Diffusion Approximation Research Articles

Stochastic Gauss-Newton method for estimating absorption and scattering in optical tomography with the Monte Carlo method for light transport.

Image reconstruction in optical tomography in the so-called transport regime, where the diffusion approximation is not valid, requires modeling of light transport using the radiative transfer equation. In this work, we approach this problem by utilizing the Monte Carlo method for light transport. In this work, we propose a methodology for absolute imaging of absorption and scattering in this regime utilizing a Monte Carlo method for light transport. The image reconstruction problem is formulated as a minimization problem that is solved using a stochastic Gauss-Newton method. In the construction of the Jacobian matrix for scattering, a perturbation approximation for Monte Carlo is utilized. The approach is evaluated with numerical simulations using an adaptive approach where the number of photon packets is adjusted during the iterations, and with different fixed numbers of photon packets. The simulations show that the Monte Carlo method for light transport can be utilized in the absolute imaging problem of optical tomography and that the absorption and scattering parameters can be estimated simultaneously with good accuracy.

Error bounds for one-dimensional constrained Langevin approximations for nearly density-dependent Markov chains

AbstractContinuous-time Markov chains are frequently used to model the stochastic dynamics of (bio)chemical reaction networks. However, except in very special cases, they cannot be analyzed exactly. Additionally, simulation can be computationally intensive. An approach to address these challenges is to consider a more tractable diffusion approximation. Leite and Williams (Ann. Appl. Prob.29, 2019) proposed a reflected diffusion as an approximation for (bio)chemical reaction networks, which they called the constrained Langevin approximation (CLA) as it extends the usual Langevin approximation beyond the first time some chemical species becomes zero in number. Further explanation and examples of the CLA can be found in Anderson et al. (SIAM Multiscale Modeling Simul.17, 2019).In this paper, we extend the approximation of Leite and Williams to (nearly) density-dependent Markov chains, as a first step to obtaining error estimates for the CLA when the diffusion state space is one-dimensional, and we provide a bound for the error in a strong approximation. We discuss some applications for chemical reaction networks and epidemic models, and illustrate these with examples. Our method of proof is designed to generalize to higher dimensions, provided there is a Lipschitz Skorokhod map defining the reflected diffusion process. The existence of such a Lipschitz map is an open problem in dimensions more than one.

Heat and mass transfer analysis of non‐miscible couple stress and micropolar fluids flow through a porous saturated channel

AbstractThis study examines the flow rate, Bejan number transportation, concentration distribution and thermal characteristics of an immiscible couple stress‐ micropolar fluids within a porous channel. The authors focus on the effects of heat radiation and an angled magnetic field on the thermal dispersion, concentration distribution and entropy formation of two different types of incompressible immiscible micropolar and couple stress fluids inside a porous channel. Here, the static walls of the channel are isothermal, and the pressure gradient in the flow domain's entrance zone is constant. In this flow problem, we tried to simulate thermal radiation in the energy equation by applying Rosseland's diffusion approximation. To solve the problem, the authors have used no‐slip conditions at the channel's immovable walls, a continuity of temperature profile, shear stresses, thermal flux, linear velocity, and micro‐rotational velocity over the fluid‐fluid interface. The equations that govern the flow of immiscible fluids are solved using a well‐defined methodology and both the temperature and flow field are then evaluated using a closed‐form solution. The mathematical results of the thermal distribution and flow velocity are used to derive the Bejan number distribution and the entropy generation number. Graphical discussions are used to illustrate the impact of different emerging factors on the model's flow and thermal properties, which describe the major impact of the proposed model. These variables involve the micropolarity parameter, Reynolds number, inclination angle parameter, radiation parameter, and Hartmann number. The outcomes of the present models are corroborated by previously established results available in the literature.

Thermodynamically consistent diffuse-interface mixture models of incompressible multicomponent fluids

The prototypical diffuse-interface model for incompressible fluid mixtures is the Navier–Stokes Cahn–Hilliard (Allen–Cahn) model. Despite its foundation in continuum mixture theory, it is not fully compatible with this theory due to the diffusive flux approximation. This paper introduces a class of thermodynamically consistent diffuse-interface incompressible fluid mixture models that is fully compatible with the continuum theory of mixtures. The proposed models can be formulated in either constituent or mixture quantities, enabling a direct comparison with the Navier–Stokes Cahn–Hilliard (Allen–Cahn) model with non-matching densities. This comparison reveals the key modelling simplifications employed in the latter.

