This paper concerns an inverse problem for the initial boundary value problem of the two-dimensional Navier-Stokes system defined in a bounded simply connected domain with slip, vorticity boundary conditions, and a global vorticity invariant constraint. We establish conditional Lipschitz stability and a local recovery for this inverse problem, where the velocity field and space-independent boundary vorticity are locally recovered from the given initial velocity field and the global vorticity invariant. Our analysis is based on well-posedness estimates and energy methods for the vorticity transport equation.

Dynamic elastography is a widely used, safe, convenient, and cost-effective method to aid in medical diagnosis. It visualizes the wave field propagating through living tissues and quantitatively determines the wave propagation speed from the acquired data, thereby enabling the extraction of the viscoelastic properties of in vivo tissues. Notably, this identification process relies on the mathematical modeling of the viscoelastic characteristics of living tissues. When living tissues are simply modeled as isotropic elastic media, J. McLaughlin and J. Yoon established the uniqueness of the identification in by reasoning that they called the “shrink and spread argument". Given the realistic viscoelastic nature of biological tissues, generalizing their results by adopting viscoelastic models is of great significance. In this paper, using their reasoning, we prove the uniqueness of identification for two typical viscoelastic media: the isotropic extended Maxwell model and the isotropic extended standard linear solid model. More precisely, we demonstrate that the shear wave speed within a region of interest Ω can be uniquely determined from a single measurement of the wave field in Ω.

No-arbitrage property provides a simple method for pricing financial derivatives. However, arbitrage opportunities exist in various fields, even for a very short time. By knowing that an arbitrage property exists, we can adopt a financial trading strategy. This paper investigates the inverse option problems (IOP) in the backward parabolic equation with a suitable initial condition in financial markets. We identify the coefficients of this problem from the measured data and attempt to find arbitrage opportunities in financial markets using a Bayesian inference approach, which is presented as an IOP solution. The posterior probability density function of the parameters is computed from the measured data. The statistics of the unknown parameters are estimated by a Markov Chain Monte Carlo (MCMC) algorithm, which exploits the posterior state space. The efficient sampling strategy of the MCMC algorithm enables us to solve inverse problems by the Bayesian inference technique. Our numerical results indicate that the Bayesian inference approach can simultaneously estimate the unknown trend and volatility coefficients from the artificial measured data and the real financial market data.

How to model the 2019 CoronaVirus (2019-nCov) spread in China is one of the most urgent and interesting problems in applied mathematics. In this paper, we propose a novel time delay dynamic system with external source to describe the trend of local outbreak for the 2019-nCoV. The external source is introduced in the newly proposed dynamic system, which can be considered as the suspected people travel to different areas. The numerical simulations exhibit the dynamic system with the external source is more reliable than the one without it, and the rate of isolation is extremely important for controlling the increase of cumulative confirmed people of 2019-nCoV. Based on our numerical simulation results with the public data, we suggest that the local government should have some more strict measures to maintain the rate of isolation. Otherwise the local cumulative confirmed people of 2019-nCoV might be out of control.

Unique continuation is one of the most important properties for the solution of partial differential equation, which means the local information of the solution can determine the global one. In this paper, we discuss a special unique continuation for Helmholtz equations on a sphere in , which is different with the classical unique continuation. The unique continuation holds only on the sphere and may not be extended to the domain in . A Hölder type conditional stability of unique continuation is proved by complex extension method. Then a Tikhonov regularized scheme is proposed and the convergence rate is obtained by using the conditional stability estimate. Numerical examples are presented to show the performance of the scheme. It should be remarked here that our results may be applied to the problem of recovering a far field pattern with its local information.

Diffuse optical tomography (DOT) is an imaging modality that uses near-infrared light. Although iterative numerical schemes are commonly used for its inverse problem, correct solutions are not obtained unless good initial guesses are chosen. We propose a numerical scheme of DOT, which works even when good initial guesses of optical parameters are not available. We use simulated annealing (SA), which is a method of the Markov-chain Monte Carlo. To implement SA for DOT, a spin Hamiltonian is introduced in the cost function, and the Metropolis algorithm or single-component Metropolis–Hastings algorithm is used. By numerical experiments, it is shown that an initial random spin configuration is brought to a converged configuration by SA, and targets in the medium are reconstructed. The proposed numerical method solves the inverse problem for DOT by finding the ground state of a spin Hamiltonian with SA.

