5 research outputs found
A Structurally Informed Data Assimilation Approach for Nonlinear Partial Differential Equations
Ensemble transform Kalman filtering (ETKF) data assimilation is often used to
combine available observations with numerical simulations to obtain
statistically accurate and reliable state representations in dynamical systems.
However, it is well known that the commonly used Gaussian distribution
assumption introduces biases for state variables that admit discontinuous
profiles, which are prevalent in nonlinear partial differential equations. This
investigation designs a new structurally informed non-Gaussian prior that
exploits statistical information from the simulated state variables. In
particular, we construct a new weighting matrix based on the second moment of
the gradient information of the state variable to replace the prior covariance
matrix used for model/data compromise in the ETKF data assimilation framework.
We further adapt our weighting matrix to include information in discontinuity
regions via a clustering technique. Our numerical experiments demonstrate that
this new approach yields more accurate estimates than those obtained using ETKF
on shallow water equations, even when ETKF is enhanced with inflation and
localization techniques
Generalized Sparse Bayesian Learning and Application to Image Reconstruction
Image reconstruction based on indirect, noisy, or incomplete data remains an important yet challenging task. While methods such as compressive sensing have demonstrated high-resolution image recovery in various settings, there remain issues of robustness due to parameter tuning. Moreover, since the recovery is limited to a point estimate, it is impossible to quantify the uncertainty, which is often desirable. Due to these inherent limitations, a sparse Bayesian learning approach is sometimes adopted to recover a posterior distribution of the unknown. Sparse Bayesian learning assumes that some linear transformation of the unknown is sparse. However, most of the methods developed are tailored to specific problems, with particular forward models and priors. Here, we present a generalized approach to sparse Bayesian learning. It has the advantage that it can be used for various types of data acquisitions and prior information. Some preliminary results on image reconstruction/recovery indicate its potential use for denoising, deblurring, and magnetic resonance imaging
Weighted inhomogeneous regularization for inverse problems with indirect and incomplete measurement data
Regularization promotes well-posedness in solving an inverse problem with
incomplete measurement data. The regularization term is typically designed
based on a priori characterization of the unknown signal, such as sparsity or
smoothness. The standard inhomogeneous regularization incorporates a spatially
changing exponent of the standard norm-based regularization to
recover a signal whose characteristic varies spatially. This study proposes a
weighted inhomogeneous regularization that extends the standard inhomogeneous
regularization through new exponent design and weighting using spatially
varying weights. The new exponent design avoids misclassification when
different characteristics stay close to each other. The weights handle another
issue when the region of one characteristic is too small to be recovered
effectively by the norm-based regularization even after identified
correctly. A suite of numerical tests shows the efficacy of the proposed
weighted inhomogeneous regularization, including synthetic image experiments
and real sea ice recovery from its incomplete wave measurements
Recent Techniques for Regularization in Partial Differential Equations and Imaging
abstract: Inverse problems model real world phenomena from data, where the data are often noisy and models contain errors. This leads to instabilities, multiple solution vectors and thus ill-posedness. To solve ill-posed inverse problems, regularization is typically used as a penalty function to induce stability and allow for the incorporation of a priori information about the desired solution. In this thesis, high order regularization techniques are developed for image and function reconstruction from noisy or misleading data. Specifically the incorporation of the Polynomial Annihilation operator allows for the accurate exploitation of the sparse representation of each function in the edge domain.
This dissertation tackles three main problems through the development of novel reconstruction techniques: (i) reconstructing one and two dimensional functions from multiple measurement vectors using variance based joint sparsity when a subset of the measurements contain false and/or misleading information, (ii) approximating discontinuous solutions to hyperbolic partial differential equations by enhancing typical solvers with l1 regularization, and (iii) reducing model assumptions in synthetic aperture radar image formation, specifically for the purpose of speckle reduction and phase error correction. While the common thread tying these problems together is the use of high order regularization, the defining characteristics of each of these problems create unique challenges.
Fast and robust numerical algorithms are also developed so that these problems can be solved efficiently without requiring fine tuning of parameters. Indeed, the numerical experiments presented in this dissertation strongly suggest that the new methodology provides more accurate and robust solutions to a variety of ill-posed inverse problems.Dissertation/ThesisDoctoral Dissertation Mathematics 201