Uniform a priori estimates for elliptic and static Maxwell interface problems
Jianguo Huang Jun Zou
Discrete & Continuous Dynamical Systems - B 2007, 7(1): 145-170 doi: 10.3934/dcdsb.2007.7.145
We present some new a priori estimates of the solutions to three-dimensional elliptic interface problems and static Maxwell interface system with variable coefficients. Different from the classical a priori estimates, the physical coefficients of the interface problems appear in these new estimates explicitly.
keywords: elliptic interface problem Maxwell interface problem. A priori estimates
Uniqueness in determining multiple polygonal scatterers of mixed type
Hongyu Liu Jun Zou
Discrete & Continuous Dynamical Systems - B 2008, 9(2): 375-396 doi: 10.3934/dcdsb.2008.9.375
We prove that a polygonal scatterer in $\mathbb{R}^2$, possibly consisting of finitely many sound-soft and sound-hard polygons, is uniquely determined by a single far-field measurement.
keywords: polygonal scatterers mixed type. Inverse obstacle scattering uniqueness
Imaging acoustic obstacles by singular and hypersingular point sources
Jingzhi Li Hongyu Liu Hongpeng Sun Jun Zou
Inverse Problems & Imaging 2013, 7(2): 545-563 doi: 10.3934/ipi.2013.7.545
We investigate a qualitative method for imaging acoustic obstacles in two and three dimensions by boundary measurements corresponding to hypersingular point sources. Rigorous mathematical justification of the imaging method is established, and numerical experiments are presented to illustrate the effectiveness of the proposed imaging scheme.
keywords: hypersingular point sources. Dirichelet-to-Neumann map sampling method indicator function Inverse acoustic scattering boundary measurment
Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems
Lijuan Wang Jun Zou
Discrete & Continuous Dynamical Systems - B 2010, 14(4): 1641-1670 doi: 10.3934/dcdsb.2010.14.1641
This work is concerned with the finite element solutions for parameter identifications in second order elliptic and parabolic systems. The $L^2$- and energy-norm error estimates of the finite element solutions are established in terms of the mesh size, time step size, regularization parameter and noise level.
keywords: error estimates. parameter identification regularization parameter Tikhonov regularization finite element method noise level
A direct sampling method for inverse scattering using far-field data
Jingzhi Li Jun Zou
Inverse Problems & Imaging 2013, 7(3): 757-775 doi: 10.3934/ipi.2013.7.757
This work is concerned with a direct sampling method (DSM) for inverse acoustic scattering problems using far-field data. Using one or few incident waves, the DSM provides quite reasonable profiles of scatterers in time-harmonic inverse acoustic scattering without a priori knowledge of either the physical properties or the number of disconnected components of the scatterer. We shall first present a novel derivation of the DSM using far-field data, then carry out a systematic evaluation of the performances and distinctions of the DSM using both near-field and far-field data. A new interpretation from the physical perspective is provided based on some numerical observations. It is shown from a variety of numerical experiments that the method has several interesting and promising potentials: a) ability to identify not only medium scatterers, but also obstacles, and even cracks, using measurement data from one or few incident directions, b) robustness with respect to large noise, and c) computational efficiency with only inner products involved.
keywords: far-field data. direct sampling method indicator function Inverse acoustic scattering
Raymond H. Chan Thomas Y. Hou Hong-Kai Zhao Haomin Zhou Jun Zou
Inverse Problems & Imaging 2013, 7(3): i-ii doi: 10.3934/ipi.2013.7.3i
In 2012, there were two scientific conferences in honor of Professor Tony F. Chan's 60th birthday. The first one was ``The International Conference on Scientific Computing'', which took place in Hong Kong from January 4-7. The second one, ``The International Conference on the Frontier of Computational and Applied Mathematics'', was held at the Institute of Pure and Applied Mathematics (IPAM) of UCLA from June 8-10. Invitations were also sent out to conference speakers, participants, Professor Chan's former colleagues, collaborators and students, to solicit for original research papers. After the standard peer review processes, we have collected 23 papers in this special issue dedicated to Professor Chan to celebrate his contribution and leadership in the area of scientific computing and image processing.