Epidemic modelling by birth-death processes with spatial scaling

In epidemic modeling, interpretation of compartment quantities, such as s, i, and r in relevant equations, is not always straightforward. Ambiguities regarding whether these quantities represent numbers or fractions of individuals in each compartment rise questions about significance of the involved parameters. In this paper, we address these challenges by considering a density-dependent epidemic modelling by a birth-death process approach inspired by Kurtz from 1970s’. In contrast to existing literature, which employs population size scaling under constant population condition, we scale with respect to the area. Namely, under the assumption of spatial homogeneity of the population, we consider the quantities of susceptible, infective and recovered per unit area. This spatial scaling allows diffusion approximation for birth-death type epidemic models with varying population size. By adopting this approach, we anticipate to contribute to a clear and transparent description of compartment quantities and parameters in epidemic modeling.

A Diffusion-Based Approach for Simulating Forward-in-Time State-Dependent Speciation and Extinction Dynamics.

We establish a general framework using a diffusion approximation to simulate forward-in-time state counts or frequencies for cladogenetic state-dependent speciation-extinction (ClaSSE) models. We apply the framework to various two- and three-region geographic-state speciation-extinction (GeoSSE) models. We show that the species range state dynamics simulated under tree-based and diffusion-based processes are comparable. We derive a method to infer rate parameters that are compatible with given observed stationary state frequencies and obtain an analytical result to compute stationary state frequencies for a given set of rate parameters. We also describe a procedure to find the time to reach the stationary frequencies of a ClaSSE model using our diffusion-based approach, which we demonstrate using a worked example for a two-region GeoSSE model. Finally, we discuss how the diffusion framework can be applied to formalize relationships between evolutionary patterns and processes under state-dependent diversification scenarios.

Formation of Giant Planets by Gas Disk Gravitational Instability on Wide Orbits around Protostars with Varied Masses. II. Quadrupled Spatial Resolution and Beta Cooling

Exoplanet demographics are sufficiently advanced to provide important constraints on theories of planet formation. While core and pebble accretion are preferred for rocky and icy planets, there appears to be a need for gas disk gravitational instability (GDGI) to play a role in the formation of M-dwarf gas giants and those orbiting at large distances. Here we present GDGI models that go beyond those presented by Boss (2011) dealing with the formation of wide-orbit gas giants. The new models use quadrupled spatial resolution, in both the radial and azimuthal directions, to reduce the effects of finite spatial resolution. The new models also employ the β cooling approximation, instead of the diffusion approximation used by Boss (2011), in order to push the models further in time. As in Boss (2011), the central protostars have masses of 0.1, 0.5, 1.0, 1.5, or 2.0 M ⊙, surrounded by disks with masses ranging from 0.019 M ⊙ to 0.21 M ⊙. For each case, two models are computed, one with an initial minimum Toomre Q stability value ranging from 1.1 to 1.7, and one with a higher initial disk temperature, resulting in the initial minimum Q ranging from 2.2 to 3.4. These new models continue to show that GDGI can explain the formation of gas giants at distances of ∼30 to ∼50 au on eccentric orbits (e less than ∼0.2), though the number formed drops to 0 as the protostar mass decreases to 0.1 M ⊙.

Evolution of cooperation in social dilemmas with assortment in finite populations

We investigate conditions for the evolution of cooperation in social dilemmas in finite populations with assortment of players by group founders and general payoff functions for cooperation and defection within groups. Using a diffusion approximation in the limit of a large population size that does not depend on the precise updating rule, we show that the first-order effect of selection on the fixation probability of cooperation when represented once can be expressed as the difference between time-averaged payoffs with respect to effective time that cooperators and defectors spend in direct competition in the different group states. Comparing this fixation probability to its value under neutrality and to the corresponding fixation probability for defection, we deduce conditions for the evolution of cooperation. We show that these conditions are generally less stringent as the level of assortment increases under a wide range of assumptions on the payoffs such as additive, synergetic or discounted benefits for cooperation, fixed cost for cooperation and threshold benefit functions. This is not necessarily the case, however, when payoffs in pairwise interactions are multiplicatively compounded within groups.

Scaling limits for the generalized Langevin equation via diffusion approximation theory

Scaling limits for the generalized Langevin equation via diffusion approximation theory

Analytical solutions in the modeling of the endovenous laser ablation

We modeled the operative treatment of incompetent truncal veins using endovenous laser ablation (EVLA). The main concern regarding the thermoablative technique is tissue damage, which is correlated with (1) the energy provided by the laser power and (2) temperature distribution. Our objective was to accurate the two functions, namely the fluence rate and the temperature, depending on the thermoablative technique and the endovenous laser treatment (ELT). First, we considered three differential equations: diffusion, heat, and bioheat equations in the endovenous-perivenous multidomain to describe the lumen, the vein wall, the tissue pad, and the skin. Second, we examined the power source according to the Beer-Lambert law in the incident beam irradiance and the heat source as the so-called absorbed optical power density. Third, we checked out the heat transfer at the skin boundary according to Newton’s law of cooling, which stands for a Robin boundary condition. For this new model, we proposed exact solutions: applying differential equations techniques, we solved (1) a diffusion approximation of the radiative transfer equation under the considered power source, and (2) the coupled heat and bioheat equations under the considered heat source accomplished with the Robin boundary condition. Then, we graphically illustrated the fluence rate profile and discussed its time dependence and steady state. Besides, we discuss thermal damage to the vein-tissue system and present open problems.

