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| 1 | 偏微分方程反问题:模型、算法和应用显示文摘偏微分方程反问题是一个重要的数学研究领域,覆盖了偏微分方程、泛函分析、非线性分析、优化算法和数值分析等不同的数学分支,在介质成像、遥感遥测和图像处理等当代重要的工程领域有广泛的应用.基于问题的不适定性,求解这类问题需要引进正则化思想.但是由于模型的复杂性和广泛性,很难建立统一的正则化框架.本文旨在对几类重要的偏微分方程反问题的研究给出一个系统的总结.在阐明偏微分方程反问题起源和特点的基础上,对以电阻抗成像、波场逆散射和介质热成像为应用背景的三类重要的偏微分方程反问题,系统阐述了核心研究问题、已有结果和方法、未来重要的研究方向.最后从反演方法有效实现的角度,对影响偏微分方程反问题数值求解精度和误差估计的主要因素给出了分析. | 程晋 刘继军 张波 | 2019 | 中国科学:数学2019，49，4： | 7 |
| 2 | Numerical Solution for the Helmholtz Equation with Mixed Boundary Condition显示文摘We consider the numerical solution for the Helmholtz equation in R^2 with mixed boundary conditions.The solvability of this mixed boundary value problem is estab- lished by the boundary integral equation method.Based on the Green formula,we express the solution in terms of the boundary data.The key to the numerical real- ization of this method is the computation of weakly singular integrals.Numerical performances show the validity and feasibility of our method.The numerical schemes proposed in this paper have been applied in the realization of probe method for inverse scattering problems. | Haibing Wang Jijun Liu | 2007 | Numerical Mathematics A Journal of Chinese Universities(English Series)2007，16，3： | 4 |
| 3 | A NEW APPROACH AND ERROR ANALYSIS FOR RECONSTRUCTING THE SCATTERED WAVE BY THE POINT SOURCE METHOD显示文摘与阻抗边界由障碍 D C 72 考虑一个反的散布问题。我们用点来源方法从它的远地模式 u<sup>∞</sup> 调查散布的地 u<sup>s</sup> 的重建。由使用边界积分方程方法,首先,我们由古典潜在的理论提供一条新途径给 Potthast 的点来源方法。这从任意的波浪散布的飞机波浪扩大点来源方法的范围。由分析 Hankel 功能的行为,第二,我们为从哪个改进了集中率的调整 paraaneter 的选择获得改进策略(与结果相比[15 ]) 为散布波浪的重建被完成。第三,数字实现被给为阻抗障碍测试倒置方法的有效性和稳定性。数学题目分类:35L, 35R。 | Jijun Liu Gen Nakamura Roland Potthast | 2007 | Journal of Computational Mathematics2007，25，2： | 2 |
| 4 | On the reconstruction of Dirichlet-to-Neumann map in inverse scattering problems with stability estimates显示文摘Consider the determination of Dirichlet-to-Neumann(D-to-N) map from the far-field pattern in inverse scattering problems,which is the key step in some recently developed inversion schemes such as probe method.Essentially,this problem is related to the reconstruction of the scattered wave from its far-field data.We firstly prove the well-known uniqueness result of the D-to-N map from the far-field pattern using a new scheme based on the mixed reciprocity principle.The advantage of this new proof scheme is that it provides an efficient algorithm for computing the D-to-N map,avoiding the numerical differentiation for the scattered wave.Then combining with the classical potential theory,a simple and feasible regularizing reconstruction scheme for the D-to-N map is proposed.Finally the stability estimate for the reconstruction with noisy input data is rigorously analyzed. | WANG HaiBing 1,2 & LIU JiJun 1 1 Department of Mathematics,Southeast University,Nanjing 210096,China 2 School of Mathematics and Computational Science,Hunan University of Science and Technology,Xiangtan 411201,China | 2010 | Science China Mathematics2010，53，8： | 2 |
| 5 | 一类带Stenfen-Boltzmann边界条件的热输运反问题的单调算法显示文摘在钢体铸造过程中,当钢铁在钢包中被加热并且融化后,如何探测接近高温铁水的钢包内壁是否出现了腐蚀现象是一个重要的问题.我们尝试通过钢包外表面的温度知道内部腐蚀的情况.本文得到了表面温度关于内部腐蚀区域的'拟单调'性质,并由此提出解决边界反演问题的数值算法.数值例子表明本文算法是有效的. | 陈贞华 陈文斌 程晋 刘源 | 2012 | 中国科学:数学2012，42，5： | 1 |
| 6 | RECONSTRUCTION OF SCATTERED FIELD FROM FAR-FIELD BY REGULARIZATION显示文摘In this paper, we consider an inverse scattering problem for an obstacle D R^2 with Robin boundary condition. By applying the point source, we give a regularizing method to recover the scattered field from the far-field pattern. Numerical implementations are also presented. | Ji-junLiu JinCheng G.Nakamura | 2004 | Journal of Computational Mathematics2004，22，3： | 1 |
| 7 | 探测法重构多个散射体的数值实现显示文摘声波障碍体的散射中(obstaucle scattering),由散射波的远场数据{u∞(■,d):■,d∈Sm-1}重构散射体是一个标准的逆散射问题.在单个散射体的情形,已经有了大量的研究工作.然而,如果所讨论的逆散射现象是由多个散射体引起的,则除了重建散射体的边界外,还需要确定不同散射体的边界类型.本文主要考虑用探测法重构两个不同类型散射体边界的数值实现.与以往单个散射体的探测方法相比,需要更为仔细地考虑针的选取和包含多个散射体的非凸性区域的构造.在构造指示函数时所需要的Neumann数据■Ω,是采取边界积分方程法直接求解Helmholtz混合边值问题得到的. | 王海兵 刘继军 | 2007 | 计算数学2007，29，2： | 1 |
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