Forward Kolmogorov Research Articles

Numerical solution of the forward Kolmogorov equations in population genetics using Eta functions

This paper introduces a new numerical method for solving forward Kolmogorov equations in population genetics. Since there's no simple analytical expression for the distribution of allele frequencies (DAF), we use these equations to derive it. The accuracy of solving these equations depends on the choice of base functions, so we use Eta-based functions for better approximations. By employing the operational matrix of integral, we simplify the partial differential equation to an algebraic one. The method's error bounds, stability, and validity are demonstrated through numerical examples. Finally, we apply this method to analyze the behavior of forward Kolmogorov equations under various evolutionary forces.

Particles with constant speed and random velocity in 3+1 space-time: separation of the space variables

We consider a particle in 3+1 space-time dimensions which constantly moves at speed of light c, randomly changing its velocity which can be represented by a point on the surface of a sphere of radius c. The velocity performs an isotropic Wiener process on this surface so that the velocity trajectories are almost everywhere continuous although not differentiable, while the position trajectories are continuous and differentiable. Together with the process that describes the particle in the ‘rest frame’, where the diffusion of velocity on the surface of the sphere is isotropic, the entire family of anisotropic processes which result from Lorentz boosts is also described. The focus of this article is on stochastic evolution in space. We identify a reduced set of variables whose stochastic evolution is autonomous from the remaining variables, but, nevertheless, carry all the relevant information concerning the spatial properties of the process. The associated stochastic equations as well the Forward Kolmogorov equation are considerably simplified compared to those of the complete set of position and velocity variables.

Data-driven methods for the inverse problem of suspension system excited by jump and diffusion stochastic track excitation

Following a lengthy tenure of service, suspension systems may exhibit inaccurate model parameters due to the presence of complex conditions, which can result in a deterioration of control and the potential for operational failures. It is therefore imperative to estimate the key parameters of suspension systems. This paper introduces two data-driven methods for addressing the inverse problem in suspension systems subjected to stochastic track excitations. By employing physics-informed neural networks (PINNs) and Monte Carlo (MC) simulation, we are able to address the resulting integro-differential equation that arises from stochastic jump processes, thus avoiding the necessity for mesh grids. In order to mitigate the numerical challenges that arise from the system parameters, a residual-based adaptive sampling method is proposed. These methods effectively infer unknown parameters, addressing scenarios where data is directly available as a probability density function (PDF) or only sparse trajectories are given. In the latter case, a novel loss function employing Kullback-Leibler divergence facilitates learning from stochastic trajectories. Both methods successfully obtain solutions to the forward Kolmogorov equation, as validated by numerical experiments testing their robustness against added noise. The results demonstrate accurate parameter estimation under varying noise intensities, highlighting the methods’ robustness.

Probability density of the solution to nonlinear systems driven by Gaussian and Poisson white noises

A new method is proposed to compute the probability density of the multi-dimensional nonlinear dynamical system perturbed by a combined excitation of Gaussian and Poisson white noises. We first deduce a probability-density solver from the Euler–Maruyama scheme of the stochastic system and the corresponding Chapman–Kolmogorov equation. This solver actually is an explicit numerical formula of the probability density of the solution to this stochastic system. To compute the probability density, we propose an efficient algorithm for this solver, which actually is the implementation of a numerical integration. Furthermore, we prove this solver is an approximated solution of the corresponding forward Kolmogorov equation. Numerical examples are conducted to illustrate our probability-density solver.

Deep neural network method to predict the dynamical system response under random excitation of combined Gaussian and Poisson white noises

Nonlinear dynamical systems excited by combined Gaussian and Poisson white noises are widely found in the scientific field. However, the introduction of Poisson white noise results in the associated forward Kolmogorov equation becoming an integro-differential equation (IDE), which makes it difficult to solve the response of these dynamical systems. At present, traditional numerical IDE solvers are limited due to both the mesh generation and high computational costs. In this context, this paper integrates the multi-element Gauss–Legendre (GL) quadrature into the physics-informed neural networks (PINNs) algorithm to predict the response of systems excited by combined Gaussian and Poisson white noises. A residual-based adaptive distribution (RAD) sampling method, which replaces Latin hypercube sampling (LHS) method, is implemented to adjust the residual points to improve the accuracy and training efficiency of the proposed algorithm. The results demonstrate that the integration of the PINNs algorithm with RAD sampling method dramatically enhances the tolerance of noisy data and expedites the convergence of the PINNs algorithm compared with the LHS sampling method. It has the ability to achieve better performance in a shorter training time and reduces the demand for data during the training process. Additionally, the Monte Carlo (MC) simulation is carried out to demonstrate the effectiveness of the numerical results in this paper.

