Research

PhD in Computer Sciences at CIMAT, México. My research area focuses on Uncertainty Quantification techniques to address the Inverse Scattering problem in a Bayesian formulation. Driving aspects of my research concerns input representation, surrogate models, numerical methods and effective dimension.

Bayesian Inverse Gravimetry

Current work involves addressing inverse Gravimetry problems, that is, shape reconstruction of source bodies, from measurements of local gravitational field (anomaly) on a surface, within Bayesian framework.

Geophysical problems offer numerical challenges since they have small scale data magnitude against large scale dimension. Fast and accurate numerical procedures, together with proper unknowns representation are mandatory for suitable Bayesian approach.

Research manuscripts on regard this work are in progress.

B-spline control points MCMC

We are currently working on developing an affine invariant MCMC algorithm dedicated to shape reconstruction problems.

Polygonal representations for shaped, within a Bayesian framework, requires to establish a correlation model for the corners of the polygon. Our contribution consists in posing the control points of an interpolating B-spline as independent random variables uniformly distributed in such a way that the correlation model is left to the interpolation process.

The source code is on development.

Affine invariant MCMC on a Point Cloud

We develop a MCMC that is invariant to affine transformations of the parameters space over a surrogate model that consists on a Point Cloud moving in a domain. Such a Point Cloud is used to generate polygonal shapes whose edges are interpolated in order to produce proposal solutions of the inverse scattering problem.

Affine invariant property of the MCMC allows to estimate the boundary of the true scatterer object regardless its position or scale.

More details can be found in the following manuscript

Palafox, A., Capistrán, M., Christen, J. A.,

"Point cloud based scatterer approximation and affine invariant sampling in the inverse scattering problem",

Accepted on Mathematical Methods in the Applied Sciences.

[pdf]

Effective Dimension

In this paper, we address the inverse scattering problem from a Bayesian approach. We pose a representation for the scatterer object in Fourier series and a MCMC sampler (namely the t-walk) is used to sample from the posterior distribution.

The main contribution of this work is to provide a way to determine the efficient number of Fourier coefficients that can be retrieved from a noisy dataset. This is done by considering different number of terms on the Fourier series expansion as different statistical models and then, the best model is obtained via Bayesian model selection.

More details can be found in the publication below.

Palafox, A., Capistrán, M., Christen, J. A.,

"Effective Parameter Dimension via Bayesian Model Selection in the Inverse Acoustic Scattering Problem"

Mathematical Problems in Engineering, 2014(Recent Theory and

Applications on Inverse Problems RTAIP14), 2014.

[pdf]

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