Particle tracking in a live cell environment is concerned with reconstructing the trajectories, locations, or velocities of the targeting particles, which holds the promise of revealing important new biological insights. The standard approach of particle tracking consists of two steps: first reconstructing statically the source locations in each time step, and second applying tracking techniques to obtain the trajectories and velocities. In contrast to the standard approach, the dynamic reconstruction seeks to simultaneously recover the source locations and velocities from all frames, which enjoys certain advantages. In this talk, we will present a rigorous mathematical analysis for the resolution limit of reconstructing source number, locations, and velocities by general dynamical reconstruction in particle tracking problems, by which we demonstrate the possibility of achieving super-resolution for dynamic reconstruction. We show that when the location-velocity pairs of the particles are separated beyond certain distances (the resolution limits), the number of particles and the location-velocity pair can be stably recovered. The resolution limits are related to the cut-off frequency of the imaging system, signal-to-noise ratio, and the sparsity of the source. By these estimates, we also derive a stability result for a sparsity-promoting dynamic reconstruction. In addition, we further show that the reconstruction of velocities has a better resolution limit which improves constantly as the particles move. The result is derived from a crucial observation that the inherent cut-off frequency for the velocity recovery can be viewed as the cut-off frequency of the imaging system multiplied by the total observation time, which may lead to a better resolution limit than the one for each diffraction-limited frame. In addition, we propose super-resolution algorithms for recovering the number and values of the velocities in the tracking problem and demonstrate theoretically or numerically their super-resolution capability.

We analyze and quantify the amount of heat generated by a nanoparticle, injected in a background medium, while excited by incident electromagnetic waves. These nanoparticles are dispersive with electric permittivity following the Lorentz model. The purpose is to determine the quantity of heat generated extremely close to the nanoparticle (at a distance proportional to the radius of the nanoparticle). We show that by exciting the medium with incident frequencies close to the Plasmonic or Dielectric resonant frequencies, we can generate any desired amount of heat close to the injected nanoparticle while the amount of heat decreases away from it. These results offer a wide range of potential applications in the areas of photo-thermal therapy, drug delivery, and material science, to cite a few. To do so, we employ time-domain integral equations and asymptotic analysis techniques to study the corresponding mathematical model for heat generation. This model is given by the heat equation where the body source term comes from the modulus of the electric field generated by the used incident electromagnetic field. Therefore, we first analyze the dominant term of this electric field by studying the full Maxwell scattering problem in the presence of Plasmonic or All-dielectric nanoparticles. As a second step, we analyze the propagation of this dominant electric field in the estimation of the heat potential. For both the electromagnetic and parabolic models, the presence of the nanoparticles is translated into the appearance of large scales in the contrasts for the heat-conductivity (for the parabolic model) and the permittivity (for the full Maxwell system) between the nanoparticle and its surrounding.

We study stability aspects for the determination of space and time-dependent lower order perturbations of the wave operator in three space dimensions with point sources. The problems under consideration here are formally determined and we establish Lipschitz stability results for these problems. The main tool in our analysis is a modified version of Bukgheĭm-Klibanov method based on Carleman estimates.

[1] V. P. Krishnan, S. Senapati, Rakesh. Stability for a formally determined inverse problem for a hyperbolic PDE with space and time dependent coefficients, SIAM J. Math. Anal. 53, no. 6, 6822–6846, 2021.

[2] V. P. Krishnan, S. Senapati, Rakesh. Point sources and stability for an inverse problem for a hyperbolic PDE with space and time dependent coefficients, J. Differential Equations 342, 622–665, 2023.

We develop fast, accurate and efficient numerical methods for solving the time harmonic scattering problem of electromagnetic waves in 3D by a multitude of obstacles for low and medium frequencies. Taking into account a large number of heterogeneities can be costly in terms of computation time and memory usage, particularly in the construction process of the matrix. We consider a multi-scale diffraction problem in low-frequency regimes in which the characteristic length of the obstacles is small compared to the incident wavelength. We use the matched asymptotic expansion method which allows for the model reduction. Two types of approximations are distinguished : near-field or quasi-static approximations that descibe the phenomenon at the microscopic scale and far-field approximations that describe the phenomenon at a long distance. In the latter one, small obstacles are no longer considered as geometric constraintsand can be modelled by equivalent point-sources which are interpreted in terms of electromagnetic multipoles.

[1] J. Labat, V. Péron, S. Tordeux. Equivalent multipolar point-source modeling of small spheres for fast and accurate electromagnetic wave scattering computations, Wave Motion 92: 102409, 2020.