Wenjia JingYau Mathematical Sciences Center, Tsinghua University, No 1. Tsinghua Yuan, Beijing 100084, ChinaPanagiotis E. SouganidisDepartment of Mathematics, The University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USAHung V. TranDepartment of Mathematics, University of Wisconsin at Madison, 480 Lincoln Drive, Madison, WI 53706, USA
Analysis of PDEsOptimization and ControlProbabilitymathscidoc:2206.03012
Discrete and Continuous Dynamical Systems - S, 11, (5), 915-939, 2018.10
We study the averaging of fronts moving with positive oscillatory normal velocity, which is periodic in space and stationary ergodic in time. The problem can be formulated as the homogenization of coercive level set Hamilton-Jacobi equations with spatio-temporal oscillations. To overcome the difficulties due to the oscillations in time and the linear growth of the Hamiltonian, we first study the long time averaged behavior of the associated reachable sets using geometric arguments. The results are new for higher than one dimensions even in the space-time periodic setting.
The focus of this work is on rigorous mathematical analysis of the topological derivative based detection algorithms for the localization of an elastic inclusion of vanishing characteristic size. A filtered quadratic misfit is considered, and the performance of the topological derivative imaging functional resulting therefrom is analyzed. Our analysis reveals that the imaging functional may not attain its maximum at the location of the inclusion. Moreover, the resolution of the image is below the diffraction limit. Both phenomena are due to the coupling of pressure and shear waves propagating with different wave speeds and polarization directions. A novel imaging functional based on the weighted Helmholtz decomposition of the topological derivative is, therefore, introduced. It is thereby substantiated that the maximum of the imaging functional is attained at the location of the inclusion and the resolution is enhanced and proves to be the diffraction limit. Finally, we investigate the stability of the proposed imaging functionals with respect to measurement and medium noises.
Habib AmmariDepartment of Mathematics and Applications, Ecole Normale Supérieure, 45 Rue d’Ulm, 75005, Paris, FranceThomas BoulierDepartment of Mathematics and Applications, Ecole Normale Supérieure, 45 Rue d’Ulm, 75005, Paris, FranceJosselin GarnierLaboratoire de Probabilités et Modèles Aléatoires & Laboratoire Jacques-Louis Lions, Université Paris VII, 75205, Paris Cedex 13, FranceWenjia JingDepartment of Mathematics and Applications, Ecole Normale Supérieure, 45 Rue d’Ulm, 75005, Paris, FranceHyeonbae KangDepartment of Mathematics, Inha University, Incheon, 402-751, KoreaHan WangDepartment of Mathematics and Applications, Ecole Normale Supérieure, 45 Rue d’Ulm, 75005, Paris, France
Analysis of PDEsMathematical PhysicsNumerical Analysis and Scientific ComputingOptimization and Controlmathscidoc:2206.03006
Foundations of Computational Mathematics, 14, 27-62, 2013.9
The aim of this paper is to provide a fast and efficient procedure for (real-time) target identification in imaging based on matching on a dictionary of precomputed generalized polarization tensors (GPTs). The approach is based on some important properties of the GPTs and new invariants. A new shape representation is given and numerically tested in the presence of measurement noise. The stability and resolution of the proposed identification algorithm is numerically quantified. We compare the proposed GPT-based shape representation with a moment-based one.