Modeling of Diffusion Transport through Oral Biofilms with the Inverse Problem Method

Aim The purpose of this study was to develop a mathe-matical model to quantitatively describe the passive trans-port of macromolecules within dental biofilms. Methodology Fluorescently labeled dextrans with different molecular mass （3 kD,10 kD,40 kD,70 kD,2 000 kD） were used as a series of diffusion probes. Streptococcus mutans,Streptococcus sanguinis,Actinomyces naeslundii and Fusobacterium nucleatum were used as inocula for biofilm formation. The diffusion processes of different probes through the in vitro biofilm were recorded with a confocal laser microscope. Results Mathematical function of biofilm penetration was constructed on the basis of the inverse problem method. Based on this function,not only the relationship between average concentration of steady-state and molecule weights can be analyzed,but also that between penetrative time and molecule weights. Conclusion This can be used to predict the effective concentration and the penetrative time of anti-biofilm medicines that can diffuse through oral biofilm. Further-more,an improved model for large molecule is proposed by considering the exchange time at the upper boundary of the dental biofilm.

TWO REGULARIZATION METHODS FOR IDENTIFYING THE SOURCE TERM PROBLEM ON THE TIME-FRACTIONAL DIFFUSION EQUATION WITH A HYPER-BESSEL OPERATOR

In this paper,we consider the inverse problem for identifying the source term of the time-fractional equation with a hyper-Bessel operator.First,we prove that this inverse problem is ill-posed,and give the conditional stability.Then,we give the optimal error bound for this inverse problem.Next,we use the fractional Tikhonov regularization method and the fractional Landweber iterative regularization method to restore the stability of the ill-posed problem,and give corresponding error estimates under different regularization parameter selection rules.Finally,we verify the effectiveness of the method through numerical examples.

A modified Tikhonov regularization method for a Cauchy problem of a time fractional diffusion equation

In this paper,we consider a Cauchy problem of the time fractional diffusion equation(TFDE)in x∈[0,L].This problem is ubiquitous in science and engineering applications.The illposedness of the Cauchy problem is explained by its solution in frequency domain.Furthermore,the problem is formulated into a minimization problem with a modified Tikhonov regularization method.The gradient of the regularization functional based on an adjoint problem is deduced and the standard conjugate gradient method is presented for solving the minimization problem.The error estimates for the regularized solutions are obtained under Hp norm priori bound assumptions.Finally,numerical examples illustrate the effectiveness of the proposed method.

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.

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.

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.

Physics-constrained neural network for solving discontinuous interface K-eigenvalue problem with application to reactor physics

Machine learning-based modeling of reactor physics problems has attracted increasing interest in recent years.Despite some progress in one-dimensional problems,there is still a paucity of benchmark studies that are easy to solve using traditional numerical methods albeit still challenging using neural networks for a wide range of practical problems.We present two networks,namely the Generalized Inverse Power Method Neural Network(GIPMNN)and Physics-Constrained GIPMNN(PC-GIPIMNN)to solve K-eigenvalue problems in neutron diffusion theory.GIPMNN follows the main idea of the inverse power method and determines the lowest eigenvalue using an iterative method.The PC-GIPMNN additionally enforces conservative interface conditions for the neutron flux.Meanwhile,Deep Ritz Method(DRM)directly solves the smallest eigenvalue by minimizing the eigenvalue in Rayleigh quotient form.A comprehensive study was conducted using GIPMNN,PC-GIPMNN,and DRM to solve problems of complex spatial geometry with variant material domains from the fleld of nuclear reactor physics.The methods were compared with the standard flnite element method.The applicability and accuracy of the methods are reported and indicate that PC-GIPMNN outperforms GIPMNN and DRM.

基于组合边界条件的固体材料热扩散系数测试方法

提出了一种组合边界条件下结合传热反问题思想测算固体材料热扩散系数的方法.建立了正问题理论模型,基于有限体积法和交替方向隐式方法的思想利用MATLAB软件对正问题温度场进行数值求解,利用共轭梯度法结合软件编程求解反问题,反演得到材料的热扩散系数.设计了实验方案并搭建了完整的测试系统,利用测试装置对亚克力板(PMMA)、硼硅玻璃(Pyrex7740)、大理石3种材料进行了综合实验,结果表明导热系数测试结果与文献值的最大相对偏差为3.45%,小于5%,验证了测试方法和装置的可行性与准确性.进一步对PMMA热扩散系数实验的不确定度进行了分析,得到扩展不确定度为4.86%,处于较低水平,说明实验数据可靠,测试方法科学.

Spectral Optimization Methods for the Time Fractional Diffusion Inverse Problem

An inverse problem of reconstructing the initial condition for a time fractional diffusion equation is investigated.On the basis of the optimal control framework,the uniqueness and first order necessary optimality condition of the minimizer for the objective functional are established,and a time-space spectral method is proposed to numerically solve the resulting minimization problem.The contribution of the paper is threefold:1)a priori error estimate for the spectral approximation is derived;2)a conjugate gradient optimization algorithm is designed to efficiently solve the inverse problem;3)some numerical experiments are carried out to show that the proposed method is capable to find out the optimal initial condition,and that the convergence rate of the method is exponential if the optimal initial condition is smooth.

Local Stability for an Inverse Coefficient Problem of a Fractional Diffusion Equation

Time-fractional diffusion equations are of great interest and importance on describing the power law decay for diffusion in porous media. In this paper, to identify the diffusion rate, i.e., the heterogeneity of medium, the authors consider an inverse coefficient problem utilizing finite measurements and obtain a local HSlder type conditional stability based upon two Carleman estimates for the corresponding differential equations of integer orders.

