===firstname:
A.
===firstname3:
L. 
===affil6:

===lastname3:
Greengard
===email:
adrianna.gillman@rice.edu
===keyword_other2:

===lastname6:

===affil5:

===lastname4:

===lastname7:

===affil7:

===postal:
Rice University
Computational and Applied Mathematics
                6100 Main Street
                M.S. 134
                Houston, Texas  77005
===ABSTRACT:
The Hierarchical Poincar\'e-Steklov (HPS) method is a recently developed 
spectral discretization for partial differential equations that 
naturally comes with an efficient direct solver.  The HPS 
method has proved extremely effective for solving free space 
scattering problems with localized variable coefficients.  For example,
for a variable media problem where the support of the variation was 
100 wavelengths in size, the HPS method required 3.6 million unknowns to 
achieve 9 digits of accuracy with the construction of the solver taking 
six minutes on a desktop computer.  Each solve only required three seconds.  In this 
talk, we present the first extension of the HPS method to inverse problems.
The problem under consideration is that of reconstructing material properties 
from scattering measurements.  We assume data is recovered from 
many plane waves impinging the medium from multiple frequencies is given.  While
this problem is ill-posed and nonlinear, it is turned into a sequence of well-posed problems via a recursive linearization technique.
This approach requires the solution to a large number of forward scattering 
problems reaching well into the mid-frequency regime making the HPS method
the ideal PDE solver.


===affil3:
Courant Institute of Mathematical Sciences
New York University
And the Simons Foundation
===title:
Inverse medium scattering using the Hierarchical Poincar\'e-Steklov method

===affil2:
Courant Institute of Mathematical Sciences
New York University
===lastname2:
Borges
===firstname4:

===keyword1:
Inverse problems, regularization
===workshop:
no
===lastname:
Gillman
===firstname5:

===keyword2:
NOT_SPECIFIED
===otherauths:

===affil4:

===competition:
no
===firstname7:

===firstname6:

===keyword_other1:

===lastname5:

===affilother:

===firstname2:
C.