# A dynamically consistent discretization method for

In applied mathematicsdiscretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. Dichotomization is the special case of discretization in which the number of discrete classes is 2, which can approximate a continuous variable as a binary variable creating a dichotomy for modeling purposes, as in binary classification.

Discretization is also related to discrete mathematicsand is an important component of granular computing. In this context, discretization may also refer to modification of variable or category granularityas when multiple discrete variables are aggregated or multiple discrete categories fused. Whenever continuous data is discretizedthere is always some amount of discretization error. The goal is to reduce the amount to a level considered negligible for the modeling purposes at hand.

The terms discretization and quantization often have the same denotation but not always identical connotations. Specifically, the two terms share a semantic field. The same is true of discretization error and quantization error. Mathematical methods relating to discretization include the Euler—Maruyama method and the zero-order hold. Discretization is also concerned with the transformation of continuous differential equations into discrete difference equationssuitable for numerical computing.

The following continuous-time state space model. The equation for the discretized measurement noise is a consequence of the continuous measurement noise being defined with a power spectral density. A clever trick to compute A d and B d in one step is by utilizing the following property:  : p. It can, however, be computed by first constructing a matrix, and computing the exponential of it .

The discretized process noise is then evaluated by multiplying the transpose of the lower-right partition of G with the upper-right partition of G :. Now we want to discretise the above expression. We assume that u is constant during each timestep. Exact discretization may sometimes be intractable due to the heavy matrix exponential and integral operations involved.

The approximate solution then becomes:.In general, mechanical energy monotonically decreases in a physically consistent system, constructed with conservative force and dissipative force. This feature is important in designing a particle method, which is a discrete system approximating continuum fluid with particles. When the discretized system can be fit into a framework of analytical mechanics, it will be a physically consistent system which prevents instability like particle scattering along with unphysical mechanical energy increase.

This is the case also in incompressible particle methods. However, most incompressible particle methods do not satisfy the physical consistency, and they need empirical relaxations to suppress the system instability due to the unphysical energy behavior. In this study, a new incompressible particle method with the physical consistency, moving particle full-implicit MPFI method, is developed, where the discretized interaction forces are related to an analytical mechanical framework for the systems with dissipation.

Moreover, a new pressure evaluation technique based on the virial theorem is proposed for the system. Using the MPFI method, static pressure, droplet extension, standing wave and dam break calculations were conducted. The capability to predict pressure and motion of incompressible free surface flow was presented, and energy dissipation property depending on the particle size and time step width was studied through the calculations.

Particle methods are widely used to calculate the complex motion of free surface flows in various engineering fields. Smoothed particle hydrodynamics SPH for weakly compressible free surface flow was proposed by Monaghan [ 1 ] as an extension from astrophysics, while moving particle semi-implicit MPS was developed by Koshizuka and Oka [ 23 ] to calculate strictly incompressible free surface flows in the nuclear engineering field.

In designing a numerical methodology for physical simulation, it is important to take fundamental physics into consideration. In general, continuum mechanics, e. However, it is not always satisfied in a discrete system approximating the continuum equations. When a discrete particle system does not satisfy the second law of thermodynamics, the mechanical energy may increase and cause instability like particle scattering. Therefore, it is important to satisfy the fundamental laws of physics in formulating interaction forces in particle methods.

The physical consistency is taken care of using analytical mechanical frameworks in various scales of calculation. For example, molecular dynamics [ 4 ] and astrodynamics [ 5 ], where the energy dissipation is negligible, are constructed following the classical analytical mechanics, and the mechanical energy is conserved in their systems.

Besides, dissipative particle dynamics DPD [ 67 ], which is for the mesoscale simulation where the thermal fluctuation is taken into consideration, is formulated based on the general equation for the nonequilibrium reversible—irreversible coupling GENERIC framework [ 89 ], and the system satisfies the first and the second laws of thermodynamics.

These systems in various scales stand on the frameworks of analytical mechanics and satisfy the fundamental laws of physics. The fundamental laws of physics are to be satisfied also in a discrete particle system for the continuum calculation.The obesity epidemic is considered a health concern of paramount importance in modern society.

In this work, a nonstandard finite difference scheme has been developed with the aim to solve numerically a mathematical model for obesity population dynamics. This interacting population model represented as a system of coupled nonlinear ordinary differential equations is used to analyze, understand, and predict the dynamics of obesity populations. The construction of the proposed discrete scheme is developed such that it is dynamically consistent with the original differential equations model.

