We study two methods to analyze spatio-temporal data. To describe data, we use principal component analysis. To predict data, we use dynamic mode decomposition. We compute numerical solutions of the complex Ginzburg-Landau equation and we use that numerical solution as data. Using principal component analysis we identify a low-dimensional subspace spanned by only 3 principal components. Using these 3 principal components we can reconstruct the original data matrix with approximately 2% error, and construct out-of-sample data with less than 3% error. Using dynamic mode decomposition we are able to predict the temporal evolution of out-of-sample data as far as 500 time steps into the future with less than 5% error. The combination of these two techniques provides robust and reliable methods to analyze complex data sets.

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## Scholarly Works (8 results)

Kubelka-Munk (KM) theory is a broadly used simplification to the radiative transfer equation (RTE) that is solvable analytically for a restricted set of very simple problems. Despite this simplicity and popularity, KM theory has never had its theoretical basis formally established. In this work, we derive KM theory systematically from the radiative transfer equation (RTE) by application of the spectrally convergent double spherical harmonics method, of order one, and analysis of the resulting, transformed, system of equations in the positive- and negative-going fluxes. We call these the generalized Kubelka-Munk (gKM) equations, and they are able to account for general boundary sources and nonhomogeneous terms. Having established theoretical footing for KM theory, we extend gKM's four-flux method to higher dimensions, applying it to a Gaussian boundary source and demonstrating the method's range of validity. Finally, we examine the application of the gKM method to the vector radiative transport equation (vRTE), allowing for the modeling of sources with polarized light. These methods offer a low cost approximation to the solutions of the scalar and vector RTE's, which we validate through comparison with benchmark solutions of the transport equation.

Radiative transport theory describes the propagation of light in random media that absorb, scatter, and emit radiation. To describe the propagation of light, the full polarization state is quantified using the Stokes parameters. For the sake of mathematical convenience, the polarization state of light is often neglected leading to the scalar radiative transport equation for the intensity only. For scalar transport theory, there is a well-established body of literature on numerical and analytic approximations to the radiative transport equation. We extend the scalar theory to the vector radiative transport equation (vRTE). In particular, we are interested in the theoretical basis for a phenomena called circular polarization memory.

Circular polarization memory is the physical phenomena whereby circular polarization retains its ellipticity and handedness when propagating in random media. This is in contrast to the propagation of linear polarization in random media, which depolarizes at a faster rate, and specular reflection of circular polarization, whereby the circular polarization handedness flips. We investigate two limits that are of known interest in the phenomena of circular polarization memory. The first limit we investigate is that of forward-peaked scattering, i.e. the limit where most scattering events occur in the forward or near-forward directions. The second limit we consider is that of strong scattering and weak absorption.

In the forward-peaked scattering limit we approximate the vRTE by a system of partial differential equations motivated by the scalar Fokker-Planck approximation. We call the leading order approximation the vector Fokker-Planck approximation. The vector Fokker Planck approximation predicts that strongly forward-peaked media exhibit circular polarization memory where the strength of the effect can be calculated from the expansion of the scattering matrix in special functions. In addition, we find in this limit that total intensity, linear polarization, and circular polarization decouple. From this result we conclude, that in the Fokker-Planck limit the scalar approximation is an appropriate leading order approximation.

In the strong scattering and weak absorbing limit the vector radiative transport equation can be analyzed using boundary layer theory. In this case, the problem of light scattering in an optically thick medium is reduced to a 1D vRTE near the boundary and a 3D diffusion equation in the interior. We develop and implement a numerical solver for the boundary layer problem by using a discrete ordinate solver in the boundary layer and a spectral method to solve the diffusion approximation in the interior. We implement the method in Fortran 95 with external dependencies on BLAS, LAPACK, and FFTW. By analyzing the spectrum of the discretized vRTE in the boundary layer, we are able to predict the presence of circular polarization memory in a given medium.

In this dissertation, we first present a model for the diffuse reflectance due to a continuous beam incident normally on a half space composed of a uniform scattering and absorbing medium. This model is the result of an asymptotic analysis of the radiative transport equation for strong scattering, weak absorption and a defined beam width. Through comparison with the diffuse reflectance computed using the numerical solution of the radiative transport equation, we show that this diffuse reflectance model gives results that are accurate for small source-detector separation distances.

We then present an explicit model for the diffuse reflectance due to a collimated beam of light incident normally on layered tissues. This model is derived using the corrected diffusion approximation applied to a layered medium, and it takes the form of a convolution with an explicit kernel and the incident beam profile. This model corrects the standard diffusion approximation over all source-detector separation distances provided the beam is sufficiently wide compared to the scattering mean-free path. We validate this model through comparison with Monte Carlo simulations. Then we use this model to estimate the optical properties of an epithelial layer from Monte Carlo simulation data. Using measurements at small source-detector separations and this model, we are able to estimate the absorption coefficient, scattering coefficient and anisotropy factor of epithelial tissues efficiently with reasonable accuracy.

Finally, we present an extension of the corrected diffusion approximation for an obliquely incident beam. This model is formed through a Fourier Series representation in the azimuthal angle which allows us to exhibit the break in axisymmetry when combined with the previous analysis. We validate this model with Monte Carlo simulations. This model can also be written in the form of a convolution of an explicit kernel with the incident beam profile. Additionally, it can be used to improve computation of the optical properties.