Uncertainty quantification for random parabolic equations with nonhomogeneous boundary conditions on a bounded domain via the approximation of the probability density function

Calatayud, Julia; López, Juan Carlos Cortés; Jornet, Marc

doi:10.1002/mma.5333

Cited by 15 publications

(22 citation statements)

References 23 publications

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“…Let (x 0 , t 0 ) = (0.25, 0.3). Figure 7.2 shows the approximations (7.9) for values of M = 9, 11,13,15,17,19,21,23. Convergence is achieved, so that the density function of u(x 0 , t 0 ) has been accurately approximated.…”

Section: Numerical Examplesmentioning

confidence: 95%

“…By Karhunen-Loève Theorem [89,Th. 5.28], the process Y = {Y (t) : t ∈ [0, T ]} can be expressed as 19) where µ Y (t) = E[Y (t)] and {ξ j } J j=1 is a sequence of random variables with zero expectation, unit variance and pairwise uncorrelated. These random variables {ξ j } J j=1 have a closed expression:…”

Section: Forcing Term Expressed As a Karhunen-loève Expansionmentioning

confidence: 99%

“…or discretizations from numerical methods are usually required. By truncating these infinite expansions or from the discretizations, one constructs a sequence of approximating processes, whose exact density functions (computed via the RVT technique) form a sequence of approximating density functions that converge rapidly [17,19,45,57]. In terms of rapidity and accuracy, this sort of approximations are preferable to Monte Carlo simulation and Kernel density estimations.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…), one usually constructs a sequence of approximating processes, whose exact density functions (computed via the RVT technique) form a sequence of approximating density functions that converge rapidly. This methodology has been carried out via numerical methods [57], Karhunen-Loève expansions [17,19] (see [89] for the theory on these expansions), polynomial expansions [14], or random power series [17,45], and it has supposed an improvement compared with Monte Carlo simulation and Kernel density estimations in terms of rate of convergence and accuracy.…”

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]. In [42], the diffusion coefficient A(t) is written as a Karhunen-Loève expansion [89]:…”

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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Section: Numerical Examplesmentioning

confidence: 95%

Section: Forcing Term Expressed As a Karhunen-loève Expansionmentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

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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Improvement of random coefficient differential models of growth of anaerobic photosynthetic bacteria by combining Bayesian inference and gPC

Calatayud

López

Jornet

2019

Math Methods in App Sciences

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The time evolution of microorganisms, such as bacteria, is of great interest in biology. In the article by D. Stanescu et al. [Electronic Transactions on Numerical Analysis, 34, 44{58 (2009)], a logistic model was proposed to model the growth of anaerobic photosynthetic bacteria. In the laboratory experiment, actual data for two species of bacteria were considered: R. capsulatus and C. vibrioforme. In this paper, we suggest a new nonlinear model by assuming that the population growth rate is not proportional to the size of the bacteria population, but to the number of interactions between the microorganisms, and by taking into account the beginning of the death phase in the kinetic curve. Stanescu et al. evaluated the eect of randomness into the model coecients by using generalized Polynomial Chaos (gPC) expansions, by setting arbitrary distributions without taking into account the likelihood of the data. By contrast, we utilize a Bayesian inverse approach for parameter estimation to obtain reliable posterior distributions for the random input coecients in both the logistic and our new model. Since our new model does not possess an explicit solution, we use gPC expansions to construct the Bayesian model and to accelerate the Markov Chain Monte Carlo algorithm for the Bayesian inference.

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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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Uncertainty quantification for random parabolic equations with nonhomogeneous boundary conditions on a bounded domain via the approximation of the probability density function

Cited by 15 publications

References 23 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

Improvement of random coefficient differential models of growth of anaerobic photosynthetic bacteria by combining Bayesian inference and gPC

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

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