On Forward and Backward Kolmogorov Equations for Pure Jump Markov Processes and Their Generalizations

On Forward and Backward Kolmogorov Equations for Pure Jump Markov Processes and Their Generalizations

Stochastic modeling of spreading an infection with standard incidence rate

This paper studies and models the random spread of an infection in a population. It extends the traditional deterministic modeling approach by incorporating discretetime stochastic modeling using Markov chains. The probability of extinction and disease persistence is then investigated using the branching chain method, with a focus on a quantity in the non-random model, which is called the basic reproductive number. Moreover, the random model is transformed into a system of stochastic differential equations by approximating the probability distribution function using the forward Kolmogorov equation. An equivalent stochastic differential system is also introduced and examined for the stochastic model. Finally, numerical simulations of the stochastic models are conducted to evaluate the theoretical findings presented in the paper.

Critical overview and comparison between models for adsorption-desorption noise in bio-chemical sensors

We critically review existing models for the adsorption-desorption noise in bio-chemical sensors, in particular the model based on simplified forward Kolmogorov equation and the models based on Langevin sources. For the latter models, we propose a generalized version to handle cases beyond the branched surface reactions (a binding site that can be alternatively occupied by different ions/molecules) and the chained reaction (a binding site that sequentially binds with ions/molecules). The models are benchmarked against kinetic Monte Carlo (kMC) simulations considering relevant case studies such as pH-sensitive ions, selective molecules binding on a functionalized surface and multi-layer adsorption on bare surfaces. It is found that although the mathematical formulation of the modeling approaches appears different, when dealing with independent binding sites, they are fully equivalent and perfectly match the kMC results. The case of competitive binding considering the correlation between the occupation of the binding sites has also been analyzed.

Fractional forward Kolmogorov equations in population genetics

This paper combines theoretical development and numerical methods to characterize the effect of heterogeneity in population genetics. To address heterogeneity in population genetics, we build the first fractional forward Kolmogorov equations in population genetics, where the distribution of the allele frequencies of a given set of loci in a population is a solution of forward Kolmogorov equations. This framework will be implemented, and the model will be studied computationally. To study the model, a new numerical method for solving the fractional partial differential equations is presented. The method is based upon the least squares approximation via Legendre polynomials. The Riemann–Liouville fractional integral operator for Legendre polynomials is utilized to reduce the solution of the fractional partial differential equations to a system of algebraic equations. The error bound and the stability of the method are presented. Illustrative examples are included to demonstrate the validity and applicability of the technique. Using the new numerical method, we derive the allele frequency distribution as a solution of the fractional forward Kolmogorov equations. We study the behavior of allele frequency distribution by considering the effect of evolutionary and demographic forces. This study shows that heterogeneity changes the behavior of the distribution of the allele frequencies.

Generalized gradient structures for measure-valued population dynamics and their large-population limit

We consider the forward Kolmogorov equation corresponding to measure-valued processes stemming from a class of interacting particle systems in population dynamics, including variations of the Bolker–Pacala–Dieckmann-Law model. Under the assumption of detailed balance, we provide a rigorous generalized gradient structure, incorporating the fluxes arising from the birth and death of the particles. Moreover, in the large population limit, we show convergence of the forward Kolmogorov equation to a Liouville equation, which is a transport equation associated with the mean-field limit of the underlying process. In addition, we show convergence of the corresponding gradient structures in the sense of Energy-Dissipation Principles, from which we establish a propagation of chaos result for the particle system and derive a generalized gradient-flow formulation for the mean-field limit.

Population processes in cyber system variability.

