APPROACHES TO ITERATIVE ALGORITHMS FOR SOLVING NONLINEAR EQUATIONS WITH AN APPLICATION IN TOMOGRAPHIC ABSORPTION SPECTROSCOPY.

In this paper we propose an approach for solving systems of nonlinear equations without computing function derivatives. Motivated by the application area of tomographic absorption spectroscopy, which is a highly-nonlinear problem with variables coupling, we consider a situation where straightforward translation to a fixed point problem is not possible because the operators that represent the relevant systems of nonlinear equations are not self-mappings, i.e., they operate between spaces of different dimensions. To overcome this difficulty we suggest an "alternating common fixed points algorithm" that acts alternatingly on the different vector variables. This approach translates the original problem to a common fixed point problem for which iterative algorithms are abound and exhibits a viable alternative to translation to an optimization problem, which usually requires derivatives information. However, to apply any of these iterative algorithms requires to ascertain the conditions that appear in their convergence theorems. To circumvent the need to verify conditions for convergence, we propose and motivate a derivative-free algorithm that better suits the tomographic absorption spectroscopy problem at hand and is even further improved by applying to it the superiorization approach. This is presented along with experimental results that demonstrate our approach.

Tomographic absorption spectroscopy based on dictionary learning.

Tomographic absorption spectroscopy (TAS) has an advantage over other optical imaging methods for practical combustor diagnostics: optical access is needed in a single plane only, and the access can be limited. However, practical TAS often suffers from limited projection data. In these cases, priors such as smoothness and sparseness can be incorporated to mitigate the ill-posedness of the inversion problem. This work investigates use of dictionary learning (DL) to effectively extract useful a priori information from the existing dataset and incorporate it in the reconstruction process to improve accuracy. We developed two DL algorithms; our numerical results suggest that they can outperform classical Tikhonov reconstruction under moderate noise conditions. Further testing with experimental data indicates that they can effectively suppress reconstruction artifacts and obtain more physically plausible solutions compared with the inverse Radon transform.

Linear multispectral absorption tomography based on regularized iterative methods.

A regularization approach of iterative algorithms was proposed to reconstruct the two-dimensional temperature and concentration distributions based on linear multispectral absorption tomography (MAT). This method introduces a secondary prior into a classical iterative algorithm via regularization to improve the reconstruction accuracy. Numerical studies revealed that the regularized iteration outperformed the classical and superiorized versions under various noisy conditions and with different number of spectral lines. The algorithms were also tested with the existing experimental data of a premixed flat flame produced by a McKenna burner. The comparison between the reconstructions and the measured temperature profile using thermocouples confirmed the superiority of our proposed regularized iterative method.

Computed tomography of chemiluminescence for the measurements of flames confined within a cylindrical glass.

Computed tomography of chemiluminescence (CTC) is one kind of volumetric tomography which can recover 3D flame structures and has found extensive applications for spatiotemporally resolved measurements of flames. However, the existing CTC techniques rely on the pinhole model and fail when the flames are confined within a cylindrical glass due to image distortion caused by the refraction on both the internal and external surfaces of the glass. In this work, a refined camera model was developed by combining the pinhole camera model with Snell's laws using a reverse ray-tracing method to incorporate the effects of refraction. A proof-of-concept demonstration of CTC based on the refined camera model was conducted on a swirl flame confined within a 20-mm-thick K9 glass. The results proved the superiority of such technique against the existing version in terms of reconstruction accuracy. This work is expected to be especially useful for the study of combustion phenomena such as combustion instability for which the flames are typically confined within cylindrical combustors.