The reciprocity gap method for a cavity in an inhomogeneous medium
Fang Zeng Xiaodong Liu Jiguang Sun Liwei Xu
Inverse Problems & Imaging 2016, 10(3): 855-868 doi: 10.3934/ipi.2016024
We consider an interior inverse medium problem of reconstructing the shape of a cavity. Both the measurement locations and point sources are inside the cavity. Due to the lack of a priori knowledge of physical prosperities of the medium inside the cavity and to avoid the computation of background Green's functions, the reciprocity gap method is employed. We prove the related theory and present some numerical examples for validation.
keywords: inhomogeneous medium reciprocity gap method. Interior inverse scattering problem
The framework of axiomatics fuzzy sets based fuzzy classifiers
Xiaodong Liu Wanquan Liu
Journal of Industrial & Management Optimization 2008, 4(3): 581-609 doi: 10.3934/jimo.2008.4.581
In this paper we will propose a new classifier design based on the AFS fuzzy theory. First, we will briefly review the current researches in data classification based on fuzzy and rough set theories and then present the AFS framework. Second, we will present new membership functions for fuzzy sets with their logic operations in the AFS framework and then tackle some theoretical and computational problems related to classifier design. Third, we will develop a new approach for fuzzy classifier design based on the proposed membership functions and their logic operations. Finally, a well-known example is used to illustrate its effectiveness. The advantage of this classifier is in two-folds. One is that it can mimic the human reasoning comprehensively and offers a far more flexible and effective way for the study of large-scale intelligent systems. The other is its simplicity in methodology and mathematical beauty in fuzzy theory.
keywords: Fuzzy classification. Sub-preference relation Molecular lattice AFS algebra AFS structure
Positive steady state of a food chain system with diffusion
Jing Liu Xiaodong Liu Sining Zheng Yanping Lin
Conference Publications 2007, 2007(Special): 667-676 doi: 10.3934/proc.2007.2007.667
This paper deals positive steady state for predator-prey microorganisms in a flow reactor with diffusion and flow. The coexistence conditions for the predator-prey populations are established. The main method used here is the fixed point index theory.
keywords: Food chain; Predator-prey; Coexistence; Diffusion; Fixed point index; Steady state; Micro-organism; Nutrient.
The factorization method for scatterers with different physical properties
Xiaodong Liu
Discrete & Continuous Dynamical Systems - S 2015, 8(3): 563-577 doi: 10.3934/dcdss.2015.8.563
The scattering of time-harmonic acoustic plane waves by a mixed type scatterer is considered. Such a scatterer is given as the union of several components with different physical properties. Some of them are impenetrable obstacles with Dirichlet or impedance boundary conditions, while the others are penetrable inhomogeneous media with compact support. This paper is concerned with modifications of the factorization method for the following two basic cases, one is the scattering by a priori separated sound-soft and sound-hard obstacles, the other one is the scattering by a scatterer with impenetrable (Dirichlet) and penetrable components. The other cases can be dealt with similarly. Finally, some numerical experiments are presented to demonstrate the feasibility and effectiveness of the modified factorization methods.
keywords: factorization method inhomogeneous medium. Acoustic scattering impenetrable obstacle mixed type scatterers
Inverse source problems in electrodynamics
Guanghui Hu Peijun Li Xiaodong Liu Yue Zhao
Inverse Problems & Imaging 2018, 12(6): 1411-1428 doi: 10.3934/ipi.2018059

This paper concerns inverse source problems for the time-dependent Maxwell equations. The electric current density is assumed to be the product of a spatial function and a temporal function. We prove uniqueness and stability in determining the spatial or temporal function from the electric field, which is measured on a sphere or at a point over a finite time interval.

keywords: Stability uniqueness inverse source problem Maxwell's equations

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