Since the total population in this mathematical model is assumed constant, the proposed scheme has been constructed to satisfy the associated conservation law and positivity condition. Numerical comparisons between the competitive nonstandard scheme developed here and Euler's method show the effectiveness of the proposed nonstandard numerical scheme.

Numerical examples show that the nonstandard difference scheme methodology is a good option to solve numerically different mathematical models where essential properties of the populations need to be satisfied in order to simulate the real world. Systems of nonlinear ordinary differential equations are used to model different kind of diseases in mathematical epidemiology see [ 1 ].

In these models, the variables commonly represent populations of susceptible, infected, recovered, transmitted disease vectors, and so forth. Since the system describes the evolution of different classes of populations, a solution over the time is required.

The ideal scenario is to obtain analytical solutions, but in most of the cases, this is not possible to achieve. Therefore, it is necessary to turn to numerical methods to obtain numerical simulations. Traditional schemes, like forward Euler, Runge-Kutta, and other numerical methods used to solve nonlinear initial value problems, sometimes fail generating oscillations, bifurcations, chaos, and false steady states [ 2 ].

Methods that use adaptive step size, also may fail [ 3 ]. One approach to avoid this class of numerical instabilities is the construction of nonstandard finite difference schemes. This technique, developed by Mickens [ 45 ], has brought a lot of applications where the nonstandard methods have been applied to various problems in science and engineering [ 6 — 14 ] in which the numerical solutions preserve properties of the solution of the continuous model, and additionally, in some cases, it is possible to use large time step sizes.

Furthermore, it has been shown that the nonstandard finite difference schemes, generated using the rules created by Mickens [ 4 ], generally provide accurate numerical solutions to differential equations. The aim of this paper is to develop a nonstandard finite difference NSFD scheme free of numerical instabilities in order to obtain the numerical solution of a mathematical model of infant obesity with constant population size presented in [ 15 ]. This interacting population model represented as a system of coupled nonlinear ordinary differential equations exhibits two steady states: a trivial steady state called obesity free equilibrium OFE and a nontrivial steady state called obesity endemic equilibrium OEE.

The general philosophy for constructing NSFD schemes is to obtain numerical solutions dynamically consistent with the underlying continuous model.

### A physically consistent particle method for incompressible fluid flow calculation

This means that all of the critical, qualitative properties of the solutions to the system of differential equations should also be satisfied by the solutions of the discrete scheme [ 16 ]. The design of a nonstandard scheme is not a straightforward task, in fact, many schemes may be developed for one model and several can fail to converge. Therefore the innovative part in this work is the construction of a NSFD scheme, such that is consistent with the properties of continuous dynamic model, as positivity and conservation of the population.

All the numerical simulations with different parameter values suggest that the NSFD scheme developed here preserves numerical stability in larger regions in comparison to the smaller regions of other standard numerical methods. However, theoretical justification of this fact is not possible due to unmanageable analytic expression of the eigenvalues of the Jacobian matrix corresponding to the linearization at the equilibrium points.

It is important to remark that most of the previous applications of nonstandard methods to ODE's have been done to one, two, and three equations, where theoretical analysis is easier.

Since subpopulations must never take on negative values, the proposed NSFD scheme is designed to satisfy positivity condition. Furthermore, due to the fact that in the mathematical model the population is assumed constant, the NSFD scheme has been developed in order to always exactly satisfy the associated conservation law [ 17 ]. The organization of this paper is as follows. In Section 2the mathematical model for the evolution of overweight and obesity population is presented.Differential equations are extensively used for modeling dynamics of physical processes in many scientific fields such as engineering, physics, and biomedical sciences.

## Discretization of continuous features in clinical datasets

Parameter estimation of differential equation models is a challenging problem because of high computational cost and high-dimensional parameter space. In this paper, we propose a novel class of methods for estimating parameters in ordinary differential equation ODE models, which is motivated by HIV dynamics modeling. The new methods exploit the form of numerical discretization algorithms for an ODE solver to formulate estimating equations.

First a penalized-spline approach is employed to estimate the state variables and the estimated state variables are then plugged in a discretization formula of an ODE solver to obtain the ODE parameter estimates via a regression approach. A higher order numerical algorithm reduces numerical error in the approximation of the derivative, which produces a more accurate estimate, but its computational cost is higher.

## A dynamically consistent nonstandard finite difference scheme for a predator–prey model

To balance the computational cost and estimation accuracy, we demonstrate, via simulation studies, that the trapezoidal discretization-based estimate is the best and is recommended for practical use. The asymptotic properties for the proposed numerical discretization-based estimators DBE are established.

Comparisons between the proposed methods and existing methods show a clear benefit of the proposed methods in regards to the trade-off between computational cost and estimation accuracy. We apply the proposed methods to an HIV study to further illustrate the usefulness of the proposed approaches.

