Mitigating Gibbs Phenomena in Uncertainty Quantification With a Stochastic Spectral Method

2017 ◽  
Vol 2 (1) ◽  
Author(s):  
Piyush M. Tagade ◽  
Han-Lim Choi
Keyword(s):  
Evolution Equations ◽  
Discontinuous Solution ◽  
Gibbs Phenomenon ◽  
Spectral Projection ◽  
Spatial Dimension ◽  
Spectral Expansion ◽  
One Dimensional ◽  
Orthogonal Bases ◽  
Stochastic Burgers Equation ◽  
Gibbs Phenomena

The use of spectral projection-based methods for simulation of a stochastic system with discontinuous solution exhibits the Gibbs phenomenon, which is characterized by oscillations near discontinuities. This paper investigates a dynamic bi-orthogonality-based approach with appropriate postprocessing for mitigating the effects of the Gibbs phenomenon. The proposed approach uses spectral decomposition of the spatial and stochastic fields in appropriate orthogonal bases, while the dynamic orthogonality (DO) condition is used to derive the resultant closed-form evolution equations. The orthogonal decomposition of the spatial field is exploited to propose a Gegenbauer reprojection-based postprocessing approach, where the orthogonal bases in spatial dimension are reprojected on the Gegenbauer polynomials in the domain of analyticity. The resultant spectral expansion in Gegenbauer series is shown to mitigate the Gibbs phenomenon. Efficacy of the proposed method is demonstrated for simulation of a one-dimensional stochastic Burgers equation and stochastic quasi-one-dimensional flow through a convergent-divergent nozzle.

scholarly journals On a stochastic Burgers equation with Dirichlet boundary conditions

2003 ◽  
Vol 2003 (43) ◽  
pp. 2735-2746 ◽  
Author(s):  
Ekaterina T. Kolkovska
Keyword(s):  
Weak Solution ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Burgers Equation ◽  
Martingale Problem ◽  
Weak Limit ◽  
Dirichlet Boundary Conditions ◽  
One Dimensional ◽  
Stochastic Burgers Equation ◽  
The One

We consider the one-dimensional Burgers equation perturbed by a white noise term with Dirichlet boundary conditions and a non-Lipschitz coefficient. We obtain existence of a weak solution proving tightness for a sequence of polygonal approximations for the equation and solving a martingale problem for the weak limit.

scholarly journals Identifying the heat sink

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
J. D. Audu ◽  
A. Boumenir ◽  
K. M. Furati ◽  
I. O. Sarumi
Keyword(s):  
Inverse Problem ◽  
Heat Equation ◽  
Temperature Profile ◽  
Spectral Methods ◽  
Heat Sink ◽  
Evolution Equations ◽  
Identification Problem ◽  
One Dimensional ◽  
Finite Dimensional ◽  
Pseudo Spectral

<p style='text-indent:20px;'>In this paper we examine the identification problem of the heat sink for a one dimensional heat equation through observations of the solution at the boundary or through a desired temperature profile to be attained at a certain given time. We make use of pseudo-spectral methods to recast the direct as well as the inverse problem in terms of linear systems in matrix form. The resulting evolution equations in finite dimensional spaces leads to fast real time algorithms which are crucial to applied control theory.</p>

scholarly journals Probing non-thermal density fluctuations in the one-dimensional Bose gas

2017 ◽  
Vol 3 (3) ◽  
Author(s):  
Jacopo De Nardis ◽  
Milosz Panfil ◽  
Andrea Gambassi ◽  
Leticia Cugliandolo ◽  
Robert Konik ◽  
...  
Keyword(s):  
Spatial Dimension ◽  
Bose Gas ◽  
Density Fluctuations ◽  
Dynamical Structure ◽  
Many Body ◽  
Initial State ◽  
One Dimensional ◽  
Fluctuation Dissipation ◽  
Quantum Integrable Models ◽  
The One

Quantum integrable models display a rich variety of non-thermal excited states with unusual properties. The most common way to probe them is by performing a quantum quench, i.e., by letting a many-body initial state unitarily evolve with an integrable Hamiltonian. At late times these systems are locally described by a generalized Gibbs ensemble with as many effective temperatures as their local conserved quantities. The experimental measurement of this macroscopic number of temperatures remains elusive. Here we show that they can be obtained for the Bose gas in one spatial dimension by probing the dynamical structure factor of the system after the quench and by employing a generalized fluctuation-dissipation theorem that we provide. Our procedure allows us to completely reconstruct the stationary state of a quantum integrable system from state-of-the-art experimental observations.

