Fokker-Planck Partial Differential Equation Research Articles

NEIM-08 MODELING COMPETITION BETWEEN SUBPOPULATIONS WITH VARIABLE DNA CONTENT IN RESOURCE LIMITED MICROENVIRONMENTS

Abstract Resource limitations shape the outcome of competitions between heterogeneous pre-malignant cells. One example of such heterogeneity is in the ploidy (DNA content) of pre-malignant cells. High-ploidy cells need more resources, to synthesize increasing amounts of DNA, RNA and proteins. To model how subpopulations with variable DNA-content compete in the resource limited environment of the human brain we developed a stochastic state-space model of the brain (S3MB). The model discretizes the brain into voxels, whereby the state of each voxel is defined by 8+ variables that are updated over time, including stiffness, glucose and proliferating cells of various DNA content. Well established Fokker-Planck partial differential equations govern the spatial distribution of resources and cells. We applied S3MB on sequencing and imaging data obtained from a primary GBM patient. We sequenced four surgical specimens collected during the 1st and 2nd surgeries of the GBM and used HATCHET to quantify its clonal composition and how it changes between the surgeries. HATCHET identified two aneuploid subpopulations of ploidy 1.98 and 2.29 respectively. The low-ploidy clone was dominant at the time of the first surgery and became even more dominant upon recurrence. MRI images were available before and after each surgery and registered to MNI space. T1 post and T2 flair scans acquired after the 1st surgery informed tumor cell densities per voxel. Magnetic Resonance Elastography scans and PET/CT scans informed stiffness and Glucose access per voxel. We performed a parameter search to recapitulate the GBM’s tumor cell density and ploidy composition before the 2nd surgery. Results suggest that the high-ploidy subpopulation had a higher Glucose-dependent proliferation rate, but a lower Glucose-dependent death rate, resulting in spatial differences in the distribution of the two subpopulations. Our results contribute to a better understanding of how genomics and microenvironments interact to shape cell fate decisions.

PDE for the joint law of the pair of a continuous diffusion and its running maximum

AbstractLet X be a d-dimensional diffusion and M the running supremum of its first component. In this paper, we show that for any $t>0,$ the density (with respect to the $(d+1)$ -dimensional Lebesgue measure) of the pair $\big(M_t,X_t\big)$ is a weak solution of a Fokker–Planck partial differential equation on the closed set $\big\{(m,x)\in \mathbb{R}^{d+1},\,{m\geq x^1}\big\},$ using an integral expansion of this density.

On the closed-loop stochastic dynamics of two-state nonlinear exothermic CSTRs with PI temperature control

Fokker-Planck (FP) partial differential equation (PDE) theory is applied to characterize the stochastic dynamics of a class of open-loop (OL) 2-state nonlinear exothermic continuous reactors with: (i) zero and time-varying mean noise disturbances, and (ii) linear proportional-integral (PI) temperature control. The characterization includes: (i) the stochastic on deterministic dynamics dependency, (ii) gain condition for robust probability density function (PDF) stability over deterministic-diffusion time biscale with stationary monomodality at prescribed most probable (MP) state, (iii) evolutions of along nearly deterministic time scale of MP state and control and their variabilities, (iv) attainment of random motion in-probability (IP) stability over deterministic-diffusion time biscale, and (v) identification of the compromise between MP state regulation speed, robustness, and control effort. The methodological developments and findings are illustrated with three indicative examples with OL complex (bimodal and vulcanoid) stationary state PDFs, including analytic assessment as well as state PDF and random motion numerical simulation.

Binary Interaction Methods for High Dimensional Global Optimization and Machine Learning

In this work we introduce a new class of gradient-free global optimization methods based on a binary interaction dynamics governed by a Boltzmann type equation. In each interaction the particles act taking into account both the best microscopic binary position and the best macroscopic collective position. For the resulting kinetic optimization methods, convergence to the global minimizer is guaranteed for a large class of functions under appropriate parameter constraints that do not depend on the dimension of the problem. In the mean-field limit we show that the resulting Fokker-Planck partial differential equations generalize the current class of consensus based optimization (CBO) methods. Algorithmic implementations inspired by the well-known direct simulation Monte Carlo methods in kinetic theory are derived and discussed. Several examples on prototype test functions for global optimization are reported including an application to machine learning.

