Analytical approximation to the multidimensional Fokker–Planck equation with steady state

The mouse skull as a source of morphometric data for phylogeny inference

Optimum signal in a diffusion leaky integrate-and-fire neuronal model

This paper is concerned with the optimal investment strategy for a defined contribution (DC) pension plan. We assumed that the financial market consists of a risk-free asset and a risky asset, where the risky asset is subject to the Ornstein–Uhlenbeck (O-U) process, and stochastic income and inflation risk were also considered in the model. We firstly derived the Hamilton–Jacobi–Bellman (HJB) equation through the stochastic control method. Secondly, under the logarithmic utility function, the closed-form solution of optimal asset allocation was obtained by using the Legendre transform method. Finally, we give several numerical examples and a financial analysis.

Application of multi-scale finite element methods to the solution of the Fokker–Planck equation

The Petroleum Recovery Institute has operated a scaled steamflood physical model since 1975. The objective of this work was to define and to evaluate the capabilities and limitations of Intercomp Ltd. "s numerical steamflood simulator and the Petroleum Recovery Institute"s physical model. Three different types of laboratory experiments were selected for numerical history matching-waterflood, hot-waterflood and steamflood. The objective was to match oil, water and steam production as well as the energy balance and temperature profiles, using a single set of physical and petrophysical parameters. In addition, sensitivity studies were computed for three different grid systems, specifically a coarse cross-sectional grid (8xlx4), a fine cross-sectional grid (l6x1x8) and a three-dimensional grid (9x5x4). Excellent matching was obtained for the fluids production and for the energy balances. However, some, difficulties were encountered during attempts to match some of the temperature profiles, especially those associated with the flow of hot water along the bottom of the reservoir. As a result of the simulations, new measurements of thermal conductivity between the model and the underlying concrete bed were performed. A more complete interpretation of the physical model was obtained. The results indicate that physical model experiments offer useful data which can enable testing and calibration of numerical models, while the latter can help in obtaining a more precise description of the former. Therefore, the physical and numerical models should be used in a mutually complementary manner. INTRODUCTION Numerical simulators for steam injection oil recovery processes are being used frequently by reservoir engineers especially in the three areas of history matching(1–6), sensitivity studies(6–7) and field design and prediction(8–10). However, numerical simulators are complicated reservoir engineering tools and one has to realize their capabilities and limitations in order to use them properly. The successful design of any enhanced oil recovery process requires that one be equipped with a reliable set of reservoir data which includes geological, physical and petrophysical properties. History matching with a numerical simulator is often used to verify the parameters that have been obtained from laboratory measurements, well logs or by correlation. Typically, it is difficult to obtain a unique result from history-matching studies of field-derived performance data because a large number of parameters, involved in the calculations, must be either estimated or determined indirectly. History matching of laboratory steam flood experiments provides a better insight into the numerical simulator"s capabilities, because one has better control over most of the parameters defining the system. An additional advantage of history matching laboratory data is that more experimental data, usually of higher quality, are available to enable comparison with numerical results. The PRI Research Division has been studying steamflood oil recovery processes using a one-twelfth of an inverted seven spot, partially scaled physical model(ll.12.l3) since 1975. Some sixty experimental runs have been made to date for various steam injection rates, steam qualities, initial water saturations and oil characteristics. Three of these runs were chosen for this study – a cold-waterflood, a hot-waterf1ood and a steamflood. (Figure in full paper) Physical Model Apparatus The experimental apparatus is shown schematically in Figure 1.