This paper deals with an inverse problem for recovering the viscoelasticity of a living body from MRE (Magnetic Resonance Elastography) data. Based on a viscoelastic partial differential equation whose solution can approximately simulate MRE data, the inverse problem is transformed to a least square variational problem. This is to search for viscoelastic coefficients of this equation such that the solution to a boundary value problem of this equation fits approximately to MRE data with respect to the least square cost function. By computing the Gateaux derivatives of the cost function, we minimize the cost function by the projected gradient method is proposed for recovering the unknown coefficients. The reconstruction results based on simulated data and real experimental data are presented and discussed.

When the inverse problem of diffuse optical tomography (DOT) is solved with the Born or Rytov approximation, the size of the matrix of the linear inverse problem becomes large if the volume (or area) of the domain in biological tissue used for reconstruction is large. The number of unknown parameters in DOT is reduced when constraints about the shape of a target are imposed for the inverse problem. Due to such constraints, the inverse problem becomes nonlinear even when the (first) Born or Rytov approximation is employed. We solve this nonlinear inverse problem by the simulated annealing, which is not trapped by local minima of the cost function.

Magnetic resonance elastography (MRE) is a developing medical imaging technique which can successfully diagnose liver diseases such as cirrhotic and fibrosis. As a hardware MRE gives an image of a viscoelastic wave in a region of interest (ROI) of a human body. Here the wave is generated by an external vibration system and further transmitted to the human body through a probe. MRE also has a software usually called elastogram which reconstructs viscoelastic modulus from the wave displacement vector. An appropriate model equation or system which connects the wave image to the modulus is known as modified stationary Stokes system if viscoelastic property of the ROI is isotropic (Ammari et al 2008 Q. Appl. Math. 66 139-175; Jiang et al 2011 SIAM J. Appl. Math. 71 1965-1989). Comparing this system with the original stationary Stokes system, it has an extra term with frequency. In most cases the elastogram has been using a more simpler model equation such as scalar model (Manduca et al 2003 Med. Image Anal. 5 237-254). In this paper we discuss about a scheme of solving the equation (*)F(A) = u, where u is the measured wave displacement and F(A) is the solution to the boundary value problem for the modified stationary Stokes system with the modulus A (see subsection 2.2 for more details). More precisely, we will show the convergence of a Newton type iteration scheme called the Levenberg-Marquardt method by proving that the nonlinear operator F satisfies the tangential cone condition. We will also provide several numerical tests even with noisy data.

In this paper, we consider the Navier-Stokes equation with memory term in a three-dimensional bounded domain. The equation is the so-called Oldroyd fluid model equation, which can describe the stress relaxation as well as the retardation of deformation due to the memory term. For this equation we considered the inverse problem for recovering the kernel of memory term in this model equation from the measurement described as the integral over determination condition. We obtained a local in time existence and uniqueness for this inverse problem.

提出了一类具有自适应参数的改进DBSCAN聚类算法,并应用于发现证券市场中关联基金账户所组成的信息群落.算法针对传统算法中半径参数ε敏感度高,对于多层密度数据集难以选择全局参数而导致聚类结果差等缺点进行了改进,此外还基于实际市场数据特征,自定义了刻画两个基金间相似程度的综合距离,使得改进算法能更好地应用在解决实际问题上.最后通过基于模拟数据和实际数据的数值实验,验证了改进算法的有效性.

COVID-19 has been impacting on the whole world critically and constantly Since December 2019. We have independently developed a novel statistical time delay dynamic model on the basis of the distribution models from CCDC. Based only on the numbers of confirmed cases in different regions in China, the model can clearly reveal that the containment of the epidemic highly depends on early and effective isolation. We apply the model on the epidemic in Japan and conclude that there could be a rapid outbreak in Japan if no effective quarantine measures are carried out immediately.