Mod-01 Lec-10 Fundamentals of Discretization: Finite Element Method

Differential equations are very popular for modeling dynamics of a physical process in many fields such as engineering, physics, and biomedical sciences. Parameter estimation of differential equation models using the observed data is an important step in characterizing the physical process under investigation. In this paper, we propose a new class of methods for estimating parameters in a nonlinear ordinary differential equation ODE model.

A general nonlinear ODE model can be written as. We assume that X t is measured with noise at time points t 1…t n and the measurement model can be written as. For presentation and notation simplicity, we mainly focus on the case of one equation i.

The non-linear least squares NLS method was the earliest and the most popular method developed for estimating the parameters of the ODE model Hemker, ; Li, Osborne and Pravan, In this case, numerical methods such as the Runge-Kutta algorithm see Hairer, Norsett and Wanner, ; Mattheij and Molenaar, have to be used to approximate the solution of the ODEs.

Xue, Miao and Wu studied the properties of the NLS estimator when both numerical error and measurement error are considered, and they showed that the NLS estimator is strongly consistent and asymptotically normal with the same asymptotic covariance as that of the case with the true ODE solution exactly known. However, the NLS method suffers from some problems such as high computational cost, unstability and convergence problems since numerical algorithms are often needed to solve the ODE repeatedly in order to obtain the NLS estimates.

In addition, it also requires the initial or boundary conditions of the state variables in order to solve the ODE numerically.Rabies is a fatal disease in dogs as well as in humans. A possible model to represent rabies transmission dynamics in human and dog populations is presented. A nonstandard finite difference scheme that replicates the dynamics of the continuous model is proposed.

Numerical tests to support the theoretical analysis are provided. This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Anguelov, R. Methods Partial Differ. Bayer, H. PhD thesis, University of Glasgow, Glasgow Brauer, F. Springer, New York Busenberg, S. Diekmann, O. Wiley, New York Fekadu, M. Google Scholar. Gantmacher, F. Chelsea, New York Hampson, K. USA 18— Haupt, W. Vaccine 17— Jackson, A. Lasalle, J. SIAMS Lubuma, J.The increasing availability of clinical data from electronic medical records EMRs has created opportunities for secondary uses of health information.

When used in machine learning classification, many data features must first be transformed by discretization. To evaluate six discretization strategies, both supervised and unsupervised, using EMR data.

Continuous features were partitioned using two supervised, and four unsupervised discretization strategies. The resulting classification accuracy was compared with that obtained with the original, continuous data.

Supervised methods were more accurate and consistent than unsupervised, but tended to produce larger decision trees. Among the unsupervised methods, equal frequency and k-means performed well overall, while equal width was significantly less accurate. This is, we believe, the first dedicated evaluation of discretization strategies using EMR data.

It is unlikely that any one discretization method applies universally to EMR data. Performance was influenced by the choice of class labels and, in the case of unsupervised methods, the number of intervals. In selecting the number of intervals there is generally a trade-off between greater accuracy and greater consistency. In general, supervised methods yield higher accuracy, but are constrained to a single specific application.

Unsupervised methods do not require class labels and can produce discretized data that can be used for multiple purposes. With the adoption of electronic medical records EMRsthe quantity and scope of clinical data available for research, quality improvement, and other secondary uses of health information will increase markedly.

Algorithms from the fields of data mining and machine learning show promise in this regard. Such methods have been successfully used with clinical data, including data from EMRs, to predict the development of retinopathy in type I diabetes, 5 the quality of glycemic control in type II diabetes, 6 and the diagnosis of pancreatic cancer.

For example, EMR data have been used to identify cohorts of patients with peripheral arterial disease, providing valuable phenotype data for genome-wide associations studies. Current issues in machine learning with clinical data include model selection, feature selection and ranking, parameter estimation, performance estimation, semantic interpretability, and algorithm optimization.

There are a few ways in which discretization can be a useful preprocessing step in machine learning and data mining tasks. First, many popular learning methods—including association rules, induction rules, and Bayesian networks—require categorical rather than continuous features.

Discretization eliminates the need for this assumption by providing a direct evaluation of the conditional probability of categorical values based on counts within the dataset. Second, widely used tree-based classifiers—including classification and regression trees CART and random forests—can be made more efficient through discretization, by obviating the need to sort continuous feature values during tree induction.

