Diffusion Approximation Research Articles

Determining sources in the bioluminescence tomography problem

Abstract In this paper, we revisit the bioluminescence tomography\,(BLT) problem, where one seeks to reconstruct bioluminescence signals (an internal light source) from external measurements of the Cauchy data. As one kind of optical imaging, the BLT has many merits such as high signal-to-noise ratio, non-destructivity and cost-effectiveness etc., and has potential applications such as cancer diagnosis, drug discovery and development as well as gene therapies and so on. In the literature, BLT is extensively studied based on diffusion approximation (DA) equation, where the distribution of peak sources is to be reconstructed and no solution uniqueness is guaranteed without adequate a priori information. Motivated by the solution uniqueness issue, several theoretical results are explored. The major contributions in this work that are new to the literature are two-fold: first, we show the theoretical uniqueness of the BLT problem where the light sources are in the shape of $C^2$ domains or polyhedral- or corona-shaped; second, we support our results with plenty of problem-orientated numerical experiments.

Establishment of a locally adaptive allele in multidimensional continuous space.

Local adaptation is widely seen when species adapt to spatially heterogeneous environments. Although many theoretical studies have investigated the dynamics of local adaptation using two-population models, there remains a need to extend the theoretical framework to continuous space settings, reflecting the real habitats of species. In this study, we use a multidimensional continuous space model and mathematically analyze the establishment process of local adaptation, with a specific emphasis on the relative roles of mutation and migration. First, the role of new mutations is evaluated by deriving the establishment probability of a locally adapted mutation using a branching process and a diffusion approximation. Next, the contribution of immigrants from a neighboring region with similar environmental conditions is considered. Theoretical predictions of the local adaptation rate agreed with the results of Wright-Fisher simulations in both mutation-driven and migration-driven cases. Evolutionary dynamics depend on several factors, including the strength of migration and selection, population density, habitat size, and spatial dimensions. These results offer a theoretical framework for assessing whether mutation or migration predominantly drives convergent local adaptation in spatially continuous environments in the presence of patchy regions with similar environmental conditions.

On magnetohydrodynamics free convective flow within coaxial cylinder due to nonlinear thermal radiation in an isothermal/isoflux condition

A theoretical exploration was reported to highlight the electrically viscous conducting fluid within a coaxial cylinder. Both the outer surface of the inner cylinder and the inner surface of the outer cylinder were porous, heated, and cool, respectively. The action of a simultaneous radial transverse magnetic field and velocity slip were imposed across the coaxial cylinder. For appropriate insight into the physical problem, the thermal Rosseland diffusion approximation was used to indicate the radiative heat flux in the energy equation. The aim is to investigate the effects of isothermal and isoflux conditions on slip-upshot heat transfer in a vertical coaxial cylinder due to nonlinear thermal radiation. The steady-state and unsteady solutions of the physical problem were provided and solved via regular perturbation and implicit finite different domains, respectively. It was noted that the magnetic and suction parameters reduced the fluid velocity and skin friction under both isothermal and isoflux conditions. Conversely, the Grashof number, thermal radiation, and injection parameters increased the fluid velocity, fluid temperature, and skin friction, with a more significant effect observed under isothermal conditions on the surfaces of the coaxial cylinder. Additionally, the sensitivity analysis revealed that the radiation parameter is highly sensitive under isothermal conditions, followed by the temperature difference parameter. In contrast, under isoflux conditions, the injection parameter shows extreme sensitivity, with the radiation parameter being the next most sensitive. The result reveals an excellent agreement between the analytical and numerical results at maximum time.

Tutorial on the Use of the Photon Diffusion Approximation for Fast Calculation of Tissue Optical Properties.

Computer simulations, which are performed at a single wavelength at a time, have been traditionally used to estimate the optical properties of tissues. The results of these simulations need to be interpolated. For a broadband estimation of tissue optical properties, the use of computer simulations becomes time consuming and computer demanding. When spectral measurements are available for a tissue, the use of the photon diffusion approximation can be done to perform simple and direct calculations to obtain the broadband spectra of some optical properties. The additional estimation of the reduced scattering coefficient at a small number of discrete wavelengths allows to perform further calculations to obtain the spectra of other optical properties. This study used spectral measurements from the heart muscle to explain the calculation pipeline to obtain a complete set of the spectral optical properties and to show its versatility for use with other tissues for various biophotonics applications.

