Diffusion Equation Approximation Research Articles

Diffusion Approximation of a Poisson Model for Cumulative Excess Returns

The paper analyzes the regressive equation for cumulative excess returns with residual conditional variances as a GARCH(1,1) process and statistical uncertainty as an AR(1) Gaussian process with correlation parameter ?. Under assumption that the lengths of time intervals between transactions are independent exponentially distributed random variables with sufficiently small mean h, we derive diffusion approximation equations. The continuous time limit equation allows concluding that a stationary conditional variance exists. Moreover, we derive this stationary distribution as inverse gamma distribution and analyze the dependence of this distribution on the correlation parameter ?.

A new nonlinear Galerkin finite element method for the computation of reaction diffusion equations

To get the stationary patterns in the approximation of reaction diffusion systems, the computation of evolution equations on large intervals of time is required. In this paper, a new nonlinear Galerkin method based on finite element discretization is presented for the approximation of reaction diffusion equations on large intervals. The new scheme is based on two different finite element spaces defined respectively on one subspace of lower-degree shape function and one of higher-degree shape function. Nonlinearity and time dependence are both treated on the lower-degree space and only a fixed stationary equation needs to be solved on the higher-degree space at each time level. The proof of the stability and convergence of the method is presented in one dimension. Numerical solutions of some well-studied reaction–diffusion systems are presented to demonstrate the effectiveness of the nonlinear Galerkin method.

SU‐D‐210‐03: Limited‐View Multi‐Source Quantitative Photoacoustic Tomography

Purpose:This work is to investigate a novel limited‐view multi‐source acquisition scheme for the direct and simultaneous reconstruction of optical coefficients in quantitative photoacoustic tomography (QPAT), which has potentially improved signal‐to‐noise ratio and reduced data acquisition time.Methods:Conventional QPAT is often considered in two steps: first to reconstruct the initial acoustic pressure from the full‐view ultrasonic data after each optical illumination, and then to quantitatively reconstruct optical coefficients (e.g., absorption and scattering coefficients) from the initial acoustic pressure, using multi‐source or multi‐wavelength scheme.Based on a novel limited‐view multi‐source scheme here, We have to consider the direct reconstruction of optical coefficients from the ultrasonic data, since the initial acoustic pressure can no longer be reconstructed as an intermediate variable due to the incomplete acoustic data in the proposed limited‐view scheme. In this work, based on a coupled photo‐acoustic forward model combining diffusion approximation and wave equation, we develop a limited‐memory Quasi‐Newton method (LBFGS) for image reconstruction that utilizes the adjoint forward problem for fast computation of gradients. Furthermore, the tensor framelet sparsity is utilized to improve the image reconstruction which is solved by Alternative Direction Method of Multipliers (ADMM).Results:The simulation was performed on a modified Shepp‐Logan phantom to validate the feasibility of the proposed limited‐view scheme and its corresponding image reconstruction algorithms.Conclusion:A limited‐view multi‐source QPAT scheme is proposed, i.e., the partial‐view acoustic data acquisition accompanying each optical illumination, and then the simultaneous rotations of both optical sources and ultrasonic detectors for next optical illumination. Moreover, LBFGS and ADMM algorithms are developed for the direct reconstruction of optical coefficients from the acoustic data.Jing Feng and Hao Gao were partially supported by the NSFC (#11405105), the 973 Program (#2015CB856000) and the Shanghai Pujiang Talent Program (#14PJ1404500).

Performance investigation of SP3 and diffusion approximation for three-dimensional whole-body optical imaging of small animals.

The third-order simplified harmonic spherical approximation (SP3) and diffusion approximation (DA) equations have been widely used in the three-dimensional (3D) whole-body optical imaging of small animals. With different types of tissues, which were classified by the ratio of µ s'/µ ɑ, the two equations have their own application scopes. However, the classification criterion was blurring and unreasonable, and the scope has not been systematically investigated until now. In this study, a new criterion for classifying tissues was established based on the absolute value of absorption and reduced scattering coefficients. Using the newly defined classification criterion, the performance and applicability of the SP3 and DA equations were evaluated with a series of investigation experiments. Extensive investigation results showed that the SP3 equation exhibited a better performance and wider applicability than the DA one in most of the observed cases, especially in tissues of low-scattering-low-absorption and low-scattering-high-absorption range. For the case of tissues with the high-scattering-low-absorption properties, a similar performance was observed for both the SP3 and the DA equations, in which case the DA was the preferred option for 3D whole-body optical imaging. Results of this study would provide significant reference for the study of hybrid light transport models.

