Overview:
My research interests span scientific computing, numerical analysis, and mathematical modeling of
physical phenomena. The lower part of this page provides pointers to
some of my work in each of these three areas.
The long term plan:
The overarching goal of my research is to develop numerical methods for
linear boundary value problems that perform far better than existing
techniques. In many environments, speed-ups of several orders of magnitude
and computational accuracy to ten digits or more can be achieved.
That such performance is in principle achievable by using integral equation
formulations has been known since the 1980's but such methods have not been
widely implemented. In my view, the core reasons for this are: (1) existing
fast solvers based on iterative methods are not robust enough for industrial
strength implementations, (2) methods for discretizing integral equations
in 3D are sorely lacking, and (3) a dearth of publicly accessible software.
My research over the last several years has been
focussed on developing fast direct solvers to overcome (1). As this
work is nearing its happy conclusion, it is time to start working on (2), and
to get serious about putting resources in place to address (3).
News:
Our work on randomized sampling techniques in linear algebra
is attracting interest. A comprehensive review of this work is given
here. A briefer treatment
can be found in
this
PNAS article.
|
Numerical Methods
My primary research interest concerns the development of fast
numerical methods for linear PDEs. Recent work has focussed on
methods based on integral equations;
introductions to such methods are given
here,
and
here.
The groundbreaking Fast Multipole Method, developed in the
early 1980s, has applications in fast solvers for PDEs, in
large particle simulations, in numerical linear algebra, etc.
The development of improved FMMs is an important part of my
research. This work is described at the
FMM resource page.
The FMM provides a tool for rapidly applying an operator
to a vector. More recently, we have developed methods for
rapidly inverting operators.
Such methods enable the development of fast
direct methods
for PDEs (as opposed to existing fast methods based on iterative
solvers). A paper,
some comments,
Work on randomized algorithms for the
approximation of matrices
is described here
and here. More at
Mark Tygert's webhome.
|
|
|
|
Applications
Recent work on modelling of ion channels and other problems
in computational biochemistry using the
techniques outlined in
this paper
is described
here[coming soon...].
Methods for solving problems defined on infinite, or large finite,
lattices are described
here. This work has applications
to the modelling of atomic crystals, large mechanical truss and frame
structures, and to random walks. A long-term to-do item for me is to
connect these methods to the fast direct solvers.
The new methods for operator compression and randomized sampling have
very interesting applications to the modelling
of heterogeneous media and the construction of scattering matrices.
The basic ideas are described
here,
and I hope to have several addition papers out soon.
Methods for engineering materials with phononic or photonic
bandgaps at prescribed locations
is described here.
|
|
|
|
Mathematical Analysis
Work on the development of lattice Greens functions (strictly
speaking, fundamental solutions) is described
here.
Some Korn-type inequalities for lattice materials
are given here. This paper also
gives a condition to distinguish between degenerate and non-degenerate infinite trusses.
Work on homogenization of periodic media is described
here. The quantitative estimates of
when to use micro-polar continuum models to treat mechanical frame structures are of particular
interest to me.
Old work on characterizing solutions near a singular point of
the eikonal equation (and related first order fully non-linear
PDEs) is described
here.
|
|