Efficient Computation of Radiative Heat Recovery from Porous Ceramic Monoliths for Efficient Solar Thermochemical Fuel Production

Solar thermochemical hydrogen (STCH) produced by heat-driven water-splitting is a promising route for producing green hydrogen and other zero-emission synfuels. However, the efficiency of STCH must be dramatically increased for it to make an impact on decarbonization efforts. We have previously presented a novel Reactor Train System (RTS) for significantly increasing the efficiency of STCH by employing heat recovery from the redox material and efficient gas exchange processes. In this paper we present a higher-fidelity model for the RTS that accommodates the slow heat diffusion through the STCH redox material. For this purpose, a novel method is introduced for transient modelling of radiative heat in participating media. This method, called GREENER: Generalized Radiation Exchange Factors and Net Radiation, combines the accuracy of Monte Carlo Ray Tracing with the low computational cost of the P1 or Rosseland diffusion approximations. Along with STCH, GREENER has application for modelling volumetric solar receivers, high temperature heat recovery systems like heat exchangers and regenerators, and packed bed reactors. Using the GREENER method, the RTS counterflow radiative heat exchanger is shown to achieve heat recovery effectiveness greater than 70%. The performance of non-uniform porous redox morphologies is evaluated, and high-performing configurations are identified.

Mutation potentiates migration swamping in polygenic local adaptation.

Locally adapted traits can exhibit a wide range of genetic architectures, from pronounced divergence at a few loci to small frequency divergence at many loci. The type of architecture that evolves depends strongly on the migration rate, as weakly selected loci experience swamping and do not make lasting contributions to divergence. Simulations from previous studies showed that even when mutations are strongly selected and should resist migration swamping, the architecture of adaptation can collapse and become transient at high mutation rates. Here, we use an analytical two-population model to study how this transition in genetic architecture depends upon population size, strength of selection, and parameters describing the mutation process. To do this, we develop a mathematical theory based on the diffusion approximation to predict the threshold mutation rate above which the transition occurs. We find that this performs well across a wide range of parameter space, based on comparisons with individual-based simulations. The threshold mutation rate depends most strongly on the average effect size of mutations, weakly on the strength of selection, and marginally on the population size. Across a wide range of the parameter space, we observe that the transition to a transient architecture occurs when the trait-wide mutation rate is 10-3-10-2, suggesting that this phenomenon is potentially relevant to complex traits with a large mutational target. On the other hand, based on the apparent stability of genetic architecture in many classic examples of local adaptation, our theory suggests that per-trait mutation rates are often relatively low.

"Development of a statistical calculation model for the carbon diffusion parameters in metals and alloys"

Carbon diffusion in metals has received a lot of attention and has been the subject of intensive theoretical investigations in recent years. The purpose of this work is development the statistical calculation model (SCM) on the diffusion of carbon in metals and its application for calculating the diffusion coefficients of carbon in alloys. It includes first-principles calculation of the diffusion coefficient according to a statistical model, physicochemical calculation of activation energies for carbon, and linear approximation of carbon diffusion in alloys. The calculated values of the diffusion coefficient for metals are within the range of the experimental values. For low-melting metals, carbon diffusion coefficients are mainly unknown from experiment, but the statistical model allows us to predict their values. The calculations are compared with known experimental data on the diffusion of carbon atoms in some metals Fe, V, Ta and W at high temperatures with fairly good agreement between the results. The SCM-model allows us to determine the influence of the alloying elements Si, Mo and Cr on the diffusion of carbon in the F – C alloy.

Velocity of ultrasound diffusion in heterogeneous media and its applications

This paper investigates the velocity of ultrasonic diffusion in heterogeneous media under the conditions where the diffuse wave approximation is valid (λ≤a). The diffusion velocity is defined as the moving speed of the peak of the energy evolution curve. The peak arrival time as a function of transport distance is calculated for infinite spaces and bounded domains. The results show that the peak arrival time is independent of the domain’s geometry, i.e. the size, shape, and boundary conditions. This is confirmed by comparing the arrival times in an infinite space and a bounded domain with a geometric feature – a surface-breaking crack of varying depth. Therefore, the velocity of ultrasonic diffusion is an intrinsic property of a medium, and the formula for the infinite three-dimensional space is sufficient to calculate the arrival time–transport distance relationship, given the diffuse properties of the medium. These findings eliminate the needs for the finite element numerical simulations in the applications to determine geometric parameters such as the crack depth using diffuse ultrasound. The diffusion velocity is a function of transport distance in general, while its far-field asymptotic value is constant regardless of the dimensionality and geometry of the domain.

