A Fokker–Planck control approach to model crowd motion is investigated. This strategy is formulated as a bilinear optimal control-constrained problem governed by the Fokker–Planck equation modeling the evolution of the probability density function of the stochastic motion of the crowd. Theoretical results on existence and regularity of controls are provided. For computational purposes, the resulting optimality system is discretized using an alternate-direction implicit Chang–Cooper scheme that guarantees conservativeness, positivity, L² stability, and second-order accuracy of the forward solution. A projected non-linear conjugate gradient scheme is used to solve the optimality system. Results of numerical experiments demonstrate the efficiency of the proposed control framework.

This paper considers the non-linear inverse problem of reconstructing an electric conductivity distribution from the interior power density in a bounded domain. Applications include the novel tomographic method known as acousto-electric tomography, in which the measurement setup in electrical impedance tomography is modulated by ultrasonic waves thus giving rise to a method potentially having both high contrast and high resolution. We formulate the inverse problem as a regularized non-linear optimization problem, show the existence of a minimizer, and derive optimality conditions. We propose a non-linear conjugate gradient scheme for finding a minimizer based on the optimality conditions. All our numerical experiments are done in two-dimensions. The experiments reveal new insight into the non-linear effects in the reconstruction. One of the interesting features we observe is that, depending on the choice of regularization, there is a trade-off between high resolution and high contrast in the reconstructed images. Our proposed non-linear optimization framework can be generalized to other hybrid imaging modalities.

The article presents an efficient image reconstruction algorithm for single scattering optical tomography (SSOT) in circular geometry of data acquisition. This novel medical imaging modality uses photons of light that scatter once in the body to recover its interior features. The mathematical model of SSOT is based on the broken ray (or V-line Radon) transform (BRT), which puts into correspondence to an image function its integrals along V-shaped piecewise linear trajectories. The process of image reconstruction in SSOT requires inversion of that transform. We implement numerical inversion of a broken ray transform in a disc with partial radial data. Our method is based on a relation between the Fourier coefficients of the image function and those of its BRT recently discovered by Ambartsoumian and Moon. The numerical algorithm requires solution of ill-conditioned matrix problems, which is accomplished using a half-rank truncated singular value decomposition method. Several numerical computations validating the inversion formula are presented, which demonstrate the accuracy, speed, and robustness of our method in the case of both noise-free and noisy data.

A new optimization approach to sparse reconstruction of log-conductivity in acousto-electric tomography is presented. This approach considers the minimization of a Tikhonov least-squares functional of measured power density together with an $L^1$-norm penalization of the log-conductivity to promote sparsity and a Perona--Malik functional to enhance edges. This minimization is performed subject to the differential constraint given by two instances of the EIT equation. The resulting infinite-dimensional optimization problem is approximated by finite differences and solved efficiently by a proximal scheme. Numerous pointers to existing literature are given to motivate the present framework and results of numerical experiments are presented that demonstrate the many advantages of the new approach.