A Fokker-Planck feedback control framework for optimal personalized therapies in colon cancer-induced angiogenesis.

In this paper, a new framework for obtaining personalized optimal treatment strategies in colon cancer-induced angiogenesis is presented. The dynamics of colon cancer is given by a Itó stochastic process, which helps in modeling the randomness present in the system. The stochastic dynamics is then represented by the Fokker-Planck (FP) partial differential equation that governs the evolution of the associated probability density function. The optimal therapies are obtained using a three step procedure. First, a finite dimensional FP-constrained optimization problem is formulated that takes input individual noisy patient data, and is solved to obtain the unknown parameters corresponding to the individual tumor characteristics. Next, a sensitivity analysis of the optimal parameter set is used to determine the parameters to be controlled, thus, helping in assessing the types of treatment therapies. Finally, a feedback FP control problem is solved to determine the optimal combination therapies. Numerical results with the combination drug, comprising of Bevacizumab and Capecitabine, demonstrate the efficiency of the proposed framework.

Reliability Assessment of Aircraft Commutators

The article describes the method of predicting the reliability and durability of an aircraft commutator, which is a primary source of electric energy in helicopters. Tests were conducted for 10 starter-generators. From this research it follows that the technical condition of brushes and bearings has a significant impact on the reliability of starter-generators. The reliability of starter-generators was determined based on the method consisting of two stages that was adopted: the first stage involved determining the density function of changes in diagnostic parameter depending on the operating time, but the second stage included the assessment of the reliability of bearings of the starter-generator taking into account the real flight profile. The first stage of the adopted method consisted of defining the dynamic model of changing the length of the starter-generator’s brush, which became the probabilistic model. Subsequently, based on differential equations, Fokker–Planck partial differential equation was derived, which describes the process of increasing the brush wear in a probabilistic way. This method enables the prediction of the residual durability of the helicopter’s starter-generator due to the change in a diagnostic parameter which is the wear of brushes during starter-generator operation. The second stage of this method allows determining the durability of starter-generator’s bearings building upon the average helicopter’s flight profile. Owing to the difficulty in measuring the wear of bearings, the relation between the durability of bearings and the temperature of surroundings can be applied by replacing the flight altitude with temperature. The reliability of the helicopter’s starter-generator was determined based on the serial-type reliability structure.

A gentle introduction to the non-equilibrium physics of trajectories: Theory, algorithms, and biomolecular applications

Despite the importance of non-equilibrium statistical mechanics in modern physics and related fields, the topic is often omitted from undergraduate and core-graduate curricula. Key aspects of non-equilibrium physics, however, can be understood with a minimum of formalism based on a rigorous trajectory picture. The fundamental object is the ensemble of trajectories, a set of independent time-evolving systems, which easily can be visualized or simulated (e.g., for protein folding) and which can be analyzed rigorously in analogy to an ensemble of static system configurations. The trajectory picture provides a straightforward basis for understanding first-passage times, "mechanisms" in complex systems, and fundamental constraints on the apparent reversibility of complex processes. Trajectories make concrete the physics underlying the diffusion and Fokker-Planck partial differential equations. Last but not least, trajectory ensembles underpin some of the most important algorithms that have provided significant advances in biomolecular studies of protein conformational and binding processes.

A Feynman-Kac based numerical method for the exit time probability of a class of transport problems

The exit time probability, which gives the likelihood that an initial condition leaves a prescribed region of the phase space of a dynamical system at, or before, a given time, is arguably one of the most natural and important transport problems. Here we present an accurate and efficient numerical method for computing this probability for systems described by non-autonomous (time-dependent) stochastic differential equations (SDEs) or their equivalent Fokker-Planck partial differential equations. The method is based on the direct approximation of the Feynman-Kac formula that establishes a link between the adjoint Fokker-Planck equation and the forward SDE. The Feynman-Kac formula is approximated using the Gauss-Hermite quadrature rules and piecewise cubic Hermite interpolating polynomials, and a GPU accelerated matrix representation is used to compute the entire time evolution of the exit time probability using a single pass of the algorithm. The method is unconditionally stable, exhibits second order convergence in space, first order convergence in time, and it is straightforward to parallelize. Applications are presented to the advection diffusion of a passive tracer in a fluid flow exhibiting chaotic advection, and to the runaway acceleration of electrons in a plasma in the presence of an electric field, collisions, and radiation damping. Benchmarks against analytical solutions as well as comparisons with explicit and implicit finite difference standard methods for the adjoint Fokker-Planck equation are presented.

