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.


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.


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

  1. 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
  2. Y. Chen (2004) On splitting for inverse scattering problems of the Helmholtz equation in two and three dimensions, in preparation.
  3. Y. Chen, and S. Shim (2002) A discrete merging formula, ps or pdf version Submitted to Advances in Computational Mathematics
  4. 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
  5. 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
  6. 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
  7. Y. Chen, Fast direct solver for the Lippmann-Schwinger equation, ps or pdf version, Advances in Computational Mathematics, vol. 16, pp. 175-190, 2002
  8. 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
  9. Y. Chen, and S. Shim (1999) Inverse scattering for lossy medium via active material, ps or pdf version (Tech report, Courant Institute, 1999)
  10. 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
  11. Y. Chen (1997) Inverse Scattering via Heisenberg's Uncertainty Principle, ps or pdf version (Inverse Problems, vol. 13, No. 2, 253-282)