Boundary behavior of diffusion approximations to Markov jump processes

This paper introduces an improved diffusion model which is accurate and fast in computation for the cases of mu(a)/mu'(s) < 0.07 as good as the conventional diffusion model for the cases of mu(a)/mu'(s) < 0.007 for surface measurement, hence more suitable than the conventional model to be the forward model used in the image reconstruction in the diffuse optical tomography. Deviation of the diffusion approximation (DA) on the medium surface is first studied in the Monte Carlo (MC) diffusion hybrid model for reflectance setup. A modification of DA and an improved MC diffusion hybrid model based on this modified DA are introduced. Numerical tests show that for media with relatively strong absorption the present modified diffusion approach can reduce the surface deviation significantly in both the hybrid and pure diffusion model, and consumes nearly no more computation time than the conventional diffusion approach.

A hybrid radiative-transfer-diffusion model for optical tomography is proposed. The light propagation is modeled with the radiative-transfer equation in the vicinity of the laser sources, and the diffusion approximation is used elsewhere in the domain. The solution of the radiative-transfer equation is used to construct a Dirichlet boundary condition for the diffusion approximation on a fictitious interface within the object. This boundary condition constitutes an approximative distributed source model for the diffusion approximation in the remaining area. The results from the proposed approach are compared with finite-element solutions of the radiative-transfer equation and the diffusion approximation and Monte Carlo simulation. The results show that the method improves the accuracy of the forward model compared with the conventional diffusion model.

This article investigates the optimal excess-loss reinsurance, investment, and the rationality of using a diffusion model to approximate a jump model. We assume that the instantaneous rate of return of the risky asset is modelled by an Ornstein-Uhlenbeck (O-U) process, and the insurance claims are modeled by a compound Poisson (CP) process and a diffusion approximation (DA) model. By using the stochastic dynamic programming method, the closed-form expressions for the optimal reinsurance and investment strategies and the corresponding value functions are derived. We find that the insurer with the CP claim model always has higher reinsurance demand than the insurer with the DA claim model. Moreover, the error of using the DA model to approximate the CP model is very much dependent on the reinsurance price. Numerical analysis shows that the insurer adjusts the investment strategy as the state of the financial market changes.

Drying kinetics of Üryani plum in a convective hot-air dryer

In this paper, a coupled radiative transfer equation and diffusion approximation model is extended for light propagation in turbid medium with low-scattering and non-scattering regions. The light propagation is modelled with the radiative transfer equation in sub-domains in which the assumptions of the diffusion approximation are not valid. The diffusion approximation is used elsewhere in the domain. The two equations are coupled through their boundary conditions and they are solved simultaneously using the finite element method. The streamline diffusion modification is used to avoid the ray-effect problem in the finite element solution of the radiative transfer equation. The proposed method is tested with simulations. The results of the coupled model are compared with the finite element solutions of the radiative transfer equation and the diffusion approximation and with results of Monte Carlo simulation. The results show that the coupled model can be used to describe photon migration in turbid medium with low-scattering and non-scattering regions more accurately than the conventional diffusion model.

A coupled radiative transfer equation and diffusion approximation model for photon migration in tissues is proposed. The light propagation is modeled with the radiative transfer equation in sub-domains in which the assumptions of the diffusion approximation are not valid and the diffusion approximation is used elsewhere in the domain. The coupled equations are solved using the finite element method. The proposed method is tested with simulations. The results of the coupled radiative transfer equation and diffusion approximation model are compared with the finite element solutions of the radiative transfer equation and the diffusion approximation. The results show that the coupled radiative transfer equation and diffusion approximation model can be used to describe photon migration in tissues more accurately than the conventional diffusion model.

A diffusion model is used to find heavy traffic approximate autocorrelation functions for several variables in queueing systems (i.e. for waiting time, system time, number in system and unfinished work). A table is given by which correlations can easily be found for each variable in anyGI/G/1 queue. Further, the infinite sum or integral of the autocorrelations is also found, and the spectral density function. The sum has applications in statistical analysis of queues.Extensive comparisons of approximate and exact correlations and their sums are reported, particularly for waiting times and system times, but also including number in system inM/M/1 queues. In general the correlations have similar accuracy to the probability distributions found by diffusion approximations. The percentage error is less for number in system than for waiting time.

Optical measurements on tissue are involved in many aspects of photobiology, e.g., identification of weak chromophores, evaluation of action spectra, and dosimetry calculations in phototherapies. Tissues are turbid materials that require different spectroscopic and analytical methods than non-scattering media. The Kubelka-Munk analysis has been employed for many years in the pigment and paper industries and to a limited extent in photobiology. It is based on the assumption that the radiant flux in a uniformly illuminated slab is uniformly diffuse and can be separated into a forward and a backward component, each subject to absorption and back-scattering (Kortum, 1969). Radiative transfer theory treats the transport of photons in terms of microscopic scattering and absorption events (Chandrasekhar, 1950). The diffusion approximation is a simplification of radiative transfer, applicable when scattering dominates over absorption (Ishimaru, 1978). In the present paper, the Kubelka-Munk and diffusion models are applied to transmission and diffuse reflection measurements on a potato layer, which was found to be a reproducible and easily handled tissue. The data were obtained with a laboratory-built spectrophotometer utilizing an integrating sphere.