Asymptotically optimal routing of a many-server parallel queueing system with long-run average criterion

This paper considers a parallel queueing system with multiple stations, each of which contains many statistically identical servers and has a dedicated queue. Upon each customer arrival, the system manager must decide to which station the customer should be routed, with the objective of minimizing the system’s long-run average delay cost. One feature of this paper is that a customer’s delay cost depends not only on his/her delay, but also on the routed station. Considering this heterogeneity across stations, we propose a routing policy, which can be regarded as an extension of the MED–FSF policy. Under this policy, any arriving customer will be routed to: (i) the station with the minimum value, which depends on the station’s expected delay and the station index when servers in all stations are fully occupied; or otherwise (ii) the station with a fastest idle server. Using asymptotic analysis, we derive diffusion limits of queue-length processes and their stationary distributions under the proposed policy in the Halfin–Whitt regime. Combined with an asymptotic lower bound result for the long-run average delay cost, we show that the proposed routing policy is asymptotically optimal under the considered objective. Finally, we provide numerical experiments to validate the accuracy of our diffusion approximation, and we compare the performance metrics under the proposed policy with those under other commonly used routing policies.

Self-chemophoresis in the thin diffuse interface approximation

Self-chemophoresis is an appealing and quite successful interpretation of the motility exhibited by certain chemically active colloidal particles suspended in a solution of their ‘fuel’: the particle has a phoretic response to self-generated, rather than externally imposed, inhomogeneities in the chemical composition of the solution. The postulated mechanism of chemophoresis is the interaction of the particle (via an adsorption potential) with the chemical inhomogeneities in the surrounding medium. When the range of this interaction is much smaller than any other relevant scale in the system, the (translational and rotational) phoretic velocities can be described in terms of a phoretic coefficient and a slip fluid velocity at the surface of the particle. Using the case of a spherical particle as a simple and physically insightful example, here we exploit an integral representation of the rigid-body motion to critically re-examine this framework. The systematic analysis highlights two steps in the approximation: first the thin–layer approximation (very large particle size), and subsequently a lubrication approximation (slow variation of the adsorption potential along the tangential direction). We also discuss how these approximations could be relaxed and the effect of this on the particle's motion.

Influence of selection on the probability of fixation at a locus with multiple alleles

BackgroundGenes exist in a population in a variety of forms (alleles), as a consequence of multiple mutation events that have arisen over the course of time. In this work we consider a locus that is subject to either multiplicative or additive selection, and has n alleles, where n can take the values 2, 3, 4, …\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ldots$$\\end{document}. We focus on determining the probability of fixation of each of the n alleles. For n=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n = 2$$\\end{document} alleles, analytical results, that are ‘exact’, under the diffusion approximation, can be found for the fixation probability. However generally there are no equally exact results for n≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge 3$$\\end{document} alleles. In the absence of such exact results, we proceed by finding results for the fixation probability, under the diffusion approximation, as a power series in scaled strengths of selection such as Ri,j=2Ne(si-sj)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R_{i,j}=2N_{e}(s_{i}-s_{j})$$\\end{document}, where Ne\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_{e}$$\\end{document} is the effective population size, while si\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s_{i}$$\\end{document} and sj\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s_{j}$$\\end{document} are the selection coefficients associated with alleles i and j, respectively.ResultsWe determined the fixation probability when all terms up to second order in the Ri,j\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R_{i,j}$$\\end{document} are kept. The truncation of the power series requires that the Ri,j\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R_{i,j}$$\\end{document} cannot be indefinitely large. For magnitudes of the Ri,j\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R_{i,j}$$\\end{document} up to a value of approximately 1, numerical evidence suggests that the results work well. Additionally, results given for the particular case of n=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n = 3$$\\end{document} alleles illustrate a general feature that holds for n≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge 3$$\\end{document} alleles, that the fixation probability of a particular allele depends on that allele’s initial frequency, but generally, this fixation probability also depends on the initial frequencies of other alleles at the locus, as well as their selective effects.ConclusionsWe have analytically exposed the leading way the probability of fixation, at a locus with multiple alleles, is affected by selection. This result may offer important insights into CDCV traits that have extreme phenotypic variance due to numerous, low-penetrance susceptibility alleles.