Parareal in time 3D numerical solver for the LWR Benchmark neutron diffusion transient model

In this paper we present a time-parallel algorithm for the 3D neutrons calculation of a transient model in a nuclear reactor core. The neutrons calculation consists in numerically solving the time dependent diffusion approximation equation, which is a simplified transport equation. The numerical resolution is done with finite elements method based on a tetrahedral meshing of the computational domain, representing the reactor core, and time discretization is achieved using a θ-scheme. The transient model presents moving control rods during the time of the reaction. Therefore, cross-sections (piecewise constants) are taken into account by interpolations with respect to the velocity of the control rods. The parallelism across the time is achieved by an adequate use of the parareal in time algorithm to the handled problem. This parallel method is a predictor corrector scheme that iteratively combines the use of two kinds of numerical propagators, one coarse and one fine. Our method is made efficient by means of a coarse solver defined with large time step and fixed position control rods model, while the fine propagator is assumed to be a high order numerical approximation of the full model.The parallel implementation of our method provides a good scalability of the algorithm. Numerical results show the efficiency of the parareal method on large light water reactor transient model corresponding to the Langenbuch–Maurer–Werner benchmark.

Approximation of stochastic advection diffusion equations with Stochastic Alternating Direction Explicit methods

The numerical solutions of stochastic partial differential equations of Itô type with time white noise process, using stable stochastic explicit finite difference methods are considered in the paper. Basically, Stochastic Alternating Direction Explicit (SADE) finite difference schemes for solving stochastic time dependent advection-diffusion and diffusion equations are represented and the main properties of these stochastic numerical methods, e.g. stability, consistency and convergence are analyzed. In particular, it is proved that when stable alternating direction explicit schemes for solving linear parabolic PDEs are developed to the stochastic case, they retain their unconditional stability properties applying to stochastic advection-diffusion and diffusion SPDEs. Numerically, unconditional stable SADE techniques are significant for approximating the solutions of the proposed SPDEs because they do not impose any restrictions for refining the computational domains. The performance of the proposed methods is tested for stochastic diffusion and advection-diffusion problems, and the accuracy and efficiency of the numerical methods are demonstrated.

Differentiable approximation of diffusion equations driven by $\alpha$-stable Lévy noise

Edward Nelson derived Brownian motion from Ornstein-Uhlenbeck theory by a scaling limit. Previously we extended the scaling limit to an Ornstein-Uhlenbeck process driven by an α-stable Levy process ...

Bioluminescence tomography for media with spatially varying refractive index

Biomedical imaging has developed into the level of molecular imaging. Bioluminescence tomography (BLT), as an optical imaging modality, is a rapidly developing new and promising field. So far, much of the theoretical analysis of BLT is based on a diffusion approximation equation for media with constant refractive index. In this article, we study the BLT problem for media with spatially varying refractive index. We introduce a general framework with Tikhonov regularization for this purpose, present its well-posedness and establish the error bounds for its numerical solution by the finite element method. Numerical results are reported on simulations of the BLT problem for media with spatially varying refractive index.

A velocity–diffusion method for a Lotka–Volterra system with nonlinear cross and self-diffusion

The aim of this paper is to introduce a deterministic particle method for the solution of two strongly coupled reaction–diffusion equations. In these equations the diffusion is nonlinear because we consider the cross and self-diffusion effects. The reaction terms on which we focus are of the Lotka–Volterra type. Our treatment of the diffusion terms is a generalization of the idea, introduced in [P. Degond, F.-J. Mustieles, A deterministic approximation of diffusion equations using particles, SIAM J. Sci. Stat. Comput. 11 (1990) 293–310] for the linear diffusion, of interpreting Fick's law in a deterministic way as a prescription on the particle velocity. Time discretization is based on the Peaceman–Rachford operator splitting scheme. Numerical experiments show good agreement with the previously appeared results. We also observe travelling front solutions, the phenomenon of pattern formation and the possibility of survival for a dominated species due to a segregation effect.