Strict dissipativity analysis for classes of optimal control problems involving probability density functions

<p style='text-indent:20px;'>Motivated by the stability and performance analysis of model predictive control schemes, we investigate strict dissipativity for a class of optimal control problems involving probability density functions. The dynamics are governed by a Fokker-Planck partial differential equation. However, for the particular classes under investigation involving linear dynamics, linear feedback laws, and Gaussian probability density functions, we are able to significantly simplify these dynamics. This enables us to perform an in-depth analysis of strict dissipativity for different cost functions.</p>

Spatial dynamics in interacting systems with discontinuous coefficients and their continuum limits

We consider a discrete model in which particles are characterized by two quantities [Formula: see text] and [Formula: see text]; both quantities evolve in time according to stochastic dynamics and the equation that governs the evolution of [Formula: see text] is also influenced by mean-field interaction between the particles. We allow for discontinuous coefficients and random initial condition and, under suitable assumptions, we prove that in the limit as the number of particles grows to infinity the dynamics of the system is described by the solution of a Fokker–Planck partial differential equation. We provide the existence and uniqueness of a solution to the latter and show that such solution arises as the limit in probability of the empirical measures of the system.

Non-stationary response statistics of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitation

An approximate analytical technique is developed for determining the non-stationary response amplitude probability density function (PDF) of nonlinear/hysteretic oscillators endowed with fractional derivative elements and subjected to evolutionary stochastic excitation. Specifically, resorting to stochastic averaging/linearization leads to a dimension reduction of the governing equation of motion and to a first-order stochastic differential equation (SDE) for the oscillator response amplitude. Associated with this first-order SDE is a Fokker–Planck partial differential equation governing the evolution in time of the non-stationary response amplitude PDF. Next, assuming an appropriately chosen time-dependent PDF form of the Rayleigh kind for the response amplitude, and substituting into the Fokker–Planck equation, yields a deterministic first-order nonlinear ordinary differential equation for the time-dependent PDF coefficient. This can be readily solved numerically via standard deterministic integration schemes. Thus, the non-stationary response amplitude PDF is approximately determined in closed-form in a computationally efficient manner. The technique can account for arbitrary excitation evolutionary power spectrum forms, even of the non-separable kind. A hardening Duffing and a bilinear hysteretic nonlinear oscillators with fractional derivative elements are considered in the numerical examples section. To assess the accuracy of the developed technique, the analytical results are compared with pertinent Monte Carlo simulation data.

Ship Stability in a Seastate by the State-Space Fokker–Planck Method

The development of a new methodology for the modeling of the nonlinear responses and stability of ships in stochastic steep waves is presented. The radiation and diffraction hydrodynamic forces are cast in a state-space form by using a panel method followed by the implementation of the estimation of signal parameters via rotational invariance techniques algorithm. Strong free-surface nonlinearities present in the hydrostatic restoring and Froude-Krylov exciting forces are modeled by the fluid impulse theory. The ambient seastate is represented by a state-space diffusion modeled after a given wave spectrum. This enables the study of the nonlinear seakeeping and stability of ships in a seastate by invoking the methods of stochastic calculus. A state-space stochastic differential equation is derived for the states governing the vessel nonlinear responses, and a linear Fokker-Planck partial differential equation is obtained for the joint probability density function of the vessel motions. 1. Introduction Ships advancing in steep random waves may experience large responses and loss of stability. The need exists for the development of rational stability criteria that will enable the derivation of the safe operation envelope of vessels operating in severe seastates. The past several decades have witnessed significant progress toward the development of potential and viscous flow computational methods for the treatment of the seakeeping hydrodynamics of ships. These methods combined with experiments have been the primary tools for the treatment of ship stability in realistic environments (Umeda et al. 2008). A comprehensive list of articles is contained in the previous reference. Yet, we still lack a general-purpose theoretical framework that may be applied for the assessment of the stability properties of ships. The study of ship stability in a seastate requires the explicit account of nonlinearities of potential and viscous origin, the coupling of all modes of motion, and the development of statistical models for the nonlinear ship responses and their extreme statistics in a stochastic seastate. The study of the stability of ships by the methods described previously is typically carried out by Monte-Carlo simulations for the determination of the safe operating envelope. This approach may, however, be very time consuming when the treatment of the six-degree-of-freedom seakeeping problem is necessary and in conditions where nonlinear coupling between the vessel responses lead to loss of stability (e.g. parametric rolling, broaching).

