Random Differential Equations in Scientific Computing

Neckel, Tobias; Rupp, Florian

doi:10.2478/9788376560267

Cited by 52 publications

(55 citation statements)

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“…This leads to the area of Random Differential Equations (RDEs). The so-called Random Mean Square Calculus provides a powerful tool to rigorously tackle RDEs [19,20]. The study of the logistic RDE, using the Mean Square Calculus and its generalizations, can be found for instance in [21,22].…”

Section: Motivation and Preliminariesmentioning

confidence: 99%

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

López

Navarro‐Quiles

Romero

et al. 2019

Communications in Nonlinear Science and Numerical Simulation

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This paper deals with the study, from a probabilistic point of view, of logistic-type differential equations with uncertainties. We assume that the initial condition is a random variable and the diffusion coefficient is a stochastic process. The main objective is to obtain the first probability density function, f 1 (p, t), of the solution stochastic process, P(t, ω). To achieve this goal, first the diffusion coefficient is represented via a truncation of order N of the Karhunen-Loève expansion, and second, the Random Variable Transformation technique is applied. In this manner, approximations, say f N 1 (p, t), of f 1 (p, t) are constructed. Afterwards, we rigorously prove that f N 1 (p, t) −→ f 1 (p, t) as N → ∞ under mild conditions assumed on input data (initial condition and diffusion coefficient). Finally, three illustrative examples are shown.

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Section: Motivation and Preliminariesmentioning

confidence: 99%

“…Again, we observe fast convergence in all these cases. We use an analogous measure to the one defined in (20), i.e., e PDF N (t) = 0.5 −0.5 f N 1 (p, t) − f N−1 1 (p, t) dp, N = 2, 3, . .…”

Section: Numerical Examplesmentioning

confidence: 99%

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

López

Navarro‐Quiles

Romero

et al. 2019

Communications in Nonlinear Science and Numerical Simulation

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“…Unlike white noise, the spectral density of real noise is not flat and the samples are correlated. Therefore, we also include colored noise in our comparison to highlight the performance under more realistic conditions [23] in which positively correlated colored noise is generated based on the work in [24]. The output SNR or SNR after de-noising, SNR o , and mean square error (MSE) are used for the performance evaluation as these metrics demonstrate the relevant improvement in SNR and difference from the original signal.…”

Section: Simulations and Resultsmentioning

confidence: 99%

A De-noising Scheme Based on the Null Hypothesis of Intrinsic Mode Functions

Al-Badrawi

Al-Jewad

Smith

et al. 2016

IEEE Signal Process. Lett.

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Empirical mode decomposition is a nonparametric adaptive tool that decomposes signals into a set of zero-mean modes called intrinsic mode functions (IMFs) that can be used to denoise a signal by selecting the relevant (noise-free) modes. In this paper, the statistical properties of IMFs, produced by a range of signal distributions, are examined. Insight from the statistical analysis is used in a null hypothesis test that validates a best fit model distribution of the IMFs. This test is then used in a de-noising scheme, which is evaluated using different test signals corrupted with white and colored noise. Simulations under different scenarios demonstrate the efficacy of the proposed method as compared to other EMD-based de-noising techniques. Index Terms-Empirical mode decomposition (EMD), intrinsic mode function (IMF), null hypothesis, signal de-noising.1070-9908

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“…'s are those in which random effects are directly manifested in its inputs parameters (initial/boundary conditions, source term and coefficients). These inputs are assumed to satisfy regularity properties such as continuity, differentiability, etc., in some adequate stochastic sense such as mean square calculus [2,3]. In [3] one can find an updated overview of the state of the art of r.d.e.'s.…”

Section: Motivationmentioning

confidence: 99%

“…These inputs are assumed to satisfy regularity properties such as continuity, differentiability, etc., in some adequate stochastic sense such as mean square calculus [2,3]. In [3] one can find an updated overview of the state of the art of r.d.e.'s. A major advantage of considering r.d.e.…”

Section: Motivationmentioning

confidence: 99%

Solving Random Homogeneous Linear Second-Order Differential Equations: A Full Probabilistic Description

et al. 2016

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Roselló, M. (2016). Solving random homogeneous linear second-order differential equations: a full probabilistic description. Mediterranean Journal of Mathematics. 13(6): 3817-3836. doi:10.1007/s00009-016-0716-6 Solving random homogeneous linear secondorder differential equations: A full probabilistic description Abstract. In this paper a complete probabilistic description for the solution of random homogeneous linear second-order differential equations via the computation of its two first probability density functions is given. As a consequence, all unidimensional and two-dimensional statistical moments can be straightforwardly determined, in particular, mean, variance and covariance functions, as well as the first-order conditional law. With the aim of providing more generality, in a first step, all involved input parameters are assumed to be statistically dependent random variables having an arbitrary joint probability density function. Secondly, the particular case that just initial conditions are random variables is also analysed. Both problems have common and distinctive feature which are highlighted in our analysis. The study is based on Random Variable Transformation method. As a consequence of our study, the well-known deterministic results are nicely generalized. Several illustrative examples are included.Mathematics Subject Classification (2010). 60H35, 60H10, 37H10. Keywords. Random Variable Transformation method, first and second probability density functions, random homogeneous linear second-order differential equations. MotivationThe quantification of uncertainty in dynamic models is currently playing an important role in many applied areas. Classical deterministic differential equations, which have demonstrated to be powerful tools for analysing problems that appear in areas such as Physics, Engineering, Chemistry, Epidemiology, etc., need to consider randomness in their formulation in order to account for measurement errors and inherent complexity of problems underThis work was completed with the support of our T E X-pert. [4][5][6]. However, it is important to point out that in the random context besides computing the solution stochastic process (s.p.), say Z(t), it is also of great interest the determination of its main statistical properties. Most of the contributions focus on the computation of the mean, µ Z (t) = E[Z(t)], and the variance, σ 2 Z (t) = V[Z(t)], functions of the solution s.p. However, a more convenient goal is the determination of its first probability density function (1-p.d.f.), f 1 (z, t), since from it one can easily compute not just these two first statistical moments,but also the one-dimensional statistical moment of any orderThe 1-p.d.f. characterizes, from a probabilistic point of view, the solution s.p. Z(t) at every time instant t. In general, more challenging is the determination of the rest of the so-called fidis (finite dimensional distributions), i.e., the n-dimensional p.d.f.'s of the solution s.p. for n ≥ 2, because it usually involves complex developments...

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Random Differential Equations in Scientific Computing

Cited by 52 publications

References 0 publications

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

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

A De-noising Scheme Based on the Null Hypothesis of Intrinsic Mode Functions

Solving Random Homogeneous Linear Second-Order Differential Equations: A Full Probabilistic Description

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