A modified perturbation method for mathematical models with randomness: An analysis through the steady-state solution to Burgers’ PDE

The variability of the data and the incomplete knowledge of the true physics require the incorporation of randomness into the formulation of mathematical models. In this setting, the deterministic numerical methods cannot capture the propagation of the uncertainty from the inputs to the model output. For some problems, such as the Burgers' equation (simplification to understand properties of the Navier–Stokes equations), a small variation in the parameters causes nonnegligible changes in the output. Thus, suitable techniques for uncertainty quantification must be used. The generalized polynomial chaos (gPC) method has been successfully applied to compute the location of the transition layer of the steady-state solution, when a small uncertainty is incorporated into the boundary. On the contrary, the classical perturbation method does not give reliable results, due to the uncertainty magnitude of the output. We propose a modification of the perturbation method that converges and is comparable with the gPC approach in terms of efficiency and rate of convergence. The method is even applicable when the input random parameters are dependent random variables

A modified perturbation method for mathematical models with randomness: An analysis through the steady-state solution to Burgers' partial differential equation

[EN] The variability of the data and the incomplete knowledge of the true physics require the incorporation of randomness into the formulation of mathematical models. In this setting, the deterministic numerical methods cannot capture the propagation of the uncertainty from the inputs to the model output. For some problems, such as the Burgers' equation (simplification to understand properties of the Navier¿Stokes equations), a small variation in the parameters causes nonnegligible changes in the output. Thus, suitable techniques for uncertainty quantification must be used. The generalized polynomial chaos (gPC) method has been successfully applied to compute the location of the transition layer of the steady-state solution, when a small uncertainty is incorporated into the boundary. On the contrary, the classical perturbation method does not give reliable results, due to the uncertainty magnitude of the output. We propose a modification of the perturbation method that converges and is comparable with the gPC approach in terms of efficiency and rate of convergence. The method is even applicable when the input random parameters are dependent random variables.This work has been supported by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), the Agencia Estatal de Investigacion (AEI), and Fondo Europeo de Desarrollo Regional (FEDER UE) Grant MTM2017-89664-P. The author Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.Calatayud, J.; Cortés, J.; Jornet, M. (2021). A modified perturbation method for mathematical models with randomness: An analysis through the steady-state solution to Burgers' partial differential equation. Mathematical Methods in the Applied Sciences. 44(15):11820-11827. https://doi.org/10.1002/mma.6420S1182011827441

Nonlinear metastability for a parabolic system of reaction-diffusion equations

We consider a system of reaction-diffusion equations in a bounded interval of the real line, with emphasis on the metastable dynamics, whereby the time-dependent solution approaches its steady state in an asymptotically exponentially long time interval as the viscosity coefficient $\varepsilon>0$ goes to zero. To rigorous describe such behavior, we analyze the dynamics of solutions in a neighborhood of a one-parameter family of approximate steady states, and we derive an ODE for the position of the internal interfaces.Comment: This paper has been withdrawn by the author due to an error in Theorem 1.1. Please refer to the paper "Slow dynamics in reaction-diffusion systems

Transition to turbulence, small disturbances, and sensitivity analysis I: A motivating problem

AbstractFor over 100 years, researchers have attempted to predict transition to turbulence in fluid flows by analyzing the spectrum of the linearized Navier–Stokes equations. However, for many simple flows this approach fails to match experimental results. Recently, new scenarios for transition have been proposed that are based on the interaction of the linearized equations of motion with small disturbances to the flow system. These new “mostly linear” theories have increased our understanding of the transition process, but the role of nonlinearity has not been explored in detail. This paper is the first of a two part work in which sensitivity analysis is used to study the effects of small disturbances on transition to turbulence. In this part, we study a highly sensitive one-dimensional Burgers' equation as a motivating problem. Sensitivity analysis is used to predict the large changes in solutions in the presence of a small disturbance. Also, sensitivity analysis is shown to provide more information about the disturbed nonlinear problem than a purely linear analysis of the problem. In the second part of this work, this analysis will be extended to the three-dimensional Navier–Stokes equations to show that small disturbances have great potential to trigger transition to turbulence

Nonlinear stability of viscous transonic flow through a nozzle.