On the Interspike Intervals Calculated from Diffusion Approximations of Stein's Neuronal Model with Reversal Potentials

In any finite genetic population, the effect of chance involved in the sampling by which one generation replaces another causes gene frequency change as long as fixation or extinction is not achieved. In genetic terminology this fluctuation in gene frequency due to the finiteness of a population is called 'random genetic drift'. Fisher (1922), using an equation of the diffusion type, was the first to make an effort to describe quantitatively the phenomenon of random genetic drift. In the corrected version of this work (1930), he found the general result that the rate of decrease of the genetic variance in the population due to drift was 1/(2N) per generation, where N is the number of individuals of mating potential. This result was confirmed by various authors using different methods (e.g. Wright, 1931, 1937; Malecot, 1944; Feller, 1950). The particular form of the diffusion model which has had the widest use was introduced into genetical literature by Wright (1945). This equation which plays a fundamental role in the theory of gene frequencies is the Kolmogorov forward equation or-as it is commonly known among physicists-the Fokker-Planck equation. Earlier Kolmogorov (1935) had introduced the steady-state form of the forward equation in genetics. The complete solutions to the problem, however, were obtained much later and are due to Kimura (1955, 1957). The problem of random genetic drift as a problem in finite Markov chains was first considered by Malecot (1944). Based on the largest non-unit characteristic root of the transition probability matrix Malecot obtained the asymptotic rate of decrease of heterozygosity. This is essentially the same as the previously known results due to Fisher and Wright for steady decay. Feller (1951) succeeded in finding a general expression for all the characteristic roots of the transition probability matrix. The use of diffusion approximations requires the assumption of an infinite population: while in fact the state space is discrete, it is treated as though it were continuous. The tacit assumption in such an approach is that for large population size the approximation of the discrete time, discrete space process by the continuous diffusion process does not result in serious error. It was only recently that a rigorous justification of the diffusion approach was given by Watterson (1962) with respect to the processes considered by Wright (1945) and Kimura (1954, 1955). The validity of the diffusion approximation as providing adequate numerical solution to the problem concerning the probability that a particular gene is lost or fixed by a certain time was one of the questions considered by Ewens (1963). Since exact expressions were not available his method of getting exact results consisted in simply powering the transition matrix using a high-speed computer and collecting the appropriate terms.

Quantifying model uncertainty in dynamical systems driven by non-Gaussian Lévy stable noise with observations on mean exit time or escape probability

A particular expression of the mutual coherence function of a wave is first derived for a system of random layers with rough boundaries, starting from the unified Bethe-Salpeter equation that involves the random medium and boundaries on exactly the same footing. An effective scattering matrix of the medium is then introduced, which is the only medium-dependent quantity involved in the expression and is obtained as a boundary-value solution of the diffusion equation. The diffusion approximation is based on an eigenfunction expansion by using a set of eigenfunctions of the medium scattering cross section that can be even rotationally variant, and the first term is good enough in the diffusion region. Here each expansion coefficient is obtained as the solution of an integral equation with terms of the boundaries, and, for the first term, the equation can be converted to a diffusion equation with a source term, subjected to boundary conditions determined by the boundary scattering cross sections. Here the condition at each boundary is valid even when the averaged refractive indices of the two media differ from each other by a large amount, as contrasted to the condition of no reflection used so far when obtaining boundary-value solutions of diffusion equations. Specific expressions are obtained for both backscattered and transmitted waves through a random layer, along with numerical examples. The boundary condition is finally generalized to meet the case of a rough boundary between two random media of different kinds, consistent with power conservation.

We use singular perturbation methods to analyze a diffusion equation that arose in studying two tandem queues. Denoting by p(n1, n2) the probability that there are n1 customers in the first queue and n2 customers in the second queue, we obtain the approximation p(n1, n2)∼ɛ2P(X, Y)=ɛ2P(ɛn1, ɛn2), where ɛ is a small parameter. The diffusion approximation P satisfies an elliptic PDE with a nondiagonal diffusion matrix and boundary conditions that involve both normal and tangential derivatives. We analyze the boundary value problem using the ray method of geometrical optics and other singular perturbation techniques. This yields the asymptotic behavior of P(X, Y) for X and/or Y large.

Experimental measurements of the diffusion constant for ultrasonic waves (around 3 MHz) propagating in water through a random set of scatterers (parallel metallic rods arranged as a slab) are presented. The slab thickness is around ten times the transport mean free path. Transmitted waves are recorded over hundreds of emitting/receiving positions in order to estimate the ensemble-averaged transmitted intensity 〈I(x,t)〉. Focused beamforming is performed on both faces of the sample in order to mimic a set of point-like sources and receivers. In theory, under the diffusion approximation, the ratio of the off-axis intensity 〈I(x,t)〉 to the on-axis intensity 〈I(0,t)〉 shows a simple gaussian dependence on the lateral dimension x, independently from absorption or boundary conditions. This yields a simple way to estimate the diffusion constant D and therefore characterize the scattering medium. Based on that method, broadband as well as frequency-resolved measurements of the diffusion constant are presented in controllable model media, such as these forests of steel rods. Experimental results and difficulties for measuring a reliable value for D on a real sample are discussed.

Optimal reinsurance under the mean–variance premium principle to minimize the probability of ruin