BIOLUMINESCENCE TOMOGRAPHY:BIOMEDICAL BACKGROUND,MATHEMATICAL THEORY,AND NUMERICAL APPROXIMATION

Over the last couple of years molecular imaging has been rapidly developed to study physiological and pathological processes in vivo at the cellular and molecular levels. Among molecular imaging modalities, optical imaging stands out for its unique advantages, especially performance and cost-effectiveness. Bioluminescence tomography （BLT） is an emerging optical imaging mode with promising biomedical advantages. In this survey paper, we explain the biomedical significance of BLT, summarize theoretical results on the analysis and numerical solution of a diffusion based BLT model, and comment on a few extensions for the study of BLT.

Numerical Approximation of Stochastic Theta Method for Random Periodic Solution of Stochastic Differential Equations

In this paper,we make use of stochastic theta method to study the existence of the numerical approximation of random periodic solution.We prove that the error between the exact random periodic solution and the approximated one is at the 1/4 order time step in mean sense when the initial time tends to∞.

Numerical Approximation of a Compressible Multiphase System

The numerical simulation of non conservative system is a difficult challenge for two reasons at least.The first one is that it is not possible to derive jump relations directly from conservation principles,so that in general,if the model description is non ambiguous for smooth solutions,this is no longer the case for discontinuous solutions.From the numerical view point,this leads to the following situation:if a scheme is stable,its limit for mesh convergence will depend on its dissipative structure.This is well known since at least[1].In this paper we are interested in the“dual”problem:given a system in non conservative form and consistent jump relations,how can we construct a numerical scheme that will,for mesh convergence,provide limit solutions that are the exact solution of the problem.In order to investigate this problem,we consider a multiphase flow model for which jump relations are known.Our scheme is an hybridation of Glimm scheme and Roe scheme.

BIFURCATION SOLUTION BRANCHESAND THEIR NUMERICAL APPROXIMATIONS OF A SEMI-LINEAR ELLIPTIC PROBLEM WITH TWO PARAMETERS

In this paper the bifurcation solution branches of a semi-linear elliptic problem are studied.

Numerical Investigation on Mixing Efficiency and Exponential Fluid Stretching in Chaotic Mixing

The stretching and folding of fluid element during chaotic mixing field is studied using numerical method. The chaotic mixing process is caused by periodic secondary flow in a twisted curved pipe. Using the nonlinear discrete velocity field as the dynamical system, the present study connects the fluid particle's stretching along its trajectory in one period to a linearized time-varying variational equation. After numerical approximation of the variational equation, fluid stretching is calculated on the whole cross section. The stretching distribution shows an exponential fluid stretching and folding, which indicates an excellent mixing performance.

A Comparison of Different Numerical Schemes to Solve Nonlinear First Order ODE

In this paper, the nonlinear first order ordinary differential equation will be considered. Three simplest numerical stencils are presented to solve this equation. We deduce that the numerical method of Trapezoidal is a good technique, which helped us to find an approximation of the exact solution with small error.

The research on the highly efficient calculation method of 3-D frequency-domain Green function

The traditional calculation method of frequency-domain Green function mainly utilizes series or asymptotic expansion to carry out numerical approximation, however, this method requires very careful zoning, thus the computing process is complex with many cycles, which has greatly affected the computing efficiency. To improve the computing efficiency, this paper introduces Gaussian integral to the numerical calculation of the frequency-domain Green function and its partial derivatives. It then compares the calculation result with that in existing references. The comparison results demonstrate that, on the basis of its sufficient accuracy, the method has greatly simplified the computing process, reduced the zoning and improved the computing efficiency.

Approximate Failures Semantics for Polynomial Labelled Transition Systems

Labelled transition systems(LTSs) are widely used to formally describe system behaviour.The labels of LTS are extended to offer a more satisfactory description of behaviour by refining the abstract labels into multivariate polynomials.These labels can be simplified by numerous numerical approximation methods.Those LTSs that can not apply failures semantics equivalence in description and verification may have a chance after using approximation on labels.The technique that combines approximation and failures semantics equivalence effectively alleviates the computational complexity and minimizes LTS.

