Session Time & Room : 1C (Aug.21, 13:20-15:00) @D405
Type : Proposal of Minisymposium
Abstract : The inverse problems of determining the trajectory of a moving target arise from many significant industrial, medical and military applications such as radar imaging, underwater sonar system， auto target recognition etc. There has been growing interest from the mathematical community, because the design of efficient and stable numerical schemes relies heavily on deep mathematical understandings. The purpose of this symposium is to bring together researchers in this area to discuss mathematical models and inverse problems (including uniqueness, stability and numerics) for identifying moving objects governed by time-dependent PDEs.
[03803] Direct reconstruction methods for moving sources in the wave equation
Format : Talk at Waseda University
Author(s) :
Takashi Ohe (Okayama University of Science)
Abstract : In this talk, we consider the reconstruction problem of moving wave sources under different observation conditions; one is observations on the boundary, and the other is observations on a small number of points. For each observation condition, we propose a direct reconstruction procedure for the parameters of moving wave sources. We also discuss the common and different issues between reconstruction procedures.
[03831] An inverse problem in mean field game from partial boundary measurement
Format : Online Talk on Zoom
Author(s) :
Yat Tin Chow (University of California, Riverside)
Samy Wu Fung (Colorado School of Mines)
Siting Liu (University of California, Los Angeles)
Levon Nurbekyan (University of California, Los Angeles)
Stanley Osher (University of California, Los Angeles)
Abstract : In this work, we consider a novel inverse problem in mean-field games (MFG). We aim to recover the MFG model parameters that govern the underlying interactions among the population based on a limited set of noisy partial observations of the population dynamics under the limited aperture. Due to its severe ill-posedness, obtaining a good quality reconstruction is very difficult. Nonetheless, it is vital to recover the model parameters stably and efficiently in order to uncover the underlying causes for population dynamics for practical needs.
Our work focuses on the simultaneous recovery of running cost and interaction energy in the MFG equations from a finite number of boundary measurements of population profile and boundary movement. To achieve this goal, we formalize the inverse problem as a constrained optimization problem of a least squares residual functional under suitable norms with L1 regularization. We then develop a fast and robust operator splitting algorithm to solve the optimization using techniques including harmonic extensions, three-operator splitting scheme, and primal-dual hybrid gradient method. Numerical experiments illustrate the effectiveness and robustness of the algorithm.
This is a joint work with Samy W. Fung (Colorado School of Mines), Siting Liu (UCLA), Levon Nurbekyan (UCLA), and Stanley J. Osher (UCLA)
[02996] Factorization method for recovering moving objects with dynamic near-field data
Format : Online Talk on Zoom
Author(s) :
Hongxia Guo (Nankai University)
Abstract : In this talk, I will present the factorization method for recovering the trajectory of a moving point source from multi-frequency data with one or sparse dynamic near-field observation points . The observable and non-observable points in the near field region are introduced. At an observable point, it is verified that the smallest annular containing the trajectory and centered at the observable point can be imaged, provided the orbit function possessing a certain property.
[04620] Imaging a moving point source from multi-frequency data measured at one and sparse observation directions (part I): far-field case
Format : Online Talk on Zoom
Author(s) :
Hongxia Guo (Nankai University)
Guanghui Hu (Nankai University, Tianjin, China)
Guanqiu Ma (Nankai University)
Abstract : We propose a multi-frequency algorithm for recovering partial information on the trajectory of a moving point source from one and sparse far-field observation directions in the frequency domain. The starting and terminal time points of the moving source are both supposed to be known.
We introduce the concept of observable directions (angles) in the far-field region and derive all observable directions (angles) for straight and circular motions. The existence of non-observable directions makes this paper much different from inverse stationary source problems.
At an observable direction, it is verified that the smallest trip containing the trajectory and perpendicular to the direction can be imaged, provided the angle between the observation direction and the velocity vector of the moving source lies in $[0,\pi/2]$.
If otherwise, one can only expect to recover a strip thinner than this smallest strip for straight and circular motions.
The far-field data measured at sparse observable directions can be used to recover the $\Theta$-convex domain of the trajectory. Both two- and three-dimensional numerical examples are implemented to show effectiveness and feasibility of the approach.
