2,189 research outputs found
Numerical methods for multiscale inverse problems
We consider the inverse problem of determining the highly oscillatory
coefficient in partial differential equations of the form
from given
measurements of the solutions. Here, indicates the smallest
characteristic wavelength in the problem (). In addition to the
general difficulty of finding an inverse, the oscillatory nature of the forward
problem creates an additional challenge of multiscale modeling, which is hard
even for forward computations. The inverse problem in its full generality is
typically ill-posed and one common approach is to replace the original problem
with an effective parameter estimation problem. We will here include microscale
features directly in the inverse problem and avoid ill-posedness by assuming
that the microscale can be accurately represented by a low-dimensional
parametrization. The basis for our inversion will be a coupling of the
parametrization to analytic homogenization or a coupling to efficient
multiscale numerical methods when analytic homogenization is not available. We
will analyze the reduced problem, , by proving uniqueness of the inverse
in certain problem classes and by numerical examples and also include numerical
model examples for medical imaging, , and exploration seismology,
Sweeping Preconditioner for the Helmholtz Equation: Moving Perfectly Matched Layers
This paper introduces a new sweeping preconditioner for the iterative
solution of the variable coefficient Helmholtz equation in two and three
dimensions. The algorithms follow the general structure of constructing an
approximate factorization by eliminating the unknowns layer by layer
starting from an absorbing layer or boundary condition. The central idea of
this paper is to approximate the Schur complement matrices of the factorization
using moving perfectly matched layers (PMLs) introduced in the interior of the
domain. Applying each Schur complement matrix is equivalent to solving a
quasi-1D problem with a banded LU factorization in the 2D case and to solving a
quasi-2D problem with a multifrontal method in the 3D case. The resulting
preconditioner has linear application cost and the preconditioned iterative
solver converges in a number of iterations that is essentially indefinite of
the number of unknowns or the frequency. Numerical results are presented in
both two and three dimensions to demonstrate the efficiency of this new
preconditioner.Comment: 25 page
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