Hydraulic River Models From ICESat‐2 Elevation and Water Surface Slope

AbstractForecasting flood and drought events requires accurate modeling tools. Hydraulic river models are based on estimates of riverbed geometry which are traditionally collected in situ. The novel Ice, Cloud and Land Elevation Satellite 2 [ICESat‐2] lidar altimetry mission with 6 simultaneous high‐resolution laser beams provides the opportunity to define river cross‐section geometries as well as observe water surface elevation [WSE] and water surface slope spatially resolved along the river chainage. This paper describes a method to utilize terrain altimetry and water surface slope estimates to define complete river geometries from ICESat‐2 data products, using the diffusive wave approximation to calculate depth in the submerged section not penetrated by the lidar. Exemplifying the method, cross‐sections are defined for a stretch of the Mekong River. Hydrodynamic model results of the stretch are compared with ICESat‐2 WSE estimates and in situ gauging station time series. Insights in river characteristics from satellite imagery and the ICESat‐2 slope estimates allow for fine‐tuning of the cross‐sections using spatially varying Manning numbers. The final model achieves a root mean square error against the ICESat‐2 WSE of 0.676 m and average Kling‐Gupta Efficiency against gauging station time series of 0.880. The method is limited by the diffusive wave approximation resulting in inaccurate cross‐section estimates in sections with supercritical flow or significant acceleration. Errors can be identified from ICESat‐2 WSE estimates and reduced with additional cross‐sections. Combined with hydrological models, the method will allow for cross‐section definition without in situ data.

Diffusion approximations for self-excited systems with applications to general branching processes

Diffusion approximations for self-excited systems with applications to general branching processes

Infrared‐aided hot‐air drying of coconut: Impact on drying kinetics and quality metrics

AbstractThis study explored various drying techniques and temperatures to analyze their effects on the drying kinetics and quality of copra. The initial moisture content of coconut kernels was 50%–55% (w.b.), which decreased to 6%–8% (w.b.) as a result of the drying process. This study focuses on evaluating the individual and hybrid effects of infrared drying (IRD) and hot‐air drying (HAD) techniques to enhance the quality of copra. Three drying methods were used: IRD, HAD, and infrared‐assisted hot‐air drying (IRAHAD). Coconut pieces were subjected to different drying temperatures (50, 60, and 70°C) with a constant air speed of 2 m/s. Optimal results were achieved by employing the IRAHAD method at 60°C, preserving a crucial fat content of 68.4% essential for increased extraction of oil from copra and comparatively high drying rates. In particular, the drying rates in IRAHAD were twice as high as those in IRD and HAD. At a drying temperature of 60°C, the logarithmic model and the diffusion approximation model were deemed the best fit for HAD and IRAHAD, respectively.Practical applicationsThis study demonstrates the efficacy of infrared‐assisted hot‐air drying (IRAHAD) at 60°C in preserving copra's crucial fat content of 68.4% for optimal oil extraction. By implementing IRAHAD, producers can efficiently dry coconut kernels while maintaining quality, enhancing extraction yields, and improving overall profitability in the coconut processing industry.

A queueing-based approach for integrated routing and appointment scheduling

This paper aims to address the integrated routing and appointment scheduling (RAS) problem for a single service provider. The RAS problem is an operational challenge faced by operators that provide services requiring home attendance, such as grocery delivery, home healthcare, or maintenance services. While considering the inherently random nature of service and travel times, the goal is to minimize a weighted sum of the operator’s travel times and idle time, and the client’s waiting times. To handle the complex search space of routing and appointment scheduling decisions, we propose a queueing-based approach to effectively deal with the appointment scheduling decisions. We use two well-known approximations from queueing theory: first, we use an approach based on phase-type distributions to accurately approximate the objective function, and second, we use an heavy-traffic approximation to derive an efficient procedure to obtain good appointment schedules. Combining these two approaches results in a fast and sufficiently accurate hybrid approximation, thus essentially reducing RAS to a routing problem. Moreover, we propose the use of a simple yet effective large neighborhood search metaheuristic to explore the space of routing decisions. The effectiveness of our proposed methodology is tested on benchmark instances with up to 40 clients, demonstrating an efficient and accurate methodology for integrated routing and appointment scheduling.

