Statistical moments of the random linear transport equation

Dorini, Fabio Antonio; Cunha, M. Cristina C.

doi:10.1016/j.jcp.2008.06.002

Cited by 24 publications

(24 citation statements)

References 10 publications

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“…× Ω) and A,B ∈ L 2 (Ω), by (12)) and 0 < a ≤ β 2 (ω) ≤ b a.s., then v N (y,t)→v(y,t) in L 2 (Ω) as N→∞.…”

Section: Approximation Of the Expectation And Variance Of The Solutmentioning

confidence: 99%

“…Proof. Since ϕ ∈ L 2 ([L 1 ,L 2 ] × Ω) and by (12), ψ ∈ L 2 ([0,1] × Ω). By hypothesis, we also have that β 2 = α 2 / (L 2 − L 1 ) 2 , A 1 and (A 2 ,…,A N ) are independent and absolutely continuous, for N ≥ 2, and ∑ ∞ n¼m ‖e −ðn 2 −2Þπ 2 β 2 t ‖ L 1 ðΩ Þ < ∞.…”

Section: Approximation Of the Probability Density Function Of The Smentioning

confidence: 99%

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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

Calatayud

López

Jornet

2018

Math Methods in App Sciences

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This paper deals with the randomized heat equation defined on a general bounded interval [L1, L2] and with nonhomogeneous boundary conditions. The solution is a stochastic process that can be related, via changes of variable, with the solution stochastic process of the random heat equation defined on [0,1] with homogeneous boundary conditions. Results in the extant literature establish conditions under which the probability density function of the solution process to the random heat equation on [0,1] with homogeneous boundary conditions can be approximated. Via the changes of variable and the Random Variable Transformation technique, we set mild conditions under which the probability density function of the solution process to the random heat equation on a general bounded interval [L1, L2] and with nonhomogeneous boundary conditions can be approximated uniformly or pointwise. Furthermore, we provide sufficient conditions in order that the expectation and the variance of the solution stochastic process can be computed from the proposed approximations of the probability density function. Numerical examples are performed in the case that the initial condition process has a certain Karhunen‐Loève expansion, being Gaussian and non‐Gaussian.

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“…× Ω) and A,B ∈ L 2 (Ω), by (12)) and 0 < a ≤ β 2 (ω) ≤ b a.s., then v N (y,t)→v(y,t) in L 2 (Ω) as N→∞.…”

Section: Approximation Of the Expectation And Variance Of The Solutmentioning

confidence: 99%

Section: Approximation Of the Probability Density Function Of The Smentioning

confidence: 99%

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

Calatayud

López

Jornet

2018

Math Methods in App Sciences

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“…In [7] the random Riemann problem for Burgers equation is solved. In [8] a numerical scheme to approximate the mth moment of the solution of the one-dimensional random linear transport equation is studied. In [9] a numerical scheme for the random linear transport equation is presented.…”

Section: Inviscid Burgers Equationmentioning

confidence: 99%

Conservation Laws for Self-Adjoint First-Order Evolution Equation

Freire¹

2021

JNMP

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We consider the problem on group classification and conservation laws for first-order evolution equations. Subclasses of these general equations which are quasi-self-adjoint and self-adjoint are obtained. By using the recent new conservation theorem due to Ibragimov, conservation laws for equations admiting self-adjoint equations are established. The results are illustrated applying them to the inviscid Burgers equation. In particular an infinite number of new symmetries of this equation are found.

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“…By means of the probability density function, all the statistical moments and confidence intervals corresponding to the solution process can be determined, provided they exist [125,. When dealing with stochastic problems with a closedform solution, the Random Variable Transformation (RVT) technique has been widely used to calculate the density function of the solution, see [23,29,52] in the setting of random differential equations, [23,53,72,73,74,146] in the context of random partial differential equations, and [44] for random difference equations. When no explicit form of the solution is available or it is given via infinite analytic expressions (such as random power series, random eigenfunctions expansions, etc.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, a major challenge is to determine the probability density function of the solution, since from it one can obtain a full characterization of all one-dimensional statistical moments of the solution (hence including, just as particular cases, the mean and the variance). We point out that the computation of the probability density function of the solution stochastic process of some random ordinary and partial differential equations describing relevant problems in Physics and Engineering has been achieved, [53,118,55,54,72,73,74,59,146].…”

Section: Introductionmentioning

confidence: 99%

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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Statistical moments of the random linear transport equation

Cited by 24 publications

References 10 publications

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

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

Conservation Laws for Self-Adjoint First-Order Evolution Equation

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

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