Hyperbolic stochastic Galerkin formulation for the p-system

Gerster, Stephan; Herty, Michaël; Sikstel, Aleksey

doi:10.1016/j.jcp.2019.05.049

Cited by 28 publications

(39 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The approximation ( 6) is a widely used strategy for computing PCE expansion of products (e.g. [9,14]). Such an operation can be cast in linear algebraic terms by considering vectors comprised of the PCE expansion coefficients.…”

Section: Polynomial Chaos Expansion -A Reviewmentioning

confidence: 99%

“…However, when the SG method is applied to a general hyperbolic system of conservation/balance laws such as the SWE, the associated SG system may lose hyperbolicity [10,18,14]. The loss of hyperbolicity may lead to unphysical solutions and prevent the use of robust numerical schemes.…”

Section: Introductionmentioning

confidence: 99%

“…But the IPMM method can be computationally expensive since one must solve an optimization problem for each cell at each discrete time level. The PCE using a Roe variable formulation is introduced in [34,14,13], where the flux of the SG system is constructed using PCE of the Roe variables. Thus, the conservative form of the system is preserved.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Hyperbolicity-Preserving and Well-Balanced Stochastic Galerkin Method for Shallow Water Equations

Dai¹,

Epshteyn²,

Narayan³

2021

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

Stochastic Galerkin formulations of the two-dimensional shallow water systems parameterized with random variables may lose hyperbolicity, and hence change the nature of the original model. In this work, we present a hyperbolicity-preserving stochastic Galerkin formulation by carefully selecting the polynomial chaos approximations to the nonlinear terms of (q x ) 2 /h, (q y ) 2 /h, and (q x q y )/h in the shallow water equations. We derive a sufficient condition to preserve the hyperbolicity of the stochastic Galerkin system which requires only a finite collection of positivity conditions on the stochastic water height at selected quadrature points in parameter space. Based on our theoretical results for the stochastic Galerkin formulation, we develop a corresponding well-balanced hyperbolicitypreserving central-upwind scheme. We demonstrate the accuracy and the robustness of the new scheme on several challenging numerical tests.

show abstract

Section: Polynomial Chaos Expansion -A Reviewmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Hyperbolicity-Preserving and Well-Balanced Stochastic Galerkin Method for Shallow Water Equations

Dai¹,

Epshteyn²,

Narayan³

2021

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

show abstract

“…Expansions of the stochastic input are substituted into the governing equations and they are projected to obtain deterministic evolution equations for the gPC coefficients. Applications of this procedure are challenging for general PDEs, since desired properties like hyperbolicity may be lost [59,19,21,22,47,58]. The resulting systems for linear hyperbolic [28,61] and kinetic equations [10,11,36,41,71,88,89] remain well-posed.…”

Section: 3mentioning

confidence: 99%

Spectral methods to study the robustness of residual neural networks with infinite layers

Trimborn¹,

Gerster²,

Visconti³

2019

Foundations of Data Science

Self Cite

View full text Add to dashboard Cite

Recently, neural networks (NN) with an infinite number of layers have been introduced. Especially for these very large NN the training procedure is very expensive. Hence, there is interest to study their robustness with respect to input data to avoid unnecessarily retraining the network. Typically, model-based statistical inference methods, e.g. Bayesian neural networks, are used to quantify uncertainties. Here, we consider a special class of residual neural networks and we study the case, when the number of layers can be arbitrarily large. Then, kinetic theory allows to interpret the network as a dynamical system, described by a partial differential equation. We study the robustness of the mean-field neural network with respect to perturbations in initial data by applying UQ approaches on the loss functions.

show abstract

“…the gradient of the deterministic entropy [11,12]. Roe variables, which include the square root of the density, preserve hyperbolicity for Euler equations [15,29,30]. The drawback of introducing auxiliary variables is an additional computational overhead that arises from an optimization problem, which is required to calculate the auxiliary variables.…”

Section: Introductionmentioning

confidence: 99%

Description of random level sets by polynomial chaos expansions

Bambach�¹,

Gerster²,

Herty³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

We present a novel approach to determine the evolution of level sets under uncertainties in the velocity fields. This leads to a stochastic description of the level sets. To compute the quantiles of random level sets, we use the stochastic Galerkin method for a hyperbolic reformulation of the level-set equations. A novel intrusive Galerkin formulation is presented and proven hyperbolic. It induces a corresponding finite-volume scheme that is specifically taylored for uncertain velocities.

show abstract

Hyperbolic stochastic Galerkin formulation for the p-system

Cited by 28 publications

References 34 publications

Hyperbolicity-Preserving and Well-Balanced Stochastic Galerkin Method for Shallow Water Equations

Hyperbolicity-Preserving and Well-Balanced Stochastic Galerkin Method for Shallow Water Equations

Spectral methods to study the robustness of residual neural networks with infinite layers

Description of random level sets by polynomial chaos expansions

Contact Info

Product

Resources

About