Coming soon.

For an overdetermined system \( \mathsf A\mathsf x \approx \mathsf b \) with \( \mathsf A \) and \( \mathsf b \) given, the least-square (LS) formulation \( \min_x \, \|\mathsf A\mathsf x-\mathsf b\|_2 \) is often used to find an acceptable solution \( \mathsf x \). The cost of solving this problem depends on the dimensions of \( \mathsf A \), which are large in many practical instances. This cost can be reduced by the use of random sketching, in which we choose a matrix \( \mathsf S \) with many fewer rows than \( \mathsf A \) and \(\mathsf b \), and solve the sketched LS problem \( \min_x \, \|\mathsf S(\mathsf A \mathsf x-\mathsf b)\|_2 \) to obtain an approximate solution to the original LS problem. Significant theoretical and practical progress has been made in the last decade in designing the appropriate structure and distribution for the sketching matrix \( \mathsf S \). When \( \mathsf A \) and \(\mathsf b \) arise from discretizations of a PDE-based inverse problem, tensor structure is often present in \( \mathsf A \) and \(\mathsf b \) For reasons of practical efficiency, \( \mathsf S \) should be designed to have a structure consistent with that of \( \mathsf A \). Can we claim similar approximation properties for the solution of the sketched LS problem with structured \( \mathsf S \) as for fully-random \( \mathsf S \)? We give estimates that relate the quality of the solution of the sketched LS problem to the size of the structured sketching matrices, for two different structures. Our results are among the first known for random sketching matrices whose structure is suitable for use in PDE inverse problems.

The process of reconstructing properties of biological tissue using measurements of incoming and outgoing light intensity is known as optical tomography. It may be described mathematically by the inverse radiative transfer equation: optical tomography amounts to reconstructing the scattering coefficient in the RTE using the boundary measurements. In the strong scattering regime, the RTE is asymptotically equivalent to the diffusion equation (DE). In the Bayesian framework, we examine the posterior distribution of the scattering coefficient after the measurements have been taken. Sampling from this distribution is computationally expensive, so we employ a two-level MCMC technique, using the DE posterior distribution to make sampling from the RTE posterior distribution computationally feasible.

Optical tomography, mostly used in medical imaging, is a technique for constructing optical properties in tested tissues via measurements of the incoming and outgoing light intensity. Mathematically, light propagation is modeled by the radiative transfer equation (RTE), and optical tomography amounts to reconstructing the scattering and the absorption coefficients in the RTE using the boundary measurements. We study this problem in the Bayesian framework, and pay special attention to the strong scattering regime. Asymptotically, when this happens, the RTE is equivalent to the diffusion equation (DE), whose inverse problem is severely ill. We study the stability deterioration as the equation changes regimes and prove the convergence of the inverse RTE to the inverse DE in both nonlinear and linear settings.

We study the motion of a particle in a particular magnetic field configuration both classically and quantum mechanically. For flux-free radially symmetric magnetic fields defined on circular regions, we establish that particle escape speeds depend, classically, on a gauge-fixed magnetic vector potential, and demonstrate some trajectories associated with this special type of magnetic field. Then we show that some of the geometric features of the classical trajectory (perpendicular exit from the field region, trapped and escape behavior) are reproduced quantum mechanically using a numerical method that extends the norm-preserving Crank-Nicolson method to problems involving magnetic fields. While there are similarities between the classical trajectory and the position expectation value of the quantum mechanical solution, there are also differences, and we demonstrate some of these.

We probe the dynamics of a modified form of the Schrodinger-Newton (SN) system of gravity coupled to single particle quantum mechanics. At the masses of interest here, the ones associated with the onset of "collapse" (where the gravitational attraction is competitive with the quantum mechanical dissipation), we show that the Schrodinger ground state energies match the Dirac ones with an error of ~ 10%. At the Planck mass scale, we predict the critical mass at which a potential collapse could occur for the self-coupled gravitational case, m ~ 3.3 Planck mass, and show that gravitational attraction opposes Gaussian spreading at around this value, which is a factor of two higher than the one predicted (and verified) for the Schrodinger-Newton system. Unlike the Schrodinger-Newton dynamics, we do not find that the self-coupled case tends to decay towards its ground state; there is no collapse in this case.