics) equations at the limit of very coarse grids.

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

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.

Multiscale MonteCarlo algorithms for eliminating both the critical slowing down and
the volume factor in increasingly advanced models of statistical physics, including
nonequilibrium models.

Electronic structures: Oneshot (no selfconsistency iterations) highly
accurate multigrid solver for allelectron (no pseudo potentials) KohnSham
equations.

Multigrid MonteCarlo approaches for solving the highdimensional (severalparticle)
Schrodinger equation by realtime path integrals.

Introducing multiscale computations to manyparticle (macromolecule or
manysmallmolecule) 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.

Multigrid methods for integral transforms and integrodifferential equations,
on adaptable grids, with applications to tribology.

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

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

Rigorous quantitative theory for predicting the performance of multigrid solvers.

Generalizing Algebraic Multigrid (AMG) fast solvers for systems of linear equations:
relaxationbased measurement of algebraic distance between variable and
leastsquare derivation of interpolation; multiple interpolation for highly
indefinite systems.

AMGinspired methods for graph problems: multiscale distances in a graph: fast
graph ordering, clustering and partitioning.

Systematic Upscaling (SU): Accurate derivation of increasingly largerscale
processing of systems, starting from their fundamental finescale equations.