Solutions and memory effect of fractional-order chaotic system:A review

Fractional calculus is a 300 years topic,which has been introduced to real physics systems modeling and engineering applications.In the last few decades,fractional-order nonlinear chaotic systems have been widely investigated.Firstly,the most used methods to solve fractional-order chaotic systems are reviewed.Characteristics and memory effect in those method are summarized.Then we discuss the memory effect in the fractional-order chaotic systems through the fractionalorder calculus and numerical solution algorithms.It shows that the integer-order derivative has full memory effect,while the fractional-order derivative has nonideal memory effect due to the kernel function.Memory loss and short memory are discussed.Finally,applications of the fractional-order chaotic systems regarding the memory effects are investigated.The work summarized in this manuscript provides reference value for the applied scientists and engineering community of fractional-order nonlinear chaotic systems.

A nearly analytic exponential time difference method for solving 2D seismic wave equations

In this paper, we propose a nearly analytic exponential time difference （NETD） method for solving the 2D acoustic and elastic wave equations. In this method, we use the nearly analytic discrete operator to approximate the high-order spatial differential operators and transform the seismic wave equations into semi-discrete ordinary differential equations （ODEs）. Then, the converted ODE system is solved by the exponential time difference （ETD） method. We investigate the properties of NETD in detail, including the stability condition for 1-D and 2-D cases, the theoretical and relative errors, the numerical dispersion relation for the 2-D acoustic case, and the computational efficiency. In order to further validate the method, we apply it to simulating acoustic/elastic wave propagation in mul- tilayer models which have strong contrasts and complex heterogeneous media, e.g., the SEG model and the Mar- mousi model. From our theoretical analyses and numerical results, the NETD can suppress numerical dispersion effectively by using the displacement and gradient to approximate the high-order spatial derivatives. In addition, because NETD is based on the structure of the Lie group method which preserves the quantitative properties of differential equations, it can achieve more accurate results than the classical methods.

Cherenkov Radiation:A Stochastic Differential Model Driven by Brownian Motions

With the development of molecular imaging,Cherenkov optical imaging technology has been widely concerned.Most studies regard the partial boundary flux as a stochastic variable and reconstruct images based on the steadystate diffusion equation.In this paper,time-variable will be considered and the Cherenkov radiation emission process will be regarded as a stochastic process.Based on the original steady-state diffusion equation,we first propose a stochastic partial differential equationmodel.The numerical solution to the stochastic partial differential model is carried out by using the finite element method.When the time resolution is high enough,the numerical solution of the stochastic diffusion equation is better than the numerical solution of the steady-state diffusion equation,which may provide a new way to alleviate the problem of Cherenkov luminescent imaging quality.In addition,the process of generating Cerenkov and penetrating in vitro imaging of 18 F radionuclide inmuscle tissue are also first proposed by GEANT4Monte Carlomethod.The result of the GEANT4 simulation is compared with the numerical solution of the corresponding stochastic partial differential equations,which shows that the stochastic partial differential equation can simulate the corresponding process.

Determining the Effective Refractive Index of AIGaAs-GaAs Slab Waveguide Based on Analytical and Finite Difference Method

Optical waveguide is the main element in integrated optics. Therefore many numerical methods are used on these elements of integrated optics. Simulation response of an optical slab waveguide used in integrated optics needs such numerical methods. These methods must be precise and useful in terms of memory capacity and time duration. In this paper, we study basic analytical and finite difference methods to determine the effective refractive index of AIGaAs-GaAs slab waveguide. Also, appropriate effective refractive index value is obtained with respect to number of grid points and number of matrix sizes. Finally, the validity of the obtained values by both methods is compared to using waveguide type.

A Convergent Numerical Algorithm for the Stochastic Growth-Fragmentation Problem

The stochastic growth-fragmentation model describes the temporal evolution of a structured cell population through a discrete-time and continuous-state Markov chain.The simulations of this stochastic process and its invariant measure are of interest.In this paper,we propose a numerical scheme for both the simulation of the process and the computation of the invariant measure,and show that under appropriate assumptions,the numerical chain converges to the continuous growth-fragmentation chain with an explicit error bound.With a triangle inequality argument,we are also able to quantitatively estimate the distance between the invariant measures of these two Markov chains.