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A parallel space-time domain decomposition method for unsteady source inversion problems
Xiaomao Deng Xiao-Chuan Cai Jun Zou
Inverse Problems & Imaging 2015, 9(4): 1069-1091 doi: 10.3934/ipi.2015.9.1069
In this paper, we propose a parallel space-time domain decomposition method for solving an unsteady source identification problem governed by the linear convection-diffusion equation. Traditional approaches require to solve repeatedly a forward parabolic system, an adjoint system and a system with respect to the unknown sources. The three systems have to be solved one after another. These sequential steps are not desirable for large scale parallel computing. A space-time restrictive additive Schwarz method is proposed for a fully implicit space-time coupled discretization scheme to recover the time-dependent pollutant source intensity functions. We show with numerical experiments that the scheme works well with noise in the observation data. More importantly it is demonstrated that the parallel space-time Schwarz preconditioner is scalable on a supercomputer with over $10^3$ processors, thus promising for large scale applications.
keywords: Domain decomposition method space-time methods parallel computation. source identification unsteady inverse problems
A degenerate evolution system modeling bean's critical-state type-II superconductors
H. M. Yin Ben Q. Li Jun Zou
Discrete & Continuous Dynamical Systems - A 2002, 8(3): 781-794 doi: 10.3934/dcds.2002.8.781
In this paper we study a degenerate evolution system $\mathbf H_t +\nabla \times [|\nabla \times \mathbf H|^{p-2}\nabla \times \mathbf H]=\mathbf F$ in a bounded domain as well as its limit as $p\to \infty$ subject to appropriate initial and boundary conditions. This system governs the evolution of the magnetic field $\mathbf H$ in a conductive medium under the influence of a system force $\mathbf F$. The system is an approximation of Bean's critical-state model for type-II superconductors. The existence, uniqueness and regularity of solutions to the system are established. Moreover, it is shown that the limit of $\mathbf H(x, t)$ as $p\to \infty$ is a solution to the Bean model.
keywords: Nonlinear evolution system Bean's critical-state model for superconductors.
Overlapping domain decomposition methods for linear inverse problems
Daijun Jiang Hui Feng Jun Zou
Inverse Problems & Imaging 2015, 9(1): 163-188 doi: 10.3934/ipi.2015.9.163
We shall derive and propose several efficient overlapping domain decomposition methods for solving some typical linear inverse problems, including the identification of the flux, the source strength and the initial temperature in second order elliptic and parabolic systems. The methods are iterative, and computationally very efficient: only local forward and adjoint problems need to be solved in each subdomain, and the local minimizations have explicit solutions. Numerical experiments are provided to demonstrate the robustness and efficiency of the methods: the algorithms converge globally, even with rather poor initial guesses; and their convergences do not deteriorate or deteriorate only slightly when the meshes are refined.
keywords: domain decomposition explicit subdomain solver. Inverse problems parameter identification
Conditional Stability and Numerical Reconstruction of Initial Temperature
Jingzhi Li Masahiro Yamamoto Jun Zou
Communications on Pure & Applied Analysis 2009, 8(1): 361-382 doi: 10.3934/cpaa.2009.8.361
In this paper, we address an inverse problem of reconstruction of the initial temperature in a heat conductive system when some measurement data of temperature are available, which may be observed in a subregion inside or on the boundary of the physical domain, along a time period which may start at some point, possibly far away from the initial time. A conditional stability estimate is first achieved by the Carleman estimate for such reconstruction. Numerical reconstruction algorithm is proposed based on the output least-squares formulation with the Tikhonov regularization using the finite element discretization, and the existence and convergence of the finite element solution are presented. Numerical experiments are carried out to demonstrate the applicability and effectiveness of the proposed method.
keywords: conditional stability Tikhonov regularization Carleman estimate finite element method backward heat problem output least-squares

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