Maarten de Hoop
Director, GMIG
mdehoop@purdue.edu
Dept of Mathematics
Purdue University
150 N Univ St
West Lafayette, IN 47907
ph. (765) 496-7678
Fax (765) 496-1169
Wave Packets: Nonlinear Approximation, Compression, Sparse Sampling
We develop new, multi-scale
approaches to nonlinear approximation, compression, sparse sampling
and reconstruction in the context of data acquisition and inverse
problems within the field of applied harmonic analysis. Results of our
research program include:
We designed almost symmetric wave packets - based on the dyadic
parabolic decomposition of phase space, like curvelets - and
developed an associated fast discrete (linear) transform, inverse
(that is, adjoint) transform pair formulated in higher
dimensions.
We used almost symmetric wave packets to construct approximate, localized solutions, and weak solutions of the wave equation. In media of limited smoothness we obtained a concentration of wave packets result.
In the nonlinear approximations and compression of functions by sums
of wave packets, one should place the wave packets on the wavefront
set of the function considered, thus not only achieving compression
but also enabling to infer geometric information using a finite
range of scales. To achieve this, we have shown it to be necessary
to have efficient non-linear algorithms for the approximation of a
function (defined in the frequency/wavenumber domain) by sparse sums
of exponentials. Designing such algorithms has its roots in the deep
results of Adamyan, Arov and Krein (AAK). We developed a
connection between elements of AAK theory and nonlinear
approximations of functions by sums of wave packets. We provided new
robust algorithms such that, given a function to be approximated and
a desired accuracy, the smallest number of wave packets needed to
obtain the given accuracy can be calculated using the singular value
decomposition of a certain block Hankel matrix.
We developed a Gaussians counterpart of our discrete almost
symmetric wave packets following the parabolic scaling, with
distinct applications.
We developed exploratory data analysis methods for uncertainty quantification of ill-posed inverse problems. These methods can be used to reveal the presence of systematic errors such as bias and discretization effects, or to validate assumptions made on the statistical model used in the analysis. The methods include: Bounds on the performance of randomized estimators of a large matrix, confidence intervals and bounds for the bias, resampling methods for model validation, and construction of training sets of functions with controlled pointwise regularity.
We used almost symmetric wave packets to construct approximate, localized solutions, and weak solutions of the wave equation. In media of limited smoothness we obtained a concentration of wave packets result.
We developed exploratory data analysis methods for uncertainty quantification of ill-posed inverse problems. These methods can be used to reveal the presence of systematic errors such as bias and discretization effects, or to validate assumptions made on the statistical model used in the analysis. The methods include: Bounds on the performance of randomized estimators of a large matrix, confidence intervals and bounds for the bias, resampling methods for model validation, and construction of training sets of functions with controlled pointwise regularity.
© 2013 Purdue University
An equal access/equal opportunity university
Copyright Complaints
Geo-Mathematical Imaging Group, Purdue University
150 N University Street, West Lafayette, IN 47907 USA Phone: (765) 496-7678 - Fax: (765) 496-1169
If you have trouble accessing this page because of a disability, please contact webmaster at:
bjd@purdue.edu