A STRONG POSITIVITY PROPERTY AND A RELATED INVERSE SOURCE PROBLEM FOR MULTI-TERM TIME-FRACTIONAL DIFFUSION EQUATIONS

In this article,we consider the diffusion equation with multi-term time-fractional derivatives.We first derive,by a subordination principle for the solution,that the solution is positive when the initial value is non-negative.As an application,we prove the uniqueness of solution to an inverse problem of determination of the temporally varying source term by integral type information in a subdomain.Finally,several numerical experiments are presented to show the accuracy and efficiency of the algorithm.

Piecewise Acoustic Source Imaging with Unknown Speed of Sound Using a Level-Set Method

We investigate the following inverse problem:starting from the acoustic wave equation,reconstruct a piecewise constant passive acoustic source from a single boundary temporal measurement without knowing the speed of sound.When the amplitudes of the source are known a priori,we prove a unique determination result of the shape and propose a level set algorithm to reconstruct the singularities.When the singularities of the source are known a priori,we show unique determination of the source amplitudes and propose a least-squares fitting algorithm to recover the source amplitudes.The analysis bridges the low-frequency source inversion problem and the inverse problem of gravimetry.The proposed algorithms are validated and quantitatively evaluated with numerical experiments in 2D and 3D.

A CONDITIONAL STABILITY FOR AN INVERSE PROBLEM ARISING IN GROUNDWATER POLLUTION

An inverse problem of determining magnitude of groundwater pollution in a hydrologic region is investigated. By applying integral identity methods, a conditional stability for the inverse problem here is constructed with aids of an optimal adjoint problem and a suitable topology.

Fundamental solution method for inverse source problem of plate equation

The elastic plate vibration model is studied under the external force. The size of the source term by the given mode of the source and some observations from the body of the plate is determined over a time interval, which is referred to be an inverse source problem of a plate equation. The uniqueness theorem for this problem is stated, and the fundamental solution to the plate equation is derived. In the case that the plate is driven by the harmonic load, the fundamental solution method （FSM） and the Tikhonov regularization technique axe used to calculate the source term. Numerical experiments of the Euler-Bernoulli beam and the Kirchhoff-Love plate show that the FSM can work well for practical use, no matter the source term is smooth or piecewise.

Global Lipschitz Stability in an Inverse Problem for the Non-stationary Transport Equation

An inverse problem of determining the source term of the non-stationary transport equation was considered.The extra lateral boundary condition was chosen as the observation data.Based on a Carleman estimate on the non-stationary transport equation,a global Lipschitz stability for the inverse problem was proved.

Inverse Source Locating Method Based on Graphical Analysis of Dye Plume Images in a Turbulent Flow

The inverse estimation of a source location of pollutant released into a turbulent flow is a probability problem instead of a deterministic one, as the turbulent flow is chaotic and irreversible. However, researches can be conducted to provide helpful instructions to the possible source location with corresponding uncertainty. This study aims to propose a method of inverse estimation of a passive-scalar source location. Experimental investigation of the dye plume characteristics released into a fully-developed turbulent flow is performed in a water channel. A planar laser-induced fluorescence (PLIF) technique is used to obtain two-dimensional images of spreading dye plumes at a bulk Reynolds number of 20,000. The distributions of high concentration areas in the PLIF images are chosen as features that characterize the traveling (diffusion) distance or time from the dye source. Graphical analysis is used to extract these high concentration areas. The procedure of graphical analysis has three steps: 1) binarization using a threshold to extract high concentration dye patches;2) labeling individual high-concentration dye patches in the binarized images;and 3) pixel-counting to measure the area and perimeter of each dye patch. We examine the variations of fractal dimension of patches, and the fractal dimension is observed to be almost constant irrespective of the distance from the source. The kurtosis of the probability density function curve of the logarithm dimensionless dye patch areas is found to be related with the downstream diffusion distance, based on which an inverse estimation method to locate a passive-scalar point source is proposed and evaluated.