A fast algorithm for fluorescence diffuse optical tomography is proposed. The algorithm is robust against the choice of initial guesses. We estimate the position of a fluorescent target by assuming a cuboid (rectangular parallelepiped) for the fluorophore target. The proposed numerical algorithm is verified by a numerical experiment and an experiment with a meat phantom. The target position is reconstructed with a cuboid from measurements in the time domain. Moreover, the long-time behavior of the emission light is investigated making use of the analytical solution to the diffusion equation.

2019年末的新型冠状病毒肺炎（简称:新冠肺炎,又称COVID–19, novel coronavirus pneumonia, NCP, 2019–nCoV）疫情得到了全球的广泛关注.文献[1–2]提出了一类新的时滞动力学系统的新冠肺炎传播模型（a time delay dynamic model for NCP,简称TDD–NCP模型）来描述疫情的传播过程.本文将这个模型用于研究部分省市的疫情传播问题,通过增加模型的源项用于模拟外来潜伏感染者对于当地疫情的影响.基于全国各级卫健委每日公布的累计确诊数与治愈数,本文有效地模拟并预测了各地疫情的发展.提出了基于TDD模型的再生数的两种计算方法,并做了估计与分析.发现疫情暴发初期再生数较大,但随着各级政府防控力度的加大而逐渐减小.最后,分析了返程潮对上海疫情发展的影响,并建议上海市政府继续加大防控力度,以防疫情二次暴发.

Optical tomography is a typical non-invasive medical imaging technique, which aims to reconstruct geometric and physical properties of tissues by passing near infrared light through tissues for obtaining the intensity measurements. Other than optical properties of tissues, we are interested in finding locations of small inclusions inside the object from boundary measurements, based on the time-dependent diffusion model. First, we analyze the asymptotic behavior of the boundary measurements weighted by the fundamental solution of a backward diffusion equation as the diameters of inclusions go to zero. Then, we derive an efficient algorithm for locating small inclusions by finite boundary measurements. This algorithm is direct, simple and easy to be implemented numerically, since it only involves matrix operations and has no iteration process. Finally, some numerical results are presented to illustrate the feasibility and robustness of the algorithm. A new observation of the algorithm is that we can take the source points and test points independently and increase the resolution of numerical results by taking more test points.

We have proposed an algorithm to identify the location of a fluorescence object in thick tissue from an epi-fluorescence measurement data on the surface of the semi-infinite geometry. We approximated the fluorescence object by a cuboid to simplify and make the inversion more robust and quick. We conducted the validation measurements with meat phantoms and have concluded the algorithm was potentially effective to identify the location and shape of the object.

In this paper, we propose a novel dynamical system with time delay to describe the outbreak of 2019-nCoV in China. One typical feature of this epidemic is that it can spread in latent period, which is therefore described by the time delay process in the differential equations. The accumulated numbers of classified populations are employed as variables, which is consistent with the official data and facilitates the parameter identification. The numerical methods for the prediction of outbreak of 2019-nCoV and parameter identification are provided, and the numerical results show that the novel dynamic system can well predict the outbreak trend so far. Based on the numerical simulations, we suggest that the transmission of individuals should be greatly controlled with high isolation rate by the government.

In this paper, we develop the Fudan-CCDC model by adding a source term to describe the imported infectors. The model is applied to analyze the situation of COVID-19 in East Asia, and then in Singapore. By data fitting, our model reveals that Singapore has a much higher isolation rate and earlier quarantine measures compared to other countries. We conclude that Singapore has been doing extraordinarily well on epidemic prevention and control. Finally we discuss the specific measures in Singapore's success, and suggest other countries to learn from the Singapore's style, so as to be well prepared in the future.

This paper deals with an inverse problem for recovering the piecewise constant viscoelasticity of a living body from MRE (Magnetic Resonance Elastography) data. Based on a scalar partial differential equation whose solution can approximately simulate MRE data, our inverse coefficient problem is considered as a statistical inverse problem of reconstructing the posterior distribution of unknown viscoelastic modulus. For sampling this distribution, one usually can use the Metropolis-Hastings Markov chain Monte Carlo (MH-MCMC) algorithm. However, without an appropriate “proposal” distribution given artificially, the MH-MCMC algorithm is hard to draw samples efficiently. To avoid this, a so-called slice sampling algorithm is introduced in this paper and applied for solving our problem. The performance of these statistical inversion algorithms is numerically tested basing on simulated data.