Physical and numerical models can be employed together to better understand the SAGD process. When there are inadequate measured physical properties as input data, and insufficient built-in mathematical processes, it is beneficial to validate a numerical model by history matching physical model results. Based on Chung's experimental data from a physicalmodel, a two-dimensional, three-phase (bitumen, water and steam) and two-component (water & bitumen) black oil numerical model for the SAGD process was developed and tested, using the CMG "STARS" thermal simulator. Both the "Spreading Steam Chamber" and the "Rising Steam Chamber" processes were simulated. The numerical results agree reasonably well with the measured data for cumulative oil production, recovery percentage and temperature profiles in the model at different times. The results from the numerical simulation were found to be relatively insensitive to the form of the relative permeability functions employed. Linear relative permeability curves were employed to simulate the experimental production results. The model can be employed as a tool to aid in sensitivity studies and in the interpretation of the laboratory results. Introduction A reservoir can only be produced once, and it is often difficult and costly to correct mistakes in the approach, once the project has been initiated. However, a model study can be conducted several times and alternatives examined. Therefore, before launching any new technology, it is advisable to conduct extensive testing, using physical and numerical models, to evaluate various scenarios for the optimization of pattern type, size, well completions and rates. It is important, technically and financially, to have a prediction of the performance that is likely to be achieved from a given proposed recovery scheme for a given reservoir. Models, both physical and numerical, improve the understanding of reservoir processes. A physical model can provide a means for checking the accuracy and assumptions of the numerical model. Validating the numerical model is advisable by history matching with a well-defined physical model to overcome some of the limitations of numerical modelling. In some recovery processes, such as the Steam-assisted Gravity Drainage (SAGD) process(l), the physics of the situation is so complex that both physical and numerical models should be employed as complementary tools to obtain insight into different mechanisms and determine which strategies will optimize the process. Several field pilots using the SAGD process are underway. These include the three Esso Resources horizontal pilot projects at Cold Lake, Alberta, the Sceptre Resources pilot project in the Tangleflags North field near Lloydminster, Saskatchewan, and the Alberta Oil Sands Technology and Research Authority's (AOSTRA) Underground Test Facility (UTF) project in the Athabasca field near Fort McMurray, Alberta. Scope and Objective of Study The scope of this numerical simulation study is to match the oil recovery and the steam temperature interface position as observed in Chung's physical model(2,6) of the SAGD process. The objective is to investigate the feasibility, using a commercial CMG simulator ("STARS"), to history match the SAGD process, particularly the "spreading steam chamber" phase and the "rising steam chamber" phase.

view Abstract Citations (10) References (18) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The distribution of stars around a massive central black hole in a spherical stellar system. I. Results for test stars with a unique mass and radius. Ipser, J. R. Abstract The Fokker-Planck equation in phase space is solved in an approximate way for the steady- state star distribution around a massive central black hole in a spherical stellar system. Our approach differs from previous ones, which have solved diffusion equations in energy (E)- angular-momentum (J) space, equations whose equivalence to the fundamental Fokker-Planck equation has not been proved for the present problem. The hole strongly influences the dynamics at r <% = 2GMH!Vc2 (MH is the hole's mass, Vc is the rms velocity in the stellar system's core), and our approximate solution there differs significantly from previous ones based on diffusion in E-J space. For r <% r /lO and rant (rorit is the minimum radius at which stars with low J have a good chance to scatter to large J and thereby avoid being consumed by the hole), we obtain a stellar density n(r) const. x r - a, with a 5/4 rather than the value 7/4 obtained in other approaches. Conversely, for r /lO <% r <% , where our results become less solidly grounded as MH approaches the largest values envisaged, we obtain a density enhancement that is steeper than that of other approaches. (Like others, we presently neglect the contribution of the stars to the gravitational field.) Roughly, we find n(r) const. x exp (3r /2r) for r > r /3 if the surrounding cluster is isothermal; and n(r) grows faster than r -2, but not faster than , for r /lO < r < r /3. The discrepancy remains unresolved. The question is raised whether it can possibly be traced to an inequivalence of the basic equations (Fokker-Planck equation in phase space versus diffusion equation in E-J space) underlying the different approaches; and/or to inequivalent applications of the boundary conditions of the problem. Analyses suggesting that the latter may be at least partially involved in a resolution are provided: evidence is presented that over a range of smaller radii, our density profile would steepen toward something resembling an a = 7/4 power law if we were to handle the "loss-cone" boundary condition governing consumption of low-J stars by the hole in the (unacceptable, we claim) way it is handled by previous calculations yielding a = 7/4. In addition, it is shown that an energy-conservation argument which previously suggested a = 7/4 suggests a < 7/4 when loss-cone consumption is accounted for. The argument suggests a 5/4 if the time scales for energy diffusion and "loss-cone diffusion" are nearly equal. Our analyses allow the possibility that a 7/4 for rant < r < r /lO, but rant < r /lO only for the largest values of MH envisaged for globular clusters and galactic nuclei. Our density profile implies a stellar consumption rate, when MH is large, that exceeds by a large amount the rates calculated previously, but the gravity of the stars would flatten our profiles and reduce our rates somewhat. Subject headings: black holes - clusters: globular - stars: stellar dynamics Publication: The Astrophysical Journal Pub Date: June 1978 DOI: 10.1086/156216 Bibcode: 1978ApJ...222..976I Keywords: Black Holes (Astronomy); Globular Clusters; Star Distribution; Stellar Systems; Angular Momentum; Fokker-Planck Equation; Galactic Nuclei; Gravitational Collapse; Iterative Solution; Quasars; Steady State; Stellar Evolution; Astrophysics; Black Hole Neighborhood:Star Distribution; Black Holes:Stellar Systems; Stellar Systems: Star Distribution full text sources ADS |

