45 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,
Fast wavelet based algorithms for linear evolution equations
A class was devised of fast wavelet based algorithms for linear evolution equations whose coefficients are time independent. The method draws on the work of Beylkin, Coifman, and Rokhlin which they applied to general Calderon-Zygmund type integral operators. A modification of their idea is applied to linear hyperbolic and parabolic equations, with spatially varying coefficients. A significant speedup over standard methods is obtained when applied to hyperbolic equations in one space dimension and parabolic equations in multidimensions
Homogenization Model for Aberrant Crypt Foci
Several explanations can be found in the literature about the origin of
colorectal cancer. There is however some agreement on the fact that the
carcinogenic process is a result of several genetic mutations of normal cells.
The colon epithelium is characterized by millions of invaginations, very small
cavities, called crypts, where most of the cellular activity occurs. It is
consensual in the medical community, that a potential first manifestation of
the carcinogenic process, observed in conventional colonoscopy images, is the
appearance of Aberrant Crypt Foci (ACF). These are clusters of abnormal crypts,
morphologically characterized by an atypical behavior of the cells that
populate the crypts. In this work an homogenization model is proposed, for
representing the cellular dynamics in the colon epithelium. The goal is to
simulate and predict, in silico, the spread and evolution of ACF, as it can be
observed in colonoscopy images. By assuming that the colon is an heterogeneous
media, exhibiting a periodic distribution of crypts, we start this work by
describing a periodic model, that represents the ACF cell-dynamics in a
two-dimensional setting. Then, homogenization techniques are applied to this
periodic model, to find a simpler model, whose solution symbolizes the averaged
behavior of ACF at the tissue level. Some theoretical results concerning the
existence of solution of the homogenized model are proven, applying a fixed
point theorem. Numerical results showing the convergence of the periodic model
to the homogenized model are presented.Comment: 26 pages, 4 figure