Xie Chunjing.Thesis (M.Phil.)--Chinese University of Hong Kong, 2004.Includes bibliographical references (leaves 65-71).Abstracts in English and Chinese.Acknowledgments --- p.iAbstract --- p.iiIntroduction --- p.3Chapter 1 --- Stability of Shock Waves in Viscous Conservation Laws --- p.10Chapter 1.1 --- Cauchy Problem for Scalar Viscous Conservation Laws and Viscous Shock Profiles --- p.10Chapter 1.2 --- Stability of Shock Waves by Energy Method --- p.15Chapter 1.3 --- Nonlinear Stability of Shock Waves by Spectrum Anal- ysis --- p.20Chapter 1.4 --- L1 Stability of Shock Waves in Scalar Viscous Con- servation Laws --- p.26Chapter 2 --- Propagation of a Viscous Shock in Bounded Domain and Half Space --- p.35Chapter 2.1 --- Slow Motion of a Viscous Shock in Bounded Domain --- p.36Chapter 2.1.1 --- Steady Problem and Projection Method --- p.36Chapter 2.1.2 --- Projection Method for Time-Dependent Prob- lem --- p.40Chapter 2.1.3 --- Super-Sensitivity of Boundary Conditions --- p.43Chapter 2.1.4 --- WKB Transformation Method --- p.45Chapter 2.2 --- Propagation of a Stationary Shock in Half Space --- p.50Chapter 2.2.1 --- Asymptotic Analysis --- p.50Chapter 2.2.2 --- Pointwise Estimate --- p.51Chapter 3 --- Nonlinear Stability of Viscous Transonic Flow Through a Nozzle --- p.58Chapter 3.1 --- Matched Asymptotic Analysis --- p.58Bibliography --- p.6

Some topics on nonlinear conservation laws.

Duan, Ben.Thesis (M.Phil.)--Chinese University of Hong Kong, 2007.Includes bibliographical references (leaves 61-67).Abstracts in English and Chinese.Acknowledgments --- p.2Abstract --- p.iIntroduction --- p.3Chapter 1 --- Stability of Shock Waves in Viscous Conservation Laws --- p.10Chapter 1.1 --- Cauchy Problem for Scalar Viscous Conservation Laws and Viscous Shock Profiles --- p.10Chapter 1.2 --- Stability of Shock Waves by Energy Method --- p.15Chapter 1.3 --- L1 Stability of Shock Waves in Scalar Viscous Con- servation Laws --- p.20Chapter 2 --- Slow Motion of a Viscous Shock --- p.29Chapter 2.1 --- Propagation of a Viscous Shock in Bounded Domain --- p.29Chapter 2.1.1 --- Steady Problem --- p.30Chapter 2.1.2 --- Time-Dependent Problem --- p.34Chapter 2.1.3 --- Super-Sensitivity of Boundary Conditions --- p.36Chapter 2.2 --- Propagation of a Stationary Shock in Half Space --- p.39Chapter 2.2.1 --- Asymptotic Analysis --- p.39Chapter 2.2.2 --- Pointwise Estimate --- p.40Chapter 3 --- Viscous Transonic Flow Through a Nozzle --- p.47Chapter 3.1 --- Nonlinear Stability and Instability of Shock Waves --- p.48Chapter 3.2 --- Asymptotic Stability and Instability --- p.49Chapter 3.3 --- Matched Asymptotic Analysis --- p.53Chapter 4 --- C --- p.60Bibliography --- p.6

Models of Delay Differential Equations

This book gathers a number of selected contributions aimed at providing a balanced picture of the main research lines in the realm of delay differential equations and their applications to mathematical modelling. The contributions have been carefully selected so that they cover interesting theoretical and practical analysis performed in the deterministic and the stochastic settings. The reader will find a complete overview of recent advances in ordinary and partial delay differential equations with applications in other multidisciplinary areas such as Finance, Epidemiology or Engineerin

Fast Adjoint-assisted Multilevel Multi delity Method for Uncertainty Quanti cation of the Aleatoric Kind

PhDIn this thesis an adjoint-based multilevel multi delity Monte Carlo (MLMF) method is proposed, analysed, and demonstrated using test problems. Firstly, a multifi delity framework using the approximate function evaluation [1] based on the adjoint error correction of Giles et al. [2] is employed as a low fidelity model. This multifi delity framework is analysed using the method proposed by Ng and Wilcox [3]. The computational cost reduction and accuracy is demonstrated using the viscous Burgers' equation subject to uncertain boundary condition. The multi fidelity framework is extended to include multilevel meshes using the MLMF of Geraci [4] called the FastUQ. Some insights on parameters affecting computational cost are shown. The implementation of FastUQ in Dakota toolkit is outlined. As a demonstration, FastUQ is used to quantify uncertainties in aerodynamic parameters due to surface variations caused by manufacturing process. A synthetic model for surface variations due to manufacturing process is proposed based on Gaussian process. The LS89 turbine cascade subject to this synthetic disturbance model at two o -design conditions is used as a test problem. Extraction of independent random modes and truncation using a goal-based principal component analysis is shown. The analysis includes truncation for problems involving multiple QoIs and test conditions. The results from FastUQ are compared to the state-of-art SMLMC method and the approximate function evaluation using adjoint error correction called the inexpensive Monte Carlo method (IMC). About 70% reduction in computational cost compared to SMLMC is achieved without any loss of accuracy. The approximate model based on the IMC has high deviations for non-linear and sensitive QoI, namely the total-pressure loss. FastUQ control variate effectively balances the low fi delity model errors and additional high fidelity evaluations to yield accurate results comparable to the high fidelity model.This work has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 642959