Thermal Conductivity of Xonotlite Insulation Material

A surface-contact hollow cubic model is developed for coupled heat transfer of gas and solid in xonotlite-type calcium silicate insulation material based on its microstructure features. Through one-dimensional heat conduction analysis in the unit cell structure, a conductive thermal conductivity expression is obtained. A transient hot strip method is used to measure the thermal conductivity of xonotlite from 300 to 970K and from 0.045Pa to atmospheric pressure. The spectral specific extinction coefficients are derived from transmission measurements on a thin xonotlite sample performed with a Fourier transform infrared (FTIR) spectrometer. The results show that the specific spectral extinction coefficients are larger than 7m2·kg−1 over the whole measured spectra, and the diffusion approximation equation is a reasonable description of radiative heat transfer in xonotlite insulation material. The effective thermal conductivity model matches extremely well with the experimental data, which is important for the thermal design and thermal analysis of xonotlite insulation material.

Thermal radiative properties of xonotlite insulation material

This paper presents experimental results of thermal radiative properties of xonotlite-type calcium silicate insulation material. Transmittance spectra were first taken using Fourier transform infrared spectrometer (FTIR) for the samples with ρ=234 kg/m3. Specific extinction coefficient spectra were then obtained by applying Beer’s law. Finally, by using the diffusion approximation, the specific Rossland mean extinction coefficients and radiative thermal conductivities were obtained for various temperatures. The results show that the specific spectral extinction coefficient of xonotlite is larger than 7 m2/kg in the whole measured spectra, and diffusion approximation equation is a reasonable description of radiative heat transfer in xonotlite insulation material. The specific Rossland mean extinction coefficient of xonotlite has a maximum value at 400 K and the radiative thermal conductivity is almost proportional to the cube of temperature.

Numerical approximation of one-dimensional stationary diffusion equations with boundary layers

This article is aimed at showing, in a very simple physical context, how one can improve the numerical solution of singular problems involving boundary layers. Taking into account information on the structure of the boundary layer can result in significant simplification and increase of accuracy. Finite elements methods are considered in this article, other discretization methods and more involved problems including higher dimensional problems and the time dependant case will be discussed in Methods and Applications of Analysis (in press), and elsewhere. Besides the numerical aspects, it is noteworthy that the approach to boundary layer expansion, presented in the Appendix, does not require matched asymptotic expansion techniques.

Principles of fluorescence laser tomography of strongly scattering media

The propagation of laser radiation and fluorescence excited by it in a strongly scattering medium are analyzed in the diffusion equation approximation. The possibility of applying efficient algorithms of the tomographic reconstruction with the use of mean paths of photons in biotissues to fluorescence laser tomography is substantiated.

Time-resolved photon-scattering measurements from scattering media fitted to non-Euclidean and conventional diffusion models

Time-resolved light scattering profiles were measured from highly forward-scattering media in the spatial range of 5--15 transport mean free paths ${(l}_{t}).$ Experimental profiles were compared with theoretical predictions based on the non-Euclidean diffusion (NED) and conventional diffusion approximation (DA) equations. The NED model was found to be better in predicting scattered photon distribution over various temporal and spatial scales than the DA approach.

Biophysical studies of pulsed photothermal radiometry in tissues and tissuelike media

Pulsed photothermal radiometry (PPTR) is a remote sensing technique which monitors the temperature transient at the surface of a sample following the absorption of a brief optical pulse. In this work, it was used to investigate optical and thermal processes occurring in laser‐irradiated, turbid, inhomogeneous media such as tissue. An optical diffusion approximation and heat conduction equation were used to derive the turbid PPTR model, for both homogeneous and layered semi‐infinite media. These models were tested experimentally in a series of aqueous gel phantoms of known optical and thermal properties, and were found to be in good agreement. A study of discrete optical absorbers in a transparent medium necessitated the inclusion of the thermal radiative effects as an additional mode of heat transfer, and a corresponding PPTR model was derived and tested experimentally. Studies of intact biomolecules and of in vivo tissue have also been performed to test and verify the derived formulations, and to determine the practical utility of PPTR for specific biomedical applications. The theoretical and experimental results presented in the thesis lay the groundwork for further inclusion of PPTR into the diagnostic arsenal of laser medicine.