First-passage and first-arrival problems in continuous-time random walks: Beyond the diffusion approximation.

Some exact solutions of the first-passage and first-arrival problems for the continuous-time random-walk model are obtained. On the basis of these exact solutions, the following has been revealed. First, for some jump-length distributions with a finite variance, the approximate solutions obtained in the diffusion approximation can differ significantly from the exact solutions. Second, for some waiting time distributions with a finite mean, the times of first passage and the times of first arrival can significantly depend on the ensemble under consideration. In particular, the mean first-passage time corresponding to the stationary ensemble can be significantly greater than the mean first-passage time corresponding to the nonaged ensemble. Third, for any continuous distribution of jump lengths, the probability of first arrival is zero for a point-like target. This last result is contrary to existing opinion, but it is consistent with the fact that a single point has a probability measure equal to zero in the probability space defined by a continuous distribution of jump lengths.

Quantum–Classical Hybrid Dynamics: Coupling Mechanisms and Diffusive Approximation

In this paper we demonstrate that any Markovian master equation defining a completely positive evolution for a quantum–classical hybrid state can always be written in terms of four basic coupling mechanisms. Each of them is characterized by a different “backaction” on each subsystem. On this basis, for each case, we find the conditions under which a diffusive limit is approached, that is, the time evolution can be approximated in terms of the first and second derivatives of the hybrid state with respect to a classical coordinate. In this limit, the restricted class of evolutions that guaranty the positivity of the hybrid state at all times (quantum Fokker-Planck master equations) emerges when the coupling mechanisms lead to infinitesimal (non-finite) changes in both the quantum and classical subsystems. A broader class of diffusive evolutions is obtained when positivity is only granted after a transient time or alternatively is granted after imposing an initial finite width on the state of the classical subsystem. A set of representative examples support these results.

Revisiting equivalent optical properties for cerebrospinal fluid to improve diffusion-based modeling accuracy in the brain.

The diffusion approximation (DA) is used in functional near-infrared spectroscopy (fNIRS) studies despite its known limitations due to the presence of cerebrospinal fluid (CSF). Nearly all of these studies rely on a set of empirical CSF optical properties, recommended by a previous simulation study, that were not selected for the purpose of minimizing DA modeling errors. We aim to directly quantify the accuracy of DA solutions in brain models by comparing those with the gold-standard solutions produced by the mesh-based Monte Carlo (MMC), based on which we derive updated recommendations. For both a 5-layer head and Colin27 atlas models, we obtain DA solutions by independently sweeping the CSF absorption ( ) and reduced scattering ( ) coefficients. Using an MMC solution with literature CSF optical properties as reference, we compute the errors for surface fluence, total brain sensitivity and brain energy-deposition, and identify the optimized settings where the such error is minimized. Our results suggest that previously recommended CSF properties can cause significant errors (8.7% to 52%) in multiple tested metrics. By simultaneously sweeping and , we can identify infinite numbers of solutions that can exactly match DA with MMC solutions for any single tested metric. Furthermore, it is also possible to simultaneously minimize multiple metrics at multiple source/detector separations, leading to our new recommendation of setting = 0.15 mm-1 while maintaining physiological for CSF in DA simulations. Our new recommendation of CSF equivalent optical properties can greatly reduce the model mismatches between DA and MMC solutions at multiple metrics without sacrificing computational speed. We also show that it is possible to eliminate such a mismatch for a single or a pair of metrics of interest.