Stochastic modeling of chemical-mechanical coupling in striated muscles.

We propose a chemical-mechanical model of myosin heads in sarcomeres, within the classical description of rigid sliding filaments. In our case, myosin heads have two mechanical degrees-of-freedom (dofs)-one of which associated with the so-called power stroke-and two possible chemical states, i.e.,bound to an actin site or not. Our major motivations are twofold: (1) to derive a multiscale coupled chemical-mechanical model and (2) to thus account-at the macroscopic scale-for mechanical phenomena that are out of reach for classical muscle models. This model is first written in the form of Langevin stochastic equations, and we are then able to obtain the corresponding Fokker-Planck partial differential equations governing the probability density functions associated with the mechanical dofs and chemical states. This second form is important, as it allows to monitor muscle energetics and also to compare our model with classical ones, such as the Huxley'57 model to which our equations are shown to reduce under two different types of simplifying assumptions. This provides insight and gives a Langevin form for Huxley'57. We then show how we can calibrate our model based on experimental data-taken here for skeletal muscles-and numerical simulations demonstrate the adequacy of the model to represent complex physiological phenomena, in particular the fast isometric transients in which the power stroke is known to have a crucial role, thus circumventing a limitation of many classical models.

Nonlinear Kinetics on Lattices Based on the Kinetic Interaction Principle.

Master equations define the dynamics that govern the time evolution of various physical processes on lattices. In the continuum limit, master equations lead to Fokker–Planck partial differential equations that represent the dynamics of physical systems in continuous spaces. Over the last few decades, nonlinear Fokker–Planck equations have become very popular in condensed matter physics and in statistical physics. Numerical solutions of these equations require the use of discretization schemes. However, the discrete evolution equation obtained by the discretization of a Fokker–Planck partial differential equation depends on the specific discretization scheme. In general, the discretized form is different from the master equation that has generated the respective Fokker–Planck equation in the continuum limit. Therefore, the knowledge of the master equation associated with a given Fokker–Planck equation is extremely important for the correct numerical integration of the latter, since it provides a unique, physically motivated discretization scheme. This paper shows that the Kinetic Interaction Principle (KIP) that governs the particle kinetics of many body systems, introduced in G. Kaniadakis, Physica A 296, 405 (2001), univocally defines a very simple master equation that in the continuum limit yields the nonlinear Fokker–Planck equation in its most general form.

Low-dimensional spike rate models derived from networks of adaptive integrate-and-fire neurons: Comparison and implementation

The spiking activity of single neurons can be well described by a nonlinear integrate-and-fire model that includes somatic adaptation. When exposed to fluctuating inputs sparsely coupled populations of these model neurons exhibit stochastic collective dynamics that can be effectively characterized using the Fokker-Planck equation. This approach, however, leads to a model with an infinite-dimensional state space and non-standard boundary conditions. Here we derive from that description four simple models for the spike rate dynamics in terms of low-dimensional ordinary differential equations using two different reduction techniques: one uses the spectral decomposition of the Fokker-Planck operator, the other is based on a cascade of two linear filters and a nonlinearity, which are determined from the Fokker-Planck equation and semi-analytically approximated. We evaluate the reduced models for a wide range of biologically plausible input statistics and find that both approximation approaches lead to spike rate models that accurately reproduce the spiking behavior of the underlying adaptive integrate-and-fire population. Particularly the cascade-based models are overall most accurate and robust, especially in the sensitive region of rapidly changing input. For the mean-driven regime, when input fluctuations are not too strong and fast, however, the best performing model is based on the spectral decomposition. The low-dimensional models also well reproduce stable oscillatory spike rate dynamics that are generated either by recurrent synaptic excitation and neuronal adaptation or through delayed inhibitory synaptic feedback. The computational demands of the reduced models are very low but the implementation complexity differs between the different model variants. Therefore we have made available implementations that allow to numerically integrate the low-dimensional spike rate models as well as the Fokker-Planck partial differential equation in efficient ways for arbitrary model parametrizations as open source software. The derived spike rate descriptions retain a direct link to the properties of single neurons, allow for convenient mathematical analyses of network states, and are well suited for application in neural mass/mean-field based brain network models.