Variability is inherent to cyber systems. Here, we introduce ideas from stochastic population biology to describe the properties of two broad kinds of cyber systems. First, we assume that each of N0 components can be in only one of two states: functional or nonfunctional. We model this situation as a Markov process that describes the transitions between functional and nonfunctional states. We derive an equation for the probability that an individual cyber component is functional and use stochastic simulation to develop intuition about the dynamics of individual cyber components. We introduce a metric of performance of the system of N0 components that depends on the numbers of functional and nonfunctional components. We numerically solve the forward Kolmogorov (or Fokker-Planck) equation for the number of functional components at time t, given the initial number of functional components. We derive a Gaussian approximation for the solution of the forward equation so that the properties of the system with many components can be determined from the transition probabilities of an individual component, allowing scaling to very large systems. Second, we consider the situation in which the operating system (OS) of cyber components is updated in time. We motivate the question of OS in use as a function of the most recent OS release with data from a network of desktop computers. We begin the analysis by specifying a temporal schedule of OS updates and the probability of transitioning from the current OS to a more recent one. We use a stochastic simulation to capture the pattern of the motivating data, and derive the forward equation for the OS of an individual computer at any time. We then include compromise of OSs to compute that a cyber component has an unexploited OS at any time. We conclude that an interdisciplinary approach to the variability of cyber systems can shed new light on the properties of those systems and offers new and exciting ways to understand them.

Jump processes as generalized gradient flows

We have created a functional framework for a class of non-metric gradient systems. The state space is a space of nonnegative measures, and the class of systems includes the Forward Kolmogorov equations for the laws of Markov jump processes on Polish spaces. This framework comprises a definition of a notion of solutions, a method to prove existence, and an archetype uniqueness result. We do this by using only the structure that is provided directly by the dissipation functional, which need not be homogeneous, and we do not appeal to any metric structure.

Solving Nonlinear Filtering Problems Using a Tensor Train Decomposition Method

In this paper, we propose an efficient numerical method to solve nonlinear filtering (NLF) problems. Specifically, we use the tensor train decomposition method to solve the forward Kolmogorov equation (FKE) arising from the NLF problem. Our method consists of offline and online stages. In the offline stage, we use the finite difference method to discretize the partial differential operators involved in the FKE and extract low-dimensional structures in the solution tensor using the tensor train decomposition method. In the online stage using the pre-computed low-rank approximation tensors, we can quickly solve the FKE given new observation data. Therefore, we can solve the NLF problem in a real-time manner. Finally, we present numerical results to show the efficiency and accuracy of the proposed method in solving up to six-dimensional NLF problems.

Factorial moments of the critical Markov branching process with geometric reproduction of particles

The factorial moments of any Markov branching process describe the behaviour of its probability generating function $F(t,s)$ in the neighbourhood of the point $s=1$. They are applied to solve the forward Kolmogorov equation for the critical Markov branching process with geometric reproduction of particles. The solution includes quickly convergent recurrent iterations of polynomials. The obtained results on factorial moments enable computation of statistical measures as shape and skewness. They are also applicable to the comparison between critical geometric branching and linear birth-death processes.

Fisheries management under incomplete information by optimal stochastic control and Hidden Markov Model filter

This paper deals with mathematical modelling of strategies in harvesting of fish populations. Since too aggressive harvesting may have severe consequences, a cautious control of the process is called for. The determination of optimal harvesting based on the biomass size is made even more challenging since there is considerable uncertainty in the estimation of the stock size, the so-called incomplete information problem.Fishery management is in this paper established as an optimal control problem for a model based on nonlinear stochastic differential equations, with economic performance as objective. The so-called certainty equivalence principle, by which the estimate is used for the purposes of optimal feedback control as if it were the certain value of the state variable, is adopted. Markov controls are identified by solving the stationary Hamilton-Jacobi-Bellman equation. State estimates are obtained by a Hidden Markov Model filter, where the forward Kolmogorov equation governs the temporal evolution of estimates with the uncertain quantities. Fishery profit over time as economic performance is computed by Monte Carlo simulations, in order to compare the performances of the strategies considering control rules and precautionary approach.Two control rules in harvest policy – harvest control rule and effort control rule – are compared, with respect to their robustness given the uncertainty. The effort control rule produces a significantly higher cumulated profits, and is less sensitive to the uncertainty in stock assessment present in the system. Moreover, a precautionary approach taking the uncertainty associated with biomass estimates into account, can be achieved via quantile estimators. This approach produces an economic gain by making a more appropriately cautious decision, and leads to sustainable harvesting of fish resources.