Environmental variation favors the evolution of phenotypic plasticity. For many species, we understand the costs and benefits of different phenotypes, but we lack a broad understanding of how plastic traits evolve across large clades. Using identical experiments conducted across North America, we examined prey responses to predator cues. We quantified five life-history traits and the magnitude of their plasticity for 23 amphibian species/populations (spanning three families and five genera) when exposed to no cues, crushed-egg cues, and predatory crayfish cues. Embryonic responses varied considerably among species and phylogenetic signal was common among the traits, whereas phylogenetic signal was rare for trait plasticities. Among trait-evolution models, the Ornstein-Uhlenbeck (OU) model provided the best fit or was essentially tied with Brownian motion. Using the best fitting model, evolutionary rates for plasticities were higher than traits for three life-history traits and lower for two. These data suggest that the evolution of life-history traits in amphibian embryos is more constrained by a species' position in the phylogeny than is the evolution of life history plasticities. The fact that an OU model of trait evolution was often a good fit to patterns of trait variation may indicate adaptive optima for traits and their plasticities.

Glaucoma, an eye disorder caused by elevated intraocular pressure (IOP), is the leading cause of irreversible blindness in humans. Understanding how IOP levels have evolved across animal species could shed light on the nature of human vulnerability to glaucoma. Here, we studied the evolution of IOP in mammals and birds and explored its life history correlates. We conducted a systematic review, to create a dataset of species-specific IOP levels and reconstructed the ancestral states of IOP using three models of evolution (Brownian, Early burst, and Ornstein-Uhlenbeck (OU)) to understand the evolution of glaucoma. Furthermore, we tested the association between life history traits (e.g., body mass, blood pressure, diet, longevity, and habitat) and IOP using phylogenetic generalized least squares (PGLS). IOP in mammals and birds evolved under the OU model, suggesting stabilizing selection toward an optimal value. Larger mammals had higher IOPs and aquatic birds had higher IOPs; no other measured life history traits, the type of tonometer used, or whether the animal was sedated when measuring IOP explained the significant variation in IOP in this dataset. Elevated IOP, which could result from physiological and anatomical processes, evolved multiple times in mammals and birds. However, we do not understand how species with high IOP avoid glaucoma. While we found very few associations between life history traits and IOP, we suggest that more detailed studies may help identify mechanisms by which IOP is decoupled from glaucoma. Importantly, species with higher IOPs (cetaceans, pinnipeds, and rhinoceros) could be good model systems for studying glaucoma-resistant adaptations.

Physically guided deep learning solver for time-dependent Fokker–Planck equation

Unified viscoelastic injectivity model: Analytical solutions predicting polymer excess pressure and fracture initiation

The primary rationale for the use of phylogenetically based statistical methods is that phylogenetic signal, the tendency for related species to resemble each other, is ubiquitous. Whether this assertion is true for a given trait in a given lineage is an empirical question, but general tools for detecting and quantifying phylogenetic signal are inadequately developed. We present new methods for continuous-valued characters that can be implemented with either phylogenetically independent contrasts or generalized least-squares models. First, a simple randomization procedure allows one to test the null hypothesis of no pattern of similarity among relatives. The test demonstrates correct Type I error rate at a nominal α = 0.05 and good power (0.8) for simulated datasets with 20 or more species. Second, we derive a descriptive statistic, K, which allows valid comparisons of the amount of phylogenetic signal across traits and trees. Third, we provide two biologically motivated branch-length transformations, one based on the Ornstein-Uhlenbeck (OU) model of stabilizing selection, the other based on a new model in which character evolution can accelerate or decelerate (ACDC) in rate (e.g., as may occur during or after an adaptive radiation). Maximum likelihood estimation of the OU (d) and ACDC (g) parameters can serve as tests for phylogenetic signal because an estimate of d or g near zero implies that a phylogeny with little hierarchical structure (a star) offers a good fit to the data. Transformations that improve the fit of a tree to comparative data will increase power to detect phylogenetic signal and may also be preferable for further comparative analyses, such as of correlated character evolution. Application of the methods to data from the literature revealed that, for trees with 20 or more species, 92% of traits exhibited significant phylogenetic signal (randomization test), including behavioral and ecological ones that are thought to be relatively evolutionarily malleable (e.g., highly adaptive) and/or subject to relatively strong environmental (nongenetic) effects or high levels of measurement error. Irrespective of sample size, most traits (but not body size, on average) showed less signal than expected given the topology, branch lengths, and a Brownian motion model of evolution (i.e., K was less than one), which may be attributed to adaptation and/or measurement error in the broad sense (including errors in estimates of phenotypes, branch lengths, and topology). Analysis of variance of log K for all 121 traits (from 35 trees) indicated that behavioral traits exhibit lower signal than body size, morphological, life-history, or physiological traits. In addition, physiological traits (corrected for body size) showed less signal than did body size itself. For trees with 20 or more species, the estimated OU (25% of traits) and/or ACDC (40%) transformation parameter differed significantly from both zero and unity, indicating that a hierarchical tree with less (or occasionally more) structure than the original better fit the data and so could be preferred for comparative analyses.

