Application of the Reduced Basis Method to Hyperspectral Diffuse Optical Tomography

Diffuse optical tomography (DOT), which uses low-energy laser light in the visible to near infrared range, has become a popular alternative to traditional medical imaging techniques such as x-ray, because it is non-ionizing and cost effective. Since DOT is especially effective in reconstructing images of soft tissue, where light penetrates more easily, one of its main applications is in breast cancer detection. Hyperspectral DOT (hyDOT) uses hundreds of optical wavelengths in the imaging process in order to improve the resolution of the image by adding new information. We develop a reduced basis method approach to solve the forward problem in hyDOT, which is to determine the measurements on the boundary of the tissue given information about the light source on the boundary, the location of any tumors, and the values of the absorption and diffusion coefficients. Our work on the forward problem is motivated by the image reconstruction problem in hyDOT which is computationally expensive because any algorithm requires solving the forward problem hundreds, if not thousands, of times. We show how the reduced basis method greatly improves the computational burden of the forward problem and thus, improves the efficiency of the reconstruction problem

The Carleman-Newton method to globally reconstruct a source term for nonlinear parabolic equation

We propose to combine the Carleman estimate and the Newton method to solve an inverse source problem for nonlinear parabolic equations from lateral boundary data. The stability of this inverse source problem is conditionally logarithmic. Hence, numerical results due to the conventional least squares optimization might not be reliable. In order to enhance the stability, we approximate this problem by truncating the high frequency terms of the Fourier series that represents the solution to the governing equation. By this, we derive a system of nonlinear elliptic PDEs whose solution consists of Fourier coefficients of the solution to the parabolic governing equation. We solve this system by the Carleman-Newton method. The Carleman-Newton method is a newly developed algorithm to solve nonlinear PDEs. The strength of the Carleman-Newton method includes (1) no good initial guess is required and (2) the computational cost is not expensive. These features are rigorously proved. Having the solutions to this system in hand, we can directly compute the solution to the proposed inverse problem. Some numerical examples are displayed

The evolution of resource distribution, slow diffusion, and dispersal strategies in heterogeneous populations

Population diffusion in river-ocean ecologies and for wild animals, including birds, mainly depends on the availability of resources and habitats. This study explores the dynamics of the resource-based competition model for two interacting species in order to investigate the spatiotemporal effects in a spatially distributed heterogeneous environment with no-flux boundary conditions. The main focus of this study is on the diffusion strategy, under conditions where the carrying capacity for two competing species is considered to be unequal. The same growth function is associated with both species, but they have different migration coefficients. The stability of global coexistence and quasi-trivial equilibria are also studied under different conditions with respect to resource function and carrying capacity. Furthermore, we investigate the case of competitive exclusion for various linear combinations of resource function and carrying capacity. Additionally, we extend the study to the instance where a higher migration rate negatively impacts population growth in competition. The efficacy of the model in the cases of one- and two-dimensional space is also demonstrated through a numerical study.AMS subject classification 201092D25, 35K57, 35K50, 37N25, 53C35

Inverse Problem In Optical Tomography Using Diffusion Approximation and Its Hopf-Cole Transformation

In this paper, we derive the Hopf-Cole transformation to the diffusion approximation. We find the analytic solution to the one dimensional diffusion approximation and its Hopf-Cole transformation for a homogenous constant background medium. We demonstrate that for a homogenous constant background medium in one dimension, the Hopf-Cole transformation improves the stability of the inverse problem. We also derive a Green's function scaling of the higher dimensional diffusion approximation for an inhomogeneous background medium and discuss a two step reconstruction algorithm

Electrical Impedance Tomography using Sparsity Regularization and Machine Learning Approach

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