Fast Reaction Limits via Gamma -Convergence of the Flux Rate Functional

We study the convergence of a sequence of evolution equations for measures supported on the nodes of a graph. The evolution equations themselves can be interpreted as the forward Kolmogorov equations of Markov jump processes, or equivalently as the equations for the concentrations in a network of linear reactions. The jump rates or reaction rates are divided in two classes; ‘slow’ rates are constant, and ‘fast’ rates are scaled as 1/epsilon , and we prove the convergence in the fast-reaction limit epsilon rightarrow 0. We establish a Gamma -convergence result for the rate functional in terms of both the concentration at each node and the flux over each edge (the level-2.5 rate function). The limiting system is again described by a functional, and characterises both fast and slow fluxes in the system. This method of proof has three advantages. First, no condition of detailed balance is required. Secondly, the formulation in terms of concentration and flux leads to a short and simple proof of the Gamma -convergence; the price to pay is a more involved compactness proof. Finally, the method of proof deals with approximate solutions, for which the functional is not zero but small, without any changes.

Wasserstein Proximal Algorithms for the Schrödinger Bridge Problem: Density Control With Nonlinear Drift

In this article, we study the Schrödinger bridge problem (SBP) with nonlinear prior dynamics. In control-theoretic language, this is a problem of minimum effort steering of a given joint state probability density function (PDF) to another over a finite-time horizon, subject to a controlled stochastic differential evolution of the state vector. As such, it can be seen as a stochastic optimal control problem in continuous time with endpoint density constraints—A topic that originated in the physics literature in 1930s, and in the recent years, has garnered burgeoning interest in the systems-control community. For generic nonlinear drift, we reduce the SBP to solving a system of forward and backward Kolmogorov partial differential equations (PDEs) that are coupled through the boundary conditions, with unknowns being the “Schrödinger factors”—so named since their product at any time yields the optimal controlled joint state PDF at that time. We show that if the drift is a gradient vector field, or is of mixed conservative–dissipative nature, then it is possible to transform these PDEs into a pair of initial value problems (IVPs) involving the same forward Kolmogorov operator. Combined with a recently proposed fixed point recursion that is contractive in the Hilbert metric, this opens up the possibility to numerically solve the SBPs in these cases by computing the Schrödinger factors via a single IVP solver for the corresponding (uncontrolled) forward Kolmogorov PDE. The flows generated by such forward Kolmogorov PDEs, for the two aforementioned types of drift, in turn, enjoy gradient descent structures on the manifold of joint PDFs with respect to suitable distance functionals. We employ a proximal algorithm developed in our prior work that exploits this geometric viewpoint, to solve these IVPs and compute the Schrödinger factors via weighted scattered point cloud evolution in the state space. We provide the algorithmic details and illustrate the proposed framework of solving the SBPs with nonlinear prior dynamics by numerical examples.

Systems of Nonlinear Backward and Forward Kolmogorov Equations: Generalized Solutions

A probabilistic approach to construction of the solution to the Cauchy problem for systems of nonlinear parabolic equations is developed. The systems under consideration can be subdivided into two ...

A Novel Real-Time Filtering Method to General Nonlinear Filtering Problem Without Memory

In this paper, the filtering problem for the general time-invariant nonlinear state-observation system is considered. Our work is based on the Yau-Yau filtering framework developed by S.-T. Yau and the third author in 2008. The key problem of Yau-Yau filtering framework is how to compute the solution to forward Kolmogorov equation (FKE) off-line effectively. Motivated by the supervised learning in machine learning, we develop an efficient method to numerically solve the FKE off-line from the point of view of optimization. Specifically, for the off-line computation part, the computation of the solution to a FKE is reduced to computing a linear system of equations by making the temporal inverse transformation and the loss function optimization, and we store the results for the preparation of on-line computation. For the on-line computation part, the unnormalized density function is approximated by a complete polynomial basis, and then the estimation of the state is computed using the stored off-line data. Our method has the merits of easily implementing, real-time and memoryless. More importantly, it can be applicable for moderate-high dimensional cases. Numerical experiments have been carried out to verify the feasibility of our method. Our algorithm outperforms extended Kalman filter, unscented Kalman filter and particle filter both in accuracy and costing time.

Systems of nonlinear backward and forward Kolmogorov equations: generalized solutions

В работе развит вероятностный подход к построению решения задачи Коши для систем нелинейных параболических уравнений. Рассматриваемые системы можно разбить на два класса, причем системы первого класса после несложного преобразования можно трактовать как системы обратных нелинейных уравнений Колмогорова, а системы второго класса интерпретируются как системы прямых нелинейных уравнений Колмогорова. Выбор соответствующей интерпретации позволяет построить стохастическую модель в терминах стохастического уравнения с коэффициентами, зависящими от решения рассматриваемой задачи Коши и замыкающего соотношения, соответствующего вероятностному представлению этого решения.