Investigation and comparison on the interaction of laser and bio-tissue based on two classical models, the beam broaden model and diffusion approximation equation model, theoretically and experimentally

Three methods are used for a numerical solution, the Monte Carlo method, diffusion approximation equation model, and beam broaden model based on Beer–Lambert’s law equation. The comparison between the first two methods is reported theoretically, and the latter is a better choice in the high-density tissue. However, the comparison between the third method and the first or the second method is rarely reported. Two classical theoretical models describing the interaction between the laser and the bio-tissue are analyzed and compared to determine which is more suitable for analyzing the interaction, the beam broaden model or diffusion approximation equation model. Intensity distribution is simulated and compared for the two models. Temperature distribution and thermal damage are investigated theoretically and experimentally for both models. The differences and the reasons are analyzed. The diffusion approximation equation model is more suitable for analyzing the mechanism between the laser and the bio-tissue based on the degree of fitting between the simulated and experimental data. Theoretical analyses for the two models are carried out in detail. The comparison between the two models is rarely reported, and it is reported in this article for the first time, theoretically and experimentally. This report provides a better choice for quickly analyzing the interaction mechanism between the laser and the bio-tissue.

Uniqueness and numerical inversion in bioluminescence tomography with time-dependent boundary measurement

In the paper, an inverse source problem in bioluminescence tomography (BLT) is investigated. BLT is a method of light imaging and offers many advantages such as sensitivity, cost-effectiveness, high signal-to-noise ratio and non-destructivity. It thus has promising prospects for many applications such as cancer diagnosis, drug discovery and development as well as gene therapies. In the literature, BLT is extensively studied based on the (stationary) diffusion approximation (DA) equation, where the distribution of peak sources is reconstructed and no solution uniqueness is guaranteed without proper a priori information. In this work, motivated by solution uniqueness, a novel dynamic coupled DA model is proposed. Theoretical analysis including the well-posedness of the forward problem and the solution uniqueness of the inverse problem are given. Based on the new model, iterative inversion algorithms under the framework of regularizing schemes are introduced and applied to reconstruct the smooth and non-smooth sources. We discretize the regularization functional with the finite element method and give the convergence rate of numerical solutions. Several numerical examples are implemented to validate the effectiveness of the new model and the proposed algorithms.

Spectra and equitemporal surfaces of the photosphere of the ultrarelativistic shell as applied to Gamma-ray bursts

We study the radiation of the ultrarelativistic shell in the diffusion approximation, which takes place at the initial stage of a gamma-ray burst. We get the effective temperature, instantaneous and time-integrated spectra for the parabolic distribution of the initial internal energy of the shell. Also we considered the types of the equitemporal surfaces with a different type of the movement of the shell.

Inverse Source Problem in Radiative Transport with Nonlinear Anisotropic Sources

An inverse source problem in the stationary radiative transport in two dimension in an absorbing and scattering medium is considered, where the source is directionally quadratic polynomial. The medium has an anisotropic scattering property that is neither negligible nor large enough for the diffusion approximation to hold. The attenuating and scattering properties of the medium are assumed known. For scattering kernels of finite Fourier coefficients in the angular variable, we show how to recover the nonlinear anisotropic radiative source from boundary measurements.

Critical Multitype Branching Processes with Random Migration

Abstract The aim of this paper is to introduce a multitype branching process with random migration following the research initiated with the Galton–Watson process with migration introduced in [N. M. Yanev and K. V. Mitov, Controlled branching processes: The case of random migration, C. R. Acad. Bulgare Sci. 33 1980, 4, 473–475]. We focus our attention in what we call the critical case. Sufficient conditions are provided for the process to have unlimited growth or not. Furthermore, using suitable normalizing sequences, we study the asymptotic distribution of the process. Finally, we obtain a Feller-type diffusion approximation.

Zastosowanie „aproksymacji dyfuzyjnej” do opisu procesu zagospodarowania złóż ropy naftowej

Analysis of the interaction of wells in the process of oil field development is based in particular on calculating the correlation coefficient of two adjacent wells. However, due to reservoir heterogeneity, this approach fails to consider the possibility of interactions between wells located at significantly greater distances. The non-equilibrium of the system is attributed to its openness, associated with environmental impacts (flooding process and changes in the stock of existing wells). The oil field development is a complex process, subject to complex works such as well grid compaction, carrying out a wide range of geological and technical operations, etc. From this point of view, the development and exploitation of fields needs a “volumetric” approach, i.e. a diffusion approximation. A crucial aspect of oil production is the timely regulation of both current production rates and water impact on the reservoir system. The oscillatory nature of the time series of measurements of oil, water, and liquid production rates carries information about the state and behavior of the reservoir system. Analysis of the features of oscillatory processes of the technological indicators of well operation across the entire area of the deposit as a whole enables early diagnosis of changes in the state of the system. The approach of the production well stock analysis in terms of the amplitude-frequency characteristics of the dynamics of oil, water, “mobile” water, and flooding, as well as the specific ratios of produced oil to the volume of water injected into the reservoir (an indicator of the effectiveness of water stimulation) allows us to consider the development as a diffusion-like process.