Discretization can derive more interpretable intervals in the data that can improve the clarity of classification models that use rule sets. Finally, by creating categorical variables, discretization enables the derivation of count data, which would otherwise not be possible with continuous data. Methods for discretization can be classified as either supervised, in which information from class labels is used to optimize the discretization, or unsupervised, in which such information is not available, or not used.

Though many different discretization algorithms have been devised and evaluated, few studies have examined the discretization of clinical data specifically.Metrics details. The interaction between prey and predator is one of the most fundamental processes in ecology. Discrete-time models are frequently used for describing the dynamics of predator and prey interaction with non-overlapping generations, such that a new generation replaces the old at regular time intervals. Keeping in view the dynamical consistency for continuous models, a nonstandard finite difference scheme is proposed for a class of predator—prey systems with Holling type-III functional response.

Positivity, boundedness, and persistence of solutions are investigated. Analysis of existence of equilibria and their stability is carried out. It is proved that a continuous system undergoes a Hopf bifurcation at its interior equilibrium, whereas the discrete-time version undergoes a Neimark—Sacker bifurcation at its interior fixed point. A numerical simulation is provided to strengthen our theoretical discussion. Many real-life biological models including prey—predator interactions often are governed by nonlinear differential equations. For these nonlinear differential systems analytical solutions are not always easy to investigate. One of the most challenging tasks is solving these nonlinear differential equations efficiently. There are several methods for converting continuous differential systems to their discrete counterparts. The most conventional way for this purpose is to implement standard difference methods such as Euler approximations and Runge—Kutta methods.

However, numerical instabilities are observed with the implementation of standard finite difference schemes. In order to get rid of these numerical instabilities, one can implement nonstandard finite difference schemes introduced by Mickens [ 1 ]. Generally, a nonstandard finite difference scheme is based on the set of rules aimed at preserving the most dynamical properties of the associated continuous-time model, such as boundedness, positivity of solutions, stability of steady states, conservation laws, and bifurcations.

In other words, the main advantage of these nonstandard finite difference schemes is to preserve the significant properties of their continuous analogs and consequently give reliable numerical results. On the other hand, the construction of these nonstandard finite difference schemes is not always a straightforward task and there are no general criteria for their construction, and these may be considered as major drawbacks for nonstandard finite difference schemes. Predator—prey interactions belong to the most important ways that species interact in ecological communities.

Predator—prey models can reasonably be seen as the building blocks for the ecosystems. Mathematical models governed by differential equations are more appropriate for the species in which populations are overlapped. In the case of non-overlapping generations, discrete-time models governed by difference equations are more suitable than differential equations.

Discretization of differential equations is one way to produce discrete-time models governed by difference equations. Numerical methods are implemented to differential equations in order to produce discrete-time models for predator—prey systems. A discrete-time model is said to be dynamically consistent with its continuous counterpart if the two demonstrate a similar dynamical behavior, such as boundedness and persistence of solutions, stability behavior of steady states, chaos, and bifurcation [ 2 ].

Forward Euler approximations and piecewise constant arguments are more frequently used methods to obtain discrete-time counterparts of predator—prey models. But both of these methods are lacking the dynamical consistency with their continuous counterparts. Ushiki [ 3 ] proposed a discrete-time predator—prey model with implementation of forward Euler approximation, and it was investigated that the discrete-time system undergoes period-doubling bifurcation and the route to chaos was also discussed.

Jing and Yang [ 4 ] implemented an Euler forward scheme to obtain a discrete version of the prey—predator system. Furthermore, in their paper they discussed period-doubling and Neimark—Sacker bifurcations.

Similarly, Liu and Xiao [ 5 ] presented complex dynamics for a discrete Lotka—Volterra system after implementation of Euler method. For a similar type of investigations related to predator—prey systems the interested reader is referred to [ 67891011121314151617 ].

All these studies reveal that the discrete predator—prey models with implementation of Euler approximation are dynamically inconsistent with their continuous counterparts. On the other hand, some other researchers implemented piecewise constant arguments to produce a discrete analog of the predator—prey system.

Jiang and Rogers [ 18 ] implemented piecewise constant arguments to study the competitive case, and Krawcewicz and Rogers [ 19 ] investigated the cooperative case. Recently, Din [ 202122 ] applied piecewise constant arguments to various classes of predator—prey system and investigated bifurcation and chaos control for discrete models. All these investigations reveal dynamical inconsistency between the discrete-time and continuous-time systems.

Keeping in view the dynamical consistency, Liu and Elaydi [ 2 ] studied competitive and cooperative systems of prey—predator type, and Al-Kahby et al. Moreover, Roeger and Allen [ 24 ], and Roeger [ 252627 ] discussed the dynamics of May-Leonard competitive models in discrete cases.