Gas emission source term estimation with 1-step nonlinear partial swarm optimization-Tikhonov regularization hybrid method

Source term identification is very important for the contaminant gas emission event. Thus, it is necessary to study the source parameter estimation method with high computation efficiency, high estimation accuracy and reasonable confidence interval. Tikhonov regularization method is a potential good tool to identify the source parameters. However, it is invalid for nonlinear inverse problem like gas emission process. 2-step nonlinear and linear PSO （partial swarm optimization）-Tikhonov regularization method proposed previously have estimated the emission source parameters successfully. But there are still some problems in computation efficiency and confidence interval. Hence, a new 1-step nonlinear method combined Tikhonov regularizafion and PSO algorithm with nonlinear forward dispersion model was proposed. First, the method was tested with simulation and experiment cases. The test results showed that 1-step nonlinear hybrid method is able to estimate multiple source parameters with reasonable confidence interval. Then, the estimation performances of different methods were compared with different cases. The estimation values with 1-step nonlinear method were close to that with 2-step nonlinear and linear PSO-Tikhonov regularization method, 1-step nonlinear method even performs better than other two methods in some cases, especially for source strength and downwind distance estimation. Compared with 2-step nonlinear method, 1-step method has higher computation efficiency. On the other hand, the confidence intervals with the method proposed in this paper seem more reasonable than that with other two methods. Finally, single PSO algorithm was compared with 1-step nonlinear PSO-Tikhonov hybrid regularization method. The results showed that the skill scores of 1-step nonlinear hybrid method to estimate source parameters were close to that of single PSO method and even better in some cases. One more important property of 1-step nonlinear PSO-Tikhonov regularization method is its reasonable confidence interval, which is not obtained by single PSO algorithm. Therefore, 1-step nonlinear hybrid regularization method proposed in this paper is a potential good method to estimate contaminant gas emission source term.

Groundwater contaminant source identification based on iterative local update ensemble smoother

Identification of the location and intensity of groundwater pollution source contributes to the effect of pollution remediation,and is called groundwater contaminant source identification.This is a kind of typical groundwater inverse problem,and the solution is usually ill-posed.Especially considering the spatial variability of hydraulic conductivity field,the identification process is more challenging.In this paper,the solution framework of groundwater contaminant source identification is composed with groundwater pollutant transport model(MT3DMS)and a data assimilation method(Iterative local update ensemble smoother,ILUES).In addition,Karhunen-Loève expansion technique is adopted as a PCA method to realize dimension reduction.In practical problems,the geostatistical method is usually used to characterize the hydraulic conductivity field,and only the contaminant source information is inversely calculated in the identification process.In this study,the identification of contaminant source information under Kriging K-field is compared with simultaneous identification of source information and K-field.The results indicate that it is necessary to carry out simultaneous identification under heterogeneous site,and ILUES has good performance in solving high-dimensional parameter inversion problems.

Compressive Sensing Approaches for Lithographic Source and Mask Joint Optimization

Source and mask joint optimization(SMO)is a widely used computational lithography method for state-of-the-art optical lithography process to improve the yield of semiconductor wafers.Nowadays,computational efficiency has become one of the most challenging issues for the development of pixelated SMO techniques.Recently,compressive sensing(CS)theory has be explored in the area of computational inverse problems.This paper proposes a CS approach to improve the computational efficiency of pixel-based SMO algorithms.To our best knowledge,this paper is the first to develop fast SMO algorithms based on the CS framework.The SMO workflow can be separated into two stages,i.e.,source optimization(SO)and mask optimization(MO).The SO and MO are formulated as the linear CS and nonlinear CS reconstruction problems,respectively.Based on the sparsity representation of the source and mask patterns on the predefined bases,the SO and MO procedures are implemented by sparse image reconstruction algorithms.A set of simulations are presented to verify the proposed CS-SMO methods.The proposed CS-SMO algorithms are shown to outperform the traditional gradient-based SMO algorithm in terms of both computational efficiency and lithography imaging performance.