The general Lie group and similarity solutions for the one-dimensional Vlasov–Maxwell equations

1985 ◽  
Vol 33 (2) ◽  
pp. 219-236 ◽  
Author(s):  
Dana Roberts
Keyword(s):  
Lie Group ◽  
Maxwell Equations ◽  
Transformation Group ◽  
Spatial Dimension ◽  
Similarity Solutions ◽  
One Dimensional ◽  
Special Cases ◽  
General Similarity ◽  
The One ◽  
The Individual

The general Lie point transformation group and the associated reduced differential equations and similarity forms for the solutions are derived here for the coupled (nonlinear) Vlasov–Maxwell equations in one spatial dimension. The case of one species in a background is shown to admit a larger group than the multi-species case. Previous exact solutions are shown to be special cases of the above solutions, and many of the new solutions are found to constrain the form of the distribution function much more than, for example, the BGK solutions do. The individual generators of the Lie group are used to find the possible subgroups. Finally, a simple physical argument is given to show that the asymptotic solution (t→∞) for a one-species, one-dimensional plasma is one of the general similarity solutions.

scholarly journals Averaging principle for one dimensional stochastic Burgers equation

2018 ◽  
Vol 265 (10) ◽  
pp. 4749-4797 ◽  
Author(s):  
Zhao Dong ◽  
Xiaobin Sun ◽  
Hui Xiao ◽  
Jianliang Zhai
Keyword(s):  
Burgers Equation ◽  
Averaging Principle ◽  
One Dimensional ◽  
Stochastic Burgers Equation

The general structure of integrable evolution equations

1979 ◽  
Vol 365 (1722) ◽  
pp. 283-311 ◽  
Keyword(s):  
Evolution Equations ◽  
Entire Functions ◽  
General Structure ◽  
Spatial Dimension ◽  
Nonlinear Ordinary Differential Equation ◽  
Singular Perturbation Theory ◽  
Field Gradients ◽  
Scattering Transform ◽  
Integrable Evolution

This paper presents some new results in connection with the structure of integrable evolution equations. It is found that the most general integrable evolution equations in one spatial dimension which is solvable using the inverse scattering transform (i. s. t.) associated with the n th order eigenvalue problem V x = ( ξR 0 + P ( x , t )) V has the simple and elegant form G ( D R , t ) P t – F ( D R , t ) x [ R 0 , P ] = Ω ( D R , t ) [ C , P ], where G , F and Ω are entire functions of an integro-differential operatos D R and the bracket refers to the commutator. The list provided by this form is not exhaustive but contains most of the known integrable equations and many new ones of both mathematical and physical significance. The simple structure allows the identification in a straightforward manner of the equation in this class which is closest to a given equation of interest. The x dependent coefficients enable the inclusion of the effects of field gradients. Furthermore when the partial derivative with respect to t is zero, the remaining equation class contains many nonlinear ordinary differential equation of importance, such as the Painleve equations of the second and third kind. The properties of the scattering matrix A( ξ , t ) corresponding to the potential P( x , t ) are investigated and in particular the time evolution of A ( ξ , t ) is found to be G ( ξ , t ) A t + F ( ξ , t ) A ξ = Ω ( ξ , t )[ C , A ], The rôle of the diagonal entries and the principal corner minors in providing the Hamiltonian structure and constants of the motion is discussed. The central rôle that certain quadratic products of the eigenfunctions play in the theory is briefly described and the necessary groundwork from a singular perturbation theory is given when n = 2 or 3.

SEMICLASSICAL APPROACH FOR NONRELATIVISTIC FERMIONS IN LOW DIMENSIONS

1991 ◽  
Vol 06 (28) ◽  
pp. 5079-5097 ◽  
Author(s):  
D. KARABALI ◽  
B. SAKITA
Keyword(s):  
Field Theory ◽  
Hermitian Matrix ◽  
Spatial Dimension ◽  
Quantum Mechanical ◽  
Semiclassical Approach ◽  
Field Representation ◽  
Low Dimensions ◽  
One Dimensional ◽  
Field Formalism ◽  
Hermitian Matrix Model

We present a collective field formalism for nonrelativistic fermions in one spatial dimension. A bosonization technique is used to convert the quantum mechanical fermionic problem to a bosonic one, which is further described as a second quantized Schrödinger field theory. A formulation in terms of current and density variables gives rise to the collective field representation. Applications of our formalism to the D=1 Hermitian matrix model and the system of one-dimensional fermions in the presence of a weak electromagnetic field are discussed.

scholarly journals Gibbs phenomena for Lq-best approximation in finite element spaces

2021 ◽  
Author(s):  
Sarah Roggendorf ◽  
Paul Houston ◽  
Kristoffer van der Zee
Keyword(s):  
Finite Element ◽  
Sufficient Conditions ◽  
Gibbs Phenomenon ◽  
Two Dimensions ◽  
Thin Layers ◽  
Jump Discontinuities ◽  
Recent Developments ◽  
Minimum Residual ◽  
Gibbs Phenomena ◽  
Theoretical Results

Recent developments in the context of minimum residual finite element methods are paving the way for designing quasi-optimal discretisation methods in non-standard function spaces, such as L q -type Sobolev spaces. For q → 1, these methods have demonstrated huge potential in avoiding the notorious Gibbs phenomena, i.e., the occurrence of spurious non-physical oscillations near thin layers and jump discontinuities. In this work we provide theoretical results that explain some of the numerical observations. In particular, we investigate the Gibbs phenomena for L q -best approximations of discontinuities in finite element spaces with 1 ≤ q < ∞. We prove sufficient conditions on meshes in one and two dimensions such that over- and undershoots vanish in the limit q → 1. Moreover, we include examples of meshes such that Gibbs phenomena remain present even for q = 1 and demonstrate that our results can be used to design meshes so as to eliminate the Gibbs phenomenon.

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