A Method for Reliability Assessment of Structural Components of Aircraft and Sea-Going Ships with Taking into Account a Given Failure Generation Model

Abstract This paper presents a metod for reliability assessment of structural components on the basis of diagnostic parameters recorded during operation of aircraft and sea going ships. It was assumed that a potential failure may simultaneously concerns surface wear process and fracture of an arbitrary structural component. In order to develop a reliability assessment model some elements of random walk theory were applied. This work covers surface wear density functions of a component. In the case of surface wear the model is based on a difference equation from which, after transformation, a Fokker-Planck partial differential equation was obtained. By solving the equation component’s surface wear density function is achieved. In the second part of the work a generalization of the model was made by introducing to it probability of disastrous fracture of a component. As a result, a generalized Fokkera-Planck’s equation was obtained. By making use of the equation, was obtained a relationship for assessment of reliability of a structural component in case when failure occurs due to surface wear with taking into account possible fracture of the component. In the end a numerical example which shows practical application of the developed method, is attached.

Positive Second Order Finite Difference Methods on Fokker-Planck Equations with Dirac Initial Data - Application in Finance

One approach to price financial derivatives in an arbitrage free manner is to work directly with the probability density function. For many models, it is possible to find an expansion of the density that follows a Fokker-Planck partial differential equation. In order to keep the arbitrage-free property numerically, it is particularly important to use a positivity preserving finite difference scheme. With a discontinuous initial condition like a Dirac, many schemes will produce oscillations or negative densities. This paper analyzes the behavior of a few simple schemes related to backward Euler, namely BDF2, Lawson-Morris and Lawson-Swaybe on the specific problem of a diffusion with Dirac initial condition. The case of the arbitrage free SABR model for interest rate derivatives is then evaluated.

An efficient computational technique for solving the Fokker–Planck equation with space and time fractional derivatives

This paper presents numerical solutions of the linear and nonlinear Fokker–Planck partial differential equations [FPPDEs] with space and time fractional derivatives through analytical solutions. These are treated by two analytical methods, namely, fractional reduced differential transform method [FRDTM] and fractional variational iteration method [FVIM] followed by some examples. Numerical results obtained by both FRDTM and FVIM are compared with some existing methods in the literature. This comparison shows the supremacy of FRDTM over FVIM and existing methods in terms of accuracy, simplicity and reliability.

Linear-Quadratic $$N$$ N -Person and Mean-Field Games: Infinite Horizon Games with Discounted Cost and Singular Limits

We consider stochastic differential games with $$N$$ nearly identical players, linear-Gaussian dynamics, and infinite horizon discounted quadratic cost. Admissible controls are feedbacks for which the system is ergodic. We first study the existence of affine Nash equilibria by means of an associated system of $$N$$ Hamilton–Jacobi–Bellman and $$N$$ Kolmogorov–Fokker–Planck partial differential equations, proving that for small discount factors quadratic-Gaussian solutions exist and are unique. Then, we prove the convergence of such solutions to the unique quadratic-Gaussian solution of the pair of Mean Field equations. We also discuss some singular limits, such as vanishing discount, vanishing noise, and cheap control.

Demand Response of Thermostatic Loads by Optimized Switching-Fraction Broadcast

Demand response is an important Smart Grid concept that aims at facilitating the integration of volatile energy resources into the electricity grid. This paper considers the problem of managing large populations of thermostat-based devices with on/off operation. The objective is to enable demand response capabilities within the intrinsic flexibility of the population. A temperature distribution model based on Fokker-Planck partial differential equations is used to capture the behavior of the population. To ensure probability conservation and high accuracy of the numerical solution, Finite Volume Method is used to spatially discretize these equations. Next, a broadcast strategy with two switching-fraction signals is proposed for actuating the population. This is applied in an open-loop scenario for tracking a power reference by running an optimization with a multilinear objective.