EXPONENTIAL TIKHONOV REGULARIZATION METHOD FOR SOLVING AN INVERSE SOURCE PROBLEM OF TIME FRACTIONAL DIFFUSION EQUATION

In this paper,we mainly study an inverse source problem of time fractional diffusion equation in a bounded domain with an over-specified terminal condition at a fixed time.A novel regularization method,which we call the exponential Tikhonov regularization method with a parameter γ,is proposed to solve the inverse source problem,and the corresponding convergence analysis is given under a-priori and a-posteriori regularization parameter choice rules.Whenγis less than or equal to zero,the optimal convergence rate can be achieved and it is independent of the value of γ.However,when γ is greater than zero,the optimal convergence rate depends on the value of γ which is related to the regularity of the unknown source.Finally,numerical experiments are conducted for showing the effectiveness of the proposed exponential regularization method.

Some Universal Properties of the Green’s Functions Associated with the Wave Equation in Bounded Partially-Homogeneous Domains and Their Use in Acoustic Tomography

Direct and inverse scattering problems connected with the wave equation in non-homogeneous bounded domains constitute challenging actual subjects for both mathematicians and engineers. Among them one can mention, for example, inverse source problems in seismology, nondestructive archeological probing, mine prospecting, inverse initial-value problems in acoustic tomography, etc. In spite of its crucial importance, almost all of the available rigorous investigations concern the case of unbounded simple domains such as layered planar or cylindrical or spherical structures. The main reason for the lack of the works related to non-homogeneous bounded structures is the extreme complexity of the explicit expressions of the Green’s functions. The aim of the present work consists in discovering some universal properties of the Green’s functions in question, which reduce enormously the difficulties arising in various applications. The universality mentioned here means that the properties are not depend on the geometrical and physical properties of the configuration. To this end one considers first the case when the domain is partially-homogeneous. Then the results are generalized to the most general case. To show the importance of the universal properties in question, they are applied to an inverse initial-value problem connected with photo-acoustic tomography.

径向热传导方程源项的分数阶Tikhonov-Landweber反演方法

本文研究了时间分数阶的径向热传导方程的源项反演问题.根据径向区域的终止时刻数据,利用分数阶Tikhonov-Landweber迭代正则化方法,求得在先验和后验的条件下的正则化解并得到相应误差.最后,数值实验表明了该正则化方法是有效的.

An Inverse Source Problem with Sparsity Constraint for the Time-Fractional Diffusion Equation

In this paper,an inverse source problem for the time-fractional diffusion equation is investigated.The observational data is on the final time and the source term is assumed to be temporally independent and with a sparse structure.Here the sparsity is understood with respect to the pixel basis,i.e.,the source has a small support.By an elastic-net regularization method,this inverse source problem is formulated into an optimization problem and a semismooth Newton(SSN)algorithm is developed to solve it.A discretization strategy is applied in the numerical realization.Several one and two dimensional numerical examples illustrate the efficiency of the proposed method.

基于可逆图扩散的网络传播溯源方法研究

随着社会的发展,各种类型网络的安全问题日益突出,尤其是网络传播问题。对网络传播的扩散源点进行准确的定位是实现控制网络传播的重要手段。对于网络传播溯源问题的研究还面临着网络结构多样、传播机制复杂等问题,因此基于图神经网络研究了网络传播溯源问题,提出了一个基于图卷积神经网络的可逆图扩散模型(GCNIGD)。在节点易感性估计阶段,考虑到网络节点之间的连接关系,结合图卷积神经网络充分利用了网络的结构信息;在节点特征构造阶段,结合了图扩展卷积对网络中信息传递进行空间局部化拓展,从而可以从多跳信息中学习来增强基于图的模型;在进行溯源阶段,将图溯源问题转化为图扩散的逆问题,构造了可逆的图网络对源节点进行准确估计,解决了网络溯源中的不适定问题。最后,在六个真实世界数据集中进行了大量的实验,实验结果表明提出的方法超越了目前已知的最先进方法。该研究对于网络中虚假信息溯源、网络攻击溯源等网络安全领域问题具有重要的指导意义。

基于Bayesian-MCMC方法的水体污染识别反问题

针对具有不适定性的环境水力学反问题,基于贝叶斯推理和二维水质模型建立水体污染识别反演模型,运用马尔科夫链蒙特卡罗法抽样获得污染源源强、污染源位置和污染泄漏时间等模型参数的后验概率分布和统计结果.实例研究结果表明,基于马尔科夫链蒙特卡罗抽样算法的贝叶斯推理可以较好地用来实现水体污染识别,具有识别精度高,误差小的特点,其可靠性和稳定性高于混合遗传模式搜索优化算法.