Low-cost solar dryer dehydrates silver banana with a prediction of diffusion approximation model

Solar dryers made from recycled materials offer a cost-effective solution for promoting the widespread adoption of this technology and reducing food production costs. In this study, a low-cost solar dryer was constructed using repurposed materials, employing indirect exposure, and its performance was evaluated through a kinetic study on banana slice drying (Musa spp.). Drying was carried out until equilibrium moisture, reaching constant mass. The drying air's temperature and relative humidity (RH, %) were monitored. Fourteen empirical models were used to fit the experimental data. The banana slices took approximately 300 min to dry, with final moisture of 1.8%. The mean operational conditions during natural drying were 59.08±9.16 °C and RH = 39±4%. The diffusion Approximation model fitted the drying curve better, as it presented the lowest reduced Chi-square (χ2 = 2.9×10-5) and a high coefficient of determination (R2 = 0.9998). The effective diffusion coefficient (Def = 5.4×10-9 m2/s, R2 = 0.9935) was determined. Thus, the solar dryer demonstrated efficient performance in the banana drying process, requiring minimal design effort. Furthermore, despite the limitations in controlling the drying conditions, most of the mathematical models successfully predicted the drying process due to the dryer's ability to maintain the continuity of the drying curve, suggesting potential viability for this low-cost dryer.

Population genomics unravels a lag phase during the global fall armyworm invasion

The time that elapsed between the initial introduction and the proliferation of an invasive species is referred to as the lag phase. The identification of the lag phase is critical for generating plans for pest management and for the prevention of biosecurity failure. However, lag phases have been identified mostly through retrospective searches of historical records. The agricultural pest fall armyworm (FAW; Spodoptera frugiperda) is native to the New World. FAW invasion was first reported from West Africa in 2016, then it spread quickly through Africa, Asia, and Oceania. Here, using population genomics approaches, we demonstrate that the FAW invasion involved an undocumented lag phase. Invasive FAW populations have negative signs of genomic Tajima’s D, and invasive population-specific genetic variations have particularly decreased Tajima’s D, supporting a substantial amount of time for the generation of new mutations in introduced FAW populations. Model-based diffusion approximations support the existence of a period with a cessation of gene flow between native and invasive FAW populations. Taken together, these results provide strong support for the presence of a lag phase during the FAW invasion. These results show the usefulness of using population genomics analyses to identify lag phases in biological invasions.

What Remote PPG Oximetry Tells Us about Pulsatile Volume?

While pulse oximetry using remote photoplethysmography (rPPG) is used in medicine and consumer health, sound theoretical foundations for this methodology are not established. Similarly to traditional pulse oximetry, rPPG oximetry uses two wavelengths to calculate the tissue oxygenation using the so-called ratio-of-ratios, R. However, the relationship between R and tissue oxygenation has not been derived analytically. As such, rPPG oximetry relies mostly on empirical methods. This article aimed to build theoretical foundations for pulse oximetry in rPPG geometry. Using the perturbation approach in diffuse approximation for light propagation in tissues, we obtained an explicit expression of the AC/DC ratio for the rPPG signal. Based on this ratio, the explicit expression for "ratio-of-ratios" was obtained. We have simulated the dependence of "ratio-of-ratios" on arterial blood saturation across a wide range (SaO2 = 70-100%) for several commonly used R/IR light sources (660/780, 660/840, 660/880, and 660/940 nm) and found that the obtained relationship can be modeled by linear functions with an extremely good fit (R2 = 0.98-0.99) for all considered R/IR pairs. Moreover, the location of the pulsatile volume can be extracted from rPPG data. From experimental data, we found that the depth of blood pulsations in the human forehead can be estimated as 0.6 mm on the arterial side, which points to the papillary dermis/subpapillary vascular plexus origin of the pulsatile volume.

Irreversible reinsurance: minimization of capital injections in presence of a fixed cost

AbstractWe propose a model in which, in exchange to the payment of a fixed transaction cost, an insurance company can choose the retention level as well as the time at which subscribing a perpetual reinsurance contract. The surplus process of the insurance company evolves according to the diffusive approximation of the Cramér-Lundberg model, claims arrive at a fixed constant rate, and the distribution of their sizes is general. Furthermore, we do not specify any particular functional form of the retention level. The aim of the company is to take actions in order to minimize the sum of the expected value of the total discounted flow of capital injections needed to avoid bankruptcy and of the fixed activation cost of the reinsurance contract. We provide an explicit solution to this problem, which involves the resolution of a static nonlinear optimization problem and of an optimal stopping problem for a reflected diffusion. We then illustrate the theoretical results in the case of proportional and excess-of-loss reinsurance, by providing a numerical study of the dependency of the optimal solution with respect to the model’s parameters.

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.