Numerical Solutions for Optimal Control of Stochastic Kolmogorov Systems

This work is concerned with controlled stochastic Kolmogorov systems.Such systems have received much attention recently owing to the wide range of applications in biology and ecology.Starting with the basic premise that the underlying system has an optimal control,this paper is devoted to designing numerical methods for approximation.Different from the existing literature on numerical methods for stochastic controls,the Kolmogorov systems take values in the first quadrant.That is,each component of the state is nonnegative.The work is designing an appropriate discrete-time controlled Markov chain to be in line with(locally consistent)the controlled diffusion.The authors demonstrate that the Kushner and Dupuis Markov chain approximation method still works.Convergence of the numerical scheme is proved under suitable conditions.

EIGENVALUES OF THE NEUMANN-POINCARE OPERATOR FOR TWO INCLUSIONS WITH CONTACT OF ORDER m： A NUMERICAL STUDY

In a composite medium that contains close-to-touching inclusions, the pointwise values of the gradient of the voltage potential may blow up as the distance S between some inclusions tends to 0 and as the conductivity contrast degenerates. In a recent paper [9], we showed that the blow-up rate of the gradient is related to how the eigenvalues of the associated Neumann-Poincare operator converge to ±1/2 as δ→ 0, and on the regularity of the contact. Here, we consider two connected 2-D inclusions, at a distance 5 〉 0 from each other. When δ=0, the contact between the inclusions is of order m 〉 2. We numerically determine the asymptotic behavior of the first eigenvalue of the Neumann- Poincare operator, in terms of 5 and rn, and we check that we recover the estimates obtained in [10].

A numerical framework for the approximate solution of fractional tumor-obesity model

In this paper,we have proposed the efficient numerical methods to solve a tumor-obesity model which involves two types of the fractional operators namely Caputo and CaputoFabrizio(CF).Stability and convergence of the proposed schemes using Caputo and CF fractional operators are analyzed.Numerical simulations are carried out to investigate the effect of low and high caloric diet on tumor dynamics of the generalized models.We perform the numerical simulations of the tumor-obesity model for different fractional order by varying immune response rate to compare the dynamics of the Caputo and CF fractional operators.

NUMERICAL HOPF BIFURCATION OF DELAY-DIFFERENTIAL EQUATIONS

In this paper we consider the numerical solution of some delay differential equations undergoing a Hopf bifurcation. We prove that if the delay differential equations have a Hopf bifurcation point atλ=λ*, then the numerical solution of the equation also has a Hopf bifurcation point atλh =λ* + O(h).

OBTAINING EXACT INTERPOLATION MULTIVARIATE POLYNOMIAL BY APPROXIMATION

In some fields such as Mathematics Mechanization, automated reasoning and Trustworthy Computing, etc., exact results are needed. Symbolic computations are used to obtain the exact results. Symbolic computations are of high complexity. In order to improve the situation, exact interpolating methods are often proposed for the exact results and approximate interpolating methods for the ap- proximate ones. In this paper, the authors study how to obtain exact interpolation polynomial with rational coefficients by approximate interpolating methods.

Multilayer perceptron neural network activated by adaptive Gaussian radial basis function and its application to predict lid-driven cavity flow

To improve the performance of multilayer perceptron(MLP)neural networks activated by conventional activation functions,this paper presents a new MLP activated by univariate Gaussian radial basis functions(RBFs)with adaptive centers and widths,which is composed of more than one hidden layer.In the hidden layer of the RBF-activated MLP network(MLPRBF),the outputs of the preceding layer are first linearly transformed and then fed into the univariate Gaussian RBF,which exploits the highly nonlinear property of RBF.Adaptive RBFs might address the issues of saturated outputs,low sensitivity,and vanishing gradients in MLPs activated by other prevailing nonlinear functions.Finally,we apply four MLP networks with the rectified linear unit(ReLU),sigmoid function(sigmoid),hyperbolic tangent function(tanh),and Gaussian RBF as the activation functions to approximate the one-dimensional(1D)sinusoidal function,the analytical solution of viscous Burgers’equation,and the two-dimensional(2D)steady lid-driven cavity flows.Using the same network structure,MLP-RBF generally predicts more accurately and converges faster than the other threeMLPs.MLP-RBF using less hidden layers and/or neurons per layer can yield comparable or even higher approximation accuracy than other MLPs equipped with more layers or neurons.

THE EFFECT OF MEMORY TERMS IN DIFFUSION PHENOMENA

In this paper the effect of integral memory terms in the behavior of diffusion phenomena is studied. The energy functional associated with different models is analyzed and stability inequalities are established. Approximation methods for the computation of the solution of the integro-differential equations are constructed. Numerical results are included.