The reaction zone structure of spontaneous reaction wave at steady subsonic and supersonic speeds is discussed using numerical simulation and theoretical approaches. The main objective of this study is to develop the mathematical model of the spontaneous reaction wave propagating at a steady speed and to clarify the relationship between the reaction zone structure and non-dimensional parameters. A one-dimensional direct numerical simulation (DNS) of the spontaneous reaction wave propagation is performed, and the modeling assumptions are discussed based on the numerical results. The non-dimensional parameters, the Damköhler number and a parameter related to compressibility defined by the Mach number of the spontaneous reaction wave velocity, are introduced into the modeling. The mathematical model is developed using asymptotic techniques, and the solution of the reaction zone structure is analytically derived using Newton’s method. The relationship between reaction zone structure of spontaneous reaction wave and non-dimensional parameters is discussed based on the analytical solutions.

Since Waddington proposed the concept of the “epigenetic landscape” in 1957, researchers have developed various methodologies to represent it in diverse processes. Studying the epigenetic landscape provides valuable qualitative information regarding cell development and the stability of phenotypic and morphogenetic patterns. Although Waddington’s original idea was a visual metaphor, a contemporary perspective relates it to the landscape formed by the basins of attraction of a dynamical system describing the temporal evolution of protein concentrations driven by a gene regulatory network. Transitions among these attractors can be driven by stochastic perturbations, with the cell state more likely to transition to the nearest attractor or to the one that presents the path of least resistance. In this study, we define the epigenetic landscape using the free energy potential obtained from the solution of the Fokker-Planck equation on the regulatory network. Specifically, we obtained a numerical approximate solution of the Fokker-Planck equation describing the Arabidopsis thaliana flower morphogenesis process. We observed good agreement between the coexpression matrix obtained from the Fokker-Planck equation and the experimental coexpression matrix. This paper proposes a method for obtaining this landscape by solving the Fokker-Planck equation (FPE) associated with a dynamical system describing the temporal evolution of protein concentrations involved in the process of interest. As these systems are high-dimensional and analytical solutions are often unfeasible, we propose a gamma mixture model to solve the FPE, transforming this problem into an optimization problem. This methodology can enhance the analysis of gene regulatory networks by directly relating theoretical mathematical models with experimental observations of coexpression matrices, thus providing a discriminating technique for competing models.

We investigate the evolutionary dynamics of a Moran model which consists of subdivided populations of individuals confined to a set of islands. In subdivided populations, migration acts with selection and genetic drift to determine the evolutionary dynamics. The individuals are assumed to be haploid with two types. They reproduce according to their fitness values, die at random, and migrate between the islands. The evolutionary dynamics of an individual based model is formulated in terms of a master equation and is approximated as the multidimensional Fokker-Planck equation (FPE) and the coupled non-linear stochastic differential equations (SDEs) with multiplicative noise. We first analyze the deterministic part of the SDEs to obtain the fixed points and determine the stability of each fixed point. We find that there is only one stable fixed point to which all populations started from any initial distribution other than the unstable absorbing points will evolve in both the symmetric and antisymmetric selection schemes. Next, we take demographic stochasticity into account and analyze the FPE by eliminating the fast variable to reduce the coupled two-variable FPE to the single-variable FPE. We derive a quasi-stationary distribution of the reduced FPE and predict the fixation probabilities and the mean fixation times to absorbing states. We also carry out numerical simulations in the form of the Gillespie algorithm and find that the results of simulations are consistent with the analytic predictions.