Analysis of random non-autonomous logistic-type differential equations via the Karhunen–Loève expansion and the Random Variable Transformation technique

López, Juan Carlos Cortés; Navarro‐Quiles, A.; Romero, Javier; Roselló, M.-D.

doi:10.1016/j.cnsns.2018.12.013

Cited by 9 publications

(36 citation statements)

References 25 publications

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“…In the recent paper [42], the authors studied the random non-autonomous logistic differential equation problem on a real interval [t 0 , T ]:…”

Section: Introductionmentioning

confidence: 99%

“…Only recently, in [42] the authors studied the non-autonomous case (3.1). The authors established conditions under which the probability density function of P (t) can be approximated, in the spirit of other previous contributions such as [43,12,19].…”

Section: Introductionmentioning

confidence: 99%

“…The authors established conditions under which the probability density function of P (t) can be approximated, in the spirit of other previous contributions such as [43,12,19]. In [42], the diffusion coefficient A(t) is written as a Karhunen-Loève expansion [89]:…”

Section: Introductionmentioning

confidence: 99%

“…, is computed via the random variable transformation technique [42,Th. 1], by assuming absolute continuity of P 0 , ξ 1 , ξ 2 , .…”

Section: Introductionmentioning

confidence: 99%

“…have support in a common bounded interval, see [20, pp. 29-31], although for a Gaussian stochastic process A(t) the exact probability density function of P (t) is already known, so the applicability of [42] is restricted.…”

Section: Introductionmentioning

confidence: 99%

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Computational methods for random differential equations: probability density function and estimation of the parameters

Gregori¹

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Mathematical models based on deterministic differential equations do not take into account the inherent uncertainty of the physical phenomenon (in a wide sense) under study. In addition, inaccuracies in the collected data often arise due to errors in the measurements. It thus becomes necessary to treat the input parameters of the model as random quantities, in the form of random variables or stochastic processes. This gives rise to the study of random ordinary and partial differential equations. The computation of the probability density function of the stochastic solution is important for uncertainty quantification of the model output. Although such computation is a difficult objective in general, certain stochastic expansions for the model coefficients allow faithful representations for the stochastic solution, which permits approximating its density function. In this regard, Karhunen-Loève and generalized polynomial chaos expansions become powerful tools for the density approximation. Also, methods based on discretizations from finite difference numerical schemes permit approximating the stochastic solution, therefore its probability density function. The main part of this dissertation aims at approximating the probability density function of important mathematical models with uncertainties in their formulation. Specifically, in this thesis we study, in the stochastic sense, the following models that arise in different scientific areas: in Physics, the model for the damped pendulum; in Biology and Epidemiology, the models for logistic growth and Bertalanffy, as well as epidemiological models; and in Thermodynamics, the heat partial differential equation. We rely on Karhunen-Loève and v generalized polynomial chaos expansions and on finite difference schemes for the density approximation of the solution. These techniques are only applicable when we have a forward model in which the input parameters have certain probability distributions already set. When the model coefficients are estimated from collected data, we have an inverse problem. The Bayesian inference approach allows estimating the probability distribution of the model parameters from their prior probability distribution and the likelihood of the data. Uncertainty quantification for the model output is then carried out using the posterior predictive distribution. In this regard, the last part of the thesis shows the estimation of the distributions of the model parameters from experimental data on bacteria growth. To do so, a hybrid method that combines Bayesian parameter estimation and generalized polynomial chaos expansions is used.

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“…In the recent paper [42], the authors studied the random non-autonomous logistic differential equation problem on a real interval [t 0 , T ]:…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…, is computed via the random variable transformation technique [42,Th. 1], by assuming absolute continuity of P 0 , ξ 1 , ξ 2 , .…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Computational methods for random differential equations: probability density function and estimation of the parameters

Gregori¹

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Improving the approximation of the probability density function of random nonautonomous logistic‐type differential equations

Calatayud

López

Jornet

2019

Math Methods in App Sciences

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In this paper, we address the problem of approximating the probability density function of the following random logistic differential equation: P ′ (t, ) = A(t, )(1 − P(t, ))P(t, ), t ∈ [t 0 , T], P(t 0 , ) = P 0 ( ), where is any outcome in the sample space Ω. In the recent contribution [Cortés, JC, et al. Commun Nonlinear Sci Numer Simulat 2019; 72: 121-138], the authors imposed conditions on the diffusion coefficient A(t) and on the initial condition P 0 to approximate the density function f 1 (p, t) of P(t): A(t) is expressed as a Karhunen-Loève expansion with absolutely continuous random coefficients that have certain growth and are independent of the absolutely continuous random variable P 0 , and the density of P 0 , P 0 , is Lipschitz on (0, 1). In this article, we tackle the problem in a different manner, by using probability tools that allow the hypotheses to be less restrictive. We only suppose that A(t) is expanded on L 2 ([t 0 , T] × Ω), so that we include other expansions such as random power series. We only require absolute continuity for P 0 , so that A(t) may be discrete or singular, due to a modified version of the random variable transformation technique. For P 0 , only almost everywhere continuity and boundedness on (0, 1) are needed. We construct an approximating sequence { N 1 (p, t)} ∞ N=1 of density functions in terms of expectations that tends to f 1 (p, t) pointwise. Numerical examples illustrate our theoretical results. KEYWORDS mean square expansion, probability density function, random logistic differential equation MSC CLASSIFICATION 34F05; 60H35; 60H10In this setting, we are assuming an underlying complete probability space (Ω,  , P), where Ω is the sample space that consists of outcomes (which might usually be omitted in evaluations),  ⊆ 2 Ω is the -algebra of events, and P is the probability measure. The initial condition P 0 is a random variable, and the diffusion coefficient A(t) is a stochastic process. In principle, these random data may take any probability distribution. The solution P(t) to (1) is also a stochastic Math Meth Appl Sci. 2019;42:7259-7267.wileyonlinelibrary.com/journal/mma

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Modeling the biological growth with a random logistic differential equation

et al. 2023

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We modeled biological growth using a random differential equation (RDE), where the initial condition is a random variable, and the growth rate is a suitable stochastic process. These assumptions let us obtain a model that represents well the random growth process observed in nature, where only a few individuals of the population reach the maximal size of the species, and the growth curve for every individual behaves randomly. Since we assumed that the initial condition is a random variable, we assigned a priori density, and we performed Bayesian inference to update the initial condition’s density of the RDE. The Karhunen–Loeve expansion was then used to approximate the random coefficient of the RDE. Then, using the RDE’s approximations, we estimated the density f(p, t). Finally, we fitted this model to the biological growth of the giant electric ray (or Cortez electric ray) Narcine entemedor. Simulations of the solution of the random logistic equation were performed to construct a curve that describes the solutions’ mean for each time. As a result, we estimated confidence intervals for the mean growth that described reasonably well the observed data. We fit the proposed model with a training dataset, and the model is tested with a different dataset. The model selection is performed with the square of the errors.

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Analysis of random non-autonomous logistic-type differential equations via the Karhunen–Loève expansion and the Random Variable Transformation technique

Cited by 9 publications

References 25 publications

Computational methods for random differential equations: probability density function and estimation of the parameters

Computational methods for random differential equations: probability density function and estimation of the parameters

Improving the approximation of the probability density function of random nonautonomous logistic‐type differential equations

Modeling the biological growth with a random logistic differential equation

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