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。

Numerical Inversion for the Initial Distribution in the Multi-TermTime-FractionalDiffusion Equation Using Final Observations

This article deals with numerical inversion for the initial distribution in the multi-term time-fractional diffusion equation using final observations.The inversion problem is of instability,but it is uniquely solvable based on the solution’s expression for the forward problem and estimation to the multivariate Mittag-Leffler function.From view point of optimality,solving the inversion problem is transformed to minimizing a cost functional,and existence of a minimum is proved by the weakly lower semi-continuity of the functional.Furthermore,the homotopy regularization algorithm is introduced based on the minimization problem to perform numerical inversions,and the inversion solutions with noisy data give good approximations to the exact initial distribution demonstrating the efficiency of the inversion algorithm.

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.

利用终端观测数据重构扩散方程中的势函数

利用终端时刻观测数据,研究扩散方程与空间相关势函数的重构问题。以一维空间域扩散模型为例,研究了一种重构势函数的单调性方法,对于多维空间域扩散模型的问题,该方法同样适用。在理论分析方面,首先推导出扩散方程正问题解的极值原理以及正则性估计。然后根据扩散方程构造一个有界算子,并证明其单调性,进而利用算子的单调性和不动点迭代,证明了势函数重构的唯一性。最后,基于算子半群理论,在终端时刻T足够大的条件下,证明了重构势函数在Hilbert空间中的条件稳定性。在数值实验方面,基于理论分析设计了合适的迭代算法,选取3个典型的数值算例进行数值实验,实验结果表明该算法是稳定有效的,且验证了单调性、唯一性、稳定性等理论结果的准确性。通过理论分析与数值实验进行研究可得,重构扩散方程势函数的单调性方法是可行的。

Reconstructions for Continuous-Wave Diffuse Optical Tomography by a Globally Convergent Method

In this paper, a novel reconstruction method is presented for Near Infrared (NIR) 2-D imaging to recover optical absorption coefficients from laboratory phantom data. The main body of this work validates a new generation of highly efficient reconstruction algorithms called “Globally Convergent Method” (GCM) based upon actual measurements taken from brain-shape phantoms. It has been demonstrated in earlier studies using computer-simulated data that this type of reconstructions is stable for imaging complex distributions of optical absorption. The results in this paper demonstrate the excellent capability of GCM in working with experimental data measured from optical phantoms mimicking a rat brain with stroke.

Study on the Shear Effect on Dye Patches Diffused in Wall-Bounded Shear Flow

This paper focuses on the high intensity filaments (dye patches) embedded in dye plumes in a wall-bounded shear flow, to investigate the shear effect on the dye patch distribution. Motivated by the widely concerned inverse estimation of the source location, we try extracting useful information to know the source location from down-stream dye patches. Accordingly, we changed the dye injection location at different distances from the wall and made observations at different downstream (diffusion) distances from the source. The orientation angle and roundness of dye patches were concerned to examine the shear effect and dye patch characteristics. To capture the dye plume images, a planar laser induced fluorescence (PLIF) technique was used. The orientation and roundness of each dye patch were calculated by least-square fitting. The statistics of both the orientation angle and the roundness were compared with those in homogeneous turbulent cases to reveal the shear effect. Different from uniformly-orientated dye patches in the homogeneous flow, larger occurrence probabilities with positive orientation angles of dye patches are observed in wall-bounded shear flow, in particular, when the injection location is near the wall. As with information extraction for the inverse estimation of source location, it is found that the orientation distribution of dye patches is independent of the diffusion distance, but related with the injection location from the wall. While for the homogeneous flow cases, a strong dependence on the diffusion distance is observed in the orientation distribution profiles. As for the roundness, similar aspects are found regarding the dependencies on the injection location in shear flow and on diffusion distance in homogeneous flow.

间断问题扩散正则化的PINN反问题求解算法

双曲守恒律方程间断问题的求解是该类方程数值求解问题研究的重点之一.采用PINN(physics-informed neural networks)求解双曲守恒律方程正问题时需要添加扩散项,但扩散项的系数很难确定,需要通过试算方法来得到,造成很大的计算浪费.为了捕捉间断并节约计算成本,对方程进行了扩散正则化处理,将正则化方程纳入损失函数中,使用守恒律方程的精确解或参考解作为训练集,学习出扩散系数,进而预测出不同时刻的解.该算法与PINN求解正问题方法相比,间断解的分辨率得到了提高,且避免了多次试算系数的麻烦.最后,通过一维和二维数值试验验证了算法的可行性,数值结果表明新算法捕捉间断能力更强、无伪振荡和抹平现象的产生,且所学习出的扩散系数为传统数值求解格式构造提供了依据.

An Energy Regularization Method for the Backward Diffusion Problem and its Applications to Image Deblurring

For the backward diffusion equation,a stable discrete energy regularization algorithm is proposed.Existence and uniqueness of the numerical solution are given.Moreover,the error between the solution of the given backward diffusion equation and the numerical solution via the regularization method can be estimated.Some numerical experiments illustrate the efficiency of the method,and its application in image deblurring.

对流-扩散方程源项识别反问题的遗传算法

给出了利用遗传算法求解对流 扩散方程源项识别反问题的一种新方法。该方法把源项反问题转化为优化问题,用遗传算法求解。它的特点在于:从多个初始点开始寻优,并借助交叉,变异算子来获得全局最优解。实例模拟结果表明,该方法具有精度高,收敛速度快且易于计算机实现等特点。

对流-扩散方程逆过程反问题的稳定性及数值求解

对流-扩散方程逆过程反问题是一不适定问题. 利用 Fourier分析理论研究了该类反问题,得到了在空间L2中的稳定性定理. 利用Tikhonov正则化方法给出了一种反演算法. 数值模拟结果表明,利用此方法求解对流 扩散方程逆过程反问题具有稳定性好、精度高的特点.