### Overview

The numerical inverse scattering project is funded by a grant from ONR.
Our main emphasis of research is to construct fast algorithms for the
forward and inverse acoustic and electromagnetic scattering problems in
two and three dimensions, for applications such as medical imaging,
non-destructive testing, and the detection or classification of
underwater objects.
### Objectives

The primary objective of this research program is to (i) develop a direct,
high-order method for the forward scattering problem cast as the Helmholtz
equation (ii) use the fast algorithm for the forward problem to solve the
inverse problem iteratively with an inversion method introduced in [11].
This direct solver will be N fold faster than any iterative methods for
an N-by-N wavelength problem in two dimensions. The fast
forward solver will be required to perform efficient variational
calculations for inversion. More recently, we discovered that inversion
based on splitting of the scattering matrix [2,5] has potential to become
a most reliable and efficient method for inverse scattering.
### Approach

The mathematical foundations and machinery for the fast direct solver
for the forward problem can be found in [7]. The main theme of this
research involves a number of extremely technical details, such as
understanding the structures
of singular solutions of the Helmholtz equation, the design of
quadratures for functions with these singularities, the strategy for
sampling on interfaces of subdomains, and the compression of the interior
and exterior Green's formulae on the interfaces [3]. For inversion, an
algorithm for the rapid evaluation of the Frechet derivative for
linearization is developed [4] for the Lippman-Schwinger equation.
### Recent Publication

- B. Alpert and Y. Chen (2002) A representation of acoustic waves
in unbounded domains,
ps or
pdf version
Tech report NISTIR 6623, National Institute of
Standards and Technology, accepted by CPAM, 2004
- Y. Chen (2004) On splitting for inverse scattering problems of
the Helmholtz equation in two and three dimensions,
in preparation.
- Y. Chen, and S. Shim (2002) A discrete merging formula,
ps or
pdf version
Submitted to Advances in Computational Mathematics
- S. Y. Shim and Y. Chen (2002) A fast algorithm for variational
calculations of the Lippmann-Schwinger equation,
ps or
pdf version,
Submitted to Advances in Computational Mathematics
- S. Y. Shim and Y. Chen (2002) Least Squares Solution of Matrix Equation
A X B* + C Y D* = E,
ps or
pdf version,
SIAM J. Matrix Anal. Appl., vol. 24, No. 3,
802 - 808, 2003
- S. Y. Shim and Y. Chen (2002) Recursive Sherman-Morrison
Factorization for Scattering Calculations,
ps or
pdf version,
Submitted to SIAM J. of Scientific Computing, 2002
- Y. Chen, Fast direct solver for the Lippmann-Schwinger equation,
ps or
pdf version,
Advances in Computational Mathematics, vol. 16, pp. 175-190, 2002
- J. C. Aguilar, Yu Chen (2002) High-Order Corrected Trapezoidal
Quadrature Rules for Functions with a Logarithmic Singularity in 2-D,
ps or
pdf version,
Computers and Mathematics with Applications, vol. 44, No. 8--9,
pp. 1031-1039, 2002
- Y. Chen, and S. Shim (1999) Inverse scattering for lossy medium
via active material,
ps or
pdf version
(Tech report, Courant Institute, 1999)
- G. Bao, F. Ma, and Y. Chen, (1999) An error estimate for recursive
linearization of the inverse scattering problems,
ps or
pdf version
J. Math. Anal. Appl. 247 (1): 255-271 JUL 1 2000
- Y. Chen (1997) Inverse Scattering via Heisenberg's
Uncertainty Principle,
ps or
pdf version
(Inverse Problems, vol. 13, No. 2, 253-282)