PROJECTING VIABILITY OFTOTOABA MACDONALDI,A POPULATION WITH UNKNOWN AGE-DEPENDENT VARIABILITY

For the endangered fish totoaba (Totoaba macdonaldi), as for other endangered species, the relative amount of environmental variability in demographic rates at various life stages is uncertain. Because of the ontogenetic migratory pattern of this species, four life history stages, (a) prerecruits, (b) juveniles, (c) pre-adults, and (d) adults, experience different environmental conditions. We used a diffusion equation approximation of random Leslie matrices to explore the effects on extinction risk of environmental variability in each of these life stages and correlation between stages. Variability in pre-adults and adults had a greater effect on the probability of extinction than did variability in prerecruits and juveniles. In general, correlation between stages increases the rate of increase of the variance of the logarithm of abundance and, hence, the probability of extinction. For example, when all life stages vary coherently, the rate of increase of the variance is the square of a weighted sum of elasticities, while when all stages vary independently, the rate of increase is the same weighted sum of the squared elasticities. The maximum extinction rate occurs when all life history rates in a Leslie matrix vary coherently. Specifically, when the coefficient of variation is the same in each parameter, the rate of increase of the standard deviation of the logarithm of abundance equals the coefficient of variation of the environmental variability. We also evaluated the accuracy of the diffusion equation approximation by comparing predicted extinction rates with those from Monte Carlo simulations of totoaba with variability in each of the four stages. The diffusion equation approximation accurately predicted probabilities of extinction in all cases except one, random variability in recruitment, where it underestimated simulation results substantially. This is apparently due to the large random jumps in abundance in this case. The diffusion equation approximation accurately predicted quasi-extinction of the adult portion of the population.

Multiple-Well Isomerization Diffusion Equation Solutions with a Shift and Invert Lanczos Algorithm

An RRKM/master equation model is constructed for the isomerization of cyclopropene. It is shown how the dense matrix of the master equation operator can be reduced to a banded form by a combination of diffusion equation approximation and equation rearrangement. The lowest eigenvalues and eigenvectors of the master equation, which are sufficient for simulation purposes, are rapidly found using shift and invert Lanczos based methods.

Analysis of a cell-vertex finite volume method for convection-diffusion problems

A cell-vertex finite volume approximation of elliptic convection-dominated diffusion equations is considered in two dimensions. The scheme is shown to be stable and second-order convergent in a mesh-dependent L 2 L_2 -norm.

Optical image reconstruction using frequency-domain data: simulations and experiments

Optical image reconstruction in a heterogeneous turbid medium with the use of frequency-domain measurements is investigated in detail. A finite-element reconstruction algorithm for optical data based on a diffusion equation approximation is presented and confirmed by a series of simulations and experiments using phantoms having optical properties in the range of those expected for tissues. Simultaneous reconstruction of absorption and scattering coefficients is achieved both theoretically and experimentally. Images with different target locations and contrast levels between target and background are also successfully recovered. All reconstructed images from both simulated and experimental data are derived directly from absolute optical data in which no differential measurement scheme is used. Results from the use of simulated and measured data suggest that quantitative images can be produced in terms of absorption and scattering coefficient values and location, size, and shape of heterogeneities within a circular background region over a range of contrast levels. Further, the effects of modulation frequency are found to be relatively modest, although boundary conditions appear to be important factors.

Analysis of a cell-vertex finite volume method for convection-diffusion problems

A cell-vertex finite volume approximation of elliptic convection-dominated diffusion equations is considered in two dimensions. The scheme is shown to be stable and second-order convergent in a mesh-dependent L 2 -norm.