脑磁源定位重建算法研究进展

脑磁图作为一种非侵入性大脑功能神经成像方法,以其高时间分辨率和无创性等特点得到临床的广泛关注。脑磁源定位研究的核心问题是利用脑外磁场数据(脑磁图)来推断脑内电流源的分布,并进一步确定神经源的位置和活动模式,这一逆问题的求解过程存在不唯一性和不适定性的难题。脑磁源定位重建方法分为分布源模型和偶极子定位。综述脑磁图和磁源成像的发展过程和研究进展,分布源模型包括最小范数估计法、低分辨率脑电磁断层扫描、欠定系统局域解法、贝叶斯推断估计法、波束成形器和稀疏源成像等;偶极子定位包括最大熵方法、最小二乘范数、多信号分类算法、神经网络和遗传算法等;论述各类算法的特色、局限性及其存在的问题;随着技术的不断进步,融合多种脑功能技术的多模态重建方法有望成为神经功能诊断领域的主导性检测技术。

用Tikhonov正则化方法同时反演对流扩散方程的对流速度和源函数

在给定两个附加观测数据的条件下,本文基于Tikhonov正则化方法研究了对流扩散方程的对流速度和源函数的同时反演问题.鉴于原问题是一个初始值非零的对流扩散方程,本文通过将初始值转化为源项得到了一个组合源项,首先将原问题转化为一个具有齐次条件的对流扩散问题.由于所得问题是不适定的,本文进而利用Tikhonov正则化方法构建了相应的极小化目标泛函,得到了问题最优解的存在性和应满足的必要条件.最后,对终端时刻较小的特殊情形,本文证明了最优解的唯一性和稳定性.

基于改进最小方差波束成形的心磁信号的源重建

由于冠心病患者的心磁图在信噪比较低时,最小方差波束成形(MV)方法的源重建效果不理想,提出一种基于改进最小方差波束成形(IMV)方法,在MV方法的基础上,使用导联矩阵和磁场信号二阶特征矩阵组成约束矩阵,构造新的空间滤波器的权矩阵,从而降低输出噪声功率增益。利用点扩散函数理论比较IMV方法和MV方法的空间谱估计的单源分辨率;利用3个电流偶极子产生36通道磁场仿真数据,采用IMV方法和MV方法在低信噪比下对仿真数据进行源重建;最后对3名冠心病患者的36通道心磁图的R波峰和T波峰时刻数据进行源重建。结果表明,IMV方法的单源分辨率更高,对仿真数据及冠心病患者心磁数据的重建精度更好。

识别Rayleigh-Stokes方程源项的分数阶Landweber迭代正则化方法

该文研究具有Riemann-Liouville时间分数阶导数的Rayleigh-Stokes方程未知源识别问题.首先证明这个问题是不适定的,并应用分数阶Landweber正则化方法求解此反问题.基于条件稳定性结果,在先验和后验正则化参数选取规则下,分别给出精确解与正则解之间的误差估计.最后通过数值例子说明此方法求解此类反问题的有效性和可行性.

基于单元辐射叠加法的水下矩形板声场重建

为解决等效源法在水下结构声源声场重建中存在的虚拟等效源配置位置难以确定、求逆过程中传递矩阵病态性过大的问题,本文提出了基于单元辐射叠加法的水下结构声源声场重建方法。该方法利用结构声源边界元法建模思想与声场波叠加原理,建立了声源表面振动与辐射声场之间的振声传递关系,得到可快速计算的振声传递函数,并利用该函数作为传递算子进行声场重建。通过对矩形板的声场重建仿真,验证了本文方法相比等效点源近场声全息具有更高的重建精度,且具有更大的有效全息距离。通过在不同阵元数、信噪比情况下的仿真分析,进一步证明了本文方法能够改善水下结构声源的声场重建精度。