Adaptive finite difference solutions of Liouville equations in computational uncertainty quantification

Razi, Mani; Attar, Peter J.; Vedula, Prakash

doi:10.1016/j.ress.2015.05.024

Cited by 6 publications

(4 citation statements)

References 40 publications

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“…In this way, the ADGFDM requires less memory cost and reduces operation time effectively over conventional non-adaptive algorithms [11] . As exemplarily shown in Fig.…”

Section: Resultsmentioning

confidence: 99%

Dynamic properties of a pulse-pumped fiber laser with a short, high-gain cavity

Chaolin

Guo

et al. 2016

Optical Fiber Technology

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We demonstrate a pulsed high-gain all-fiber laser without intracavity modulators, where a short and heavily Erbium-doped fiber is used as the gain medium in a ring cavity. By pulsed-pumping this short high gain cavity and tuning an intracavity variable optical coupler, the laser generates optical pulses with a pulse-width of s at a repetition rate in the order of kHz down to one-shot operation. Furthermore, dynamic properties of this laser are investigated theoretically based on a travelling-wave-model, in which an adaptive-discrete-grid-finite-difference-method is applied. The simulation results validate the experimental results. The demonstrated pulsed laser is compact, flexible and cost-effective, which will have great potential for applications in all-optical sensing and communication systems.

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“…In this way, the ADGFDM requires less memory cost and reduces operation time effectively over conventional non-adaptive algorithms [11] . As exemplarily shown in Fig.…”

Section: Resultsmentioning

confidence: 99%

Dynamic properties of a pulse-pumped fiber laser with a short, high-gain cavity

Chaolin

Guo

et al. 2016

Optical Fiber Technology

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“…As the domain for the continuous problem is infinite, at each time-step the grid is found using Eqs. 10 and 11 and the time-dependent boundary S Ω (t) is determined based upon the tails of the response [58,55]. For comparison purposes, numerical solutions are also generated with the standard upwinding scheme on a uniform grid within S Ω (t).…”

Section: Test Problem 1: Liouville Equationmentioning

confidence: 99%

Grid adaptation and non-iterative defect correction for improved accuracy of numerical solutions of PDEs

Razi

Attar

Vedula

2015

Applied Mathematics and Computation

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In this work we present a computational approach for improving the order of accuracy of a given finite difference method for solution of linear and nonlinear hyperbolic partial differential equations. The methodology consists of analysis of leading order terms in the discretization error of any given finite difference method, leading to a modified version of the original partial differential equation. Singular perturbations of this modified equation are regularized using an adaptive grid distribution and a non-iterative defect correction method is used to eliminate the leading order, regular perturbation terms in the modified equation. Implementation of this approach on a low order finite difference scheme not only results in an increase in its order of accuracy but also results in an improvement in its numerical stability due to the regularization of singular perturbations. The proposed approach is applied to four different canonical problems including the numerical solution of (1) Liouville equation, (2) inviscid Burgers equation, (3) nonlinear reaction-advection equation and a (4) system of hyperbolic PDEs. When compared to exact solutions, the numerical results demonstrate the ability of this method in both boosting the accuracy of finite difference schemes up to the desired order and also providing a fully stable numerical solution.

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“…Future work will consider non-uniform meshes determined using various standard 37,43,48,49 and non-standard methods. 29 As stated in the previous paragraph the computational domain is resized at each time-step and therefore finite difference computations are performed only over a small region of the random response variable space. Consequently, when the adaptive moving uniform grid distribution is applied, for the same number of grid points, the spatial increments are much smaller than the spatial increments of a fixed uniform computational domain that covers the entire area in which the conditional density convects.…”

Section: Time-varying Computational Domain/mesh Adaptationmentioning

confidence: 99%

“…In this work, a novel intrusive approach based upon the formulation of UQ problems in the terms of Liouville equation 29 is applied to a number of problems with varying state dimension. While maintaining a reasonable level of accuracy as system evolves in time, this approach is capable of dealing with UQ problems involving moderate state dimension.…”

Section: Introductionmentioning

confidence: 99%

Uncertainty Quantification for Multidimensional Dynamical Systems Based on Adaptive Numerical Solution of Liouville Equation

Razi

Attar

Vedula

2013

54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference

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I. AbstractComputational uncertainty quantification of multidimensional dynamical systems is investigated through an approach based upon a finite difference solution of the Liouville equation, which governs the evolution of the response conditional density function. The computational performance of the numerical solution is improved by (i) quadrature-based sampling of the random input parameters and (ii) a time-adaptive method for determining the computational grid in the space of random response variables. The proposed intrusive approach is applied to four test problems including the uncertainty quantification of a spring-mass system, Van der Pol Oscillator, linear double spring-mass system and the flow induced vibrations of an airfoil containing cubic nonlinearity in an incompressible flow. Comparisons of the numerical solutions obtained from both fixed and adaptive time-varying grids with analytical solutions (when they exist) and Monte Carlo simulation results demonstrate the capability of the proposed methodology for the accurate prediction of the long time statistics of dynamical systems of moderate dimension. When compared to the fixed grid solution, the adaptive grid solution demonstrates considerable reduction in computational costs for equivalent accuracy. II. IntroductionMeasurement of the effect of randomness in computational model input parameters on output variables is obtained through the process of computational uncertainty quantification. The results produced by such analysis play a crucial role in the quantitative reliability assessment of the system. The computational uncertainty analysis of a computational model as a system is studied through the propagation of model input uncertainty through the computational model. This analysis gives the statistics of the model output which are essential for determining the probability of undesirable events (output) and consequently reliability assessment of the physical system being modeled. Computational uncertainty quantification (CUQ) is used for wide variety of engineering applications such as computational fluid dynamics, 6 chemical systems, 9 scientific computing 10 and structural mechanics. 11 Moreover, in the Aerospace engineering literature, CUQ has been applied to problems such as the prediction of limit cycle oscillations, 1 transonic flutter, 31 hypersonic aerothermoelastic analysis 24 and rocket simulation. 23 According to the classification proposed by Melchers 13, 14 there are three types of uncertainties including aleatory, epistemic uncertainty and uncertainty due to human error. In this work only aleatory uncertainty is considered. Unlike epistemic uncertainty, which is due to the lack of knowledge about the true physics of the system being modeled, aleatory or irreducible uncertainty is concerned with inherent parameter randomness in the system.Computational uncertainty quantification approaches which deal with aleatory uncertainty can be classified into two major groups including intrusive or non-intrusive methods. "Black box" treatment...

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Adaptive finite difference solutions of Liouville equations in computational uncertainty quantification

Cited by 6 publications

References 40 publications

Dynamic properties of a pulse-pumped fiber laser with a short, high-gain cavity

Dynamic properties of a pulse-pumped fiber laser with a short, high-gain cavity

Grid adaptation and non-iterative defect correction for improved accuracy of numerical solutions of PDEs

Uncertainty Quantification for Multidimensional Dynamical Systems Based on Adaptive Numerical Solution of Liouville Equation

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