Multi-level Monte Carlo weak Galerkin method for elliptic equations with stochastic jump coefficients

Li, Jingshi; Wang, Xiaoshen; Zhang, Kai

doi:10.1016/j.amc.2015.11.064

Cited by 20 publications

(18 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The concept of multilevel Monte Carlo simulation has been developed in [29] to calculate parametric integrals and has been rediscovered in [26] to estimate the expected value of functionals of stochastic differential equations. Ever since, multilevel Monte Carlo techniques have been successfully applied to various problems, for instance in the context of elliptic random PDEs in [1,12,19,31,38] to just name a few. These sampling algorithms are fundamentally different from approaches using Polynomial Chaos.…”

mentioning

confidence: 99%

“…As we show in the numerical examples, this jump-diffusion coefficient can be used to model a wide array of scenarios. This generalizes the work in [31] and uses partly [30]. To approximate the expectation of the solution we develop and test, further, variants of the multilevel Monte Carlo method which are tailored to our problem: namely adaptive and bootstrapping multilevel Monte Carlo methods.…”

mentioning

confidence: 99%

See 1 more Smart Citation

A Study of Elliptic Partial Differential Equations with Jump Diffusion Coefficients

Barth¹,

Stein²

2018

SIAM/ASA J. Uncertainty Quantification

View full text Add to dashboard Cite

As a simplified model for subsurface flows elliptic equations may be utilized. Insufficient measurements or uncertainty in those are commonly modeled by a random coefficient, which then accounts for the uncertain permeability of a given medium. As an extension of this methodology to flows in heterogeneous\fractured\porous media, we incorporate jumps in the diffusion coefficient. These discontinuities then represent transitions in the media. More precisely, we consider a second order elliptic problem where the random coefficient is given by the sum of a (continuous) Gaussian random field and a (discontinuous) jump part. To estimate moments of the solution to the resulting random partial differential equation, we use a pathwise numerical approximation combined with multilevel Monte Carlo sampling. In order to account for the discontinuities and improve the convergence of the pathwise approximation, the spatial domain is decomposed with respect to the jump positions in each sample, leading to path-dependent grids. Hence, it is not possible to create a sequence of grids which is suitable for each sample path a-priori. We address this issue by an adaptive multilevel algorithm, where the discretization on each level is sample-dependent and fulfills given refinement conditions.

show abstract

mentioning

confidence: 99%

mentioning

confidence: 99%

A Study of Elliptic Partial Differential Equations with Jump Diffusion Coefficients

Barth¹,

Stein²

2018

SIAM/ASA J. Uncertainty Quantification

View full text Add to dashboard Cite

show abstract

“…Moreover, WGMFEM is developed in second-order elliptic equations with Robin boundary conditions [32], parabolic differential equations with memory [33], Helmholtz equations with large wave numbers [34], and biharmonic equations [35]. In addition to solving deterministic problems, there has also been some progress in the WG method for stochastic problems, such as stochastic Brinkman problems [36], elliptic problems with stochastic jump coefficients [37], and random interface grating problems [38].…”

Section: Introductionmentioning

confidence: 99%

Mixed Weak Galerkin Method for Heat Equation with Random Initial Condition

Zhou

Zou

Chai

et al. 2020

Mathematical Problems in Engineering

View full text Add to dashboard Cite

This paper is devoted to the numerical analysis of weak Galerkin mixed finite element method (WGMFEM) for solving a heat equation with random initial condition. To set up the finite element spaces, we choose piecewise continuous polynomial functions of degree j+1 with j≥0 for the primary variables and piecewise discontinuous vector-valued polynomial functions of degree j for the flux ones. We further establish the stability analysis of both semidiscrete and fully discrete WGMFE schemes. In addition, we prove the optimal order convergence estimates in L2 norm for scalar solutions and triple-bar norm for vector solutions and statistical variance-type convergence estimates. Ultimately, we provide a few numerical experiments to illustrate the efficiency of the proposed schemes and theoretical analysis.

show abstract

“…This makes the computations of the WGFEM and that of the conforming finite element are comparable. The WGFEM has been developed for various PDEs and we refer to Stokes' equations [22,31,28], the Brinkman equations [20,15,32], stochastic jump coefficients problem [18], Maxwell's equations [19], the Navier-Stokes Equations [17,37], and references therein.…”

Section: Introductionmentioning

confidence: 99%

An adaptive weak Galerkin finite element method with hierarchical bases for the elliptic problem

Zhang

et al. 2020

Numerical Methods Partial

Self Cite

View full text Add to dashboard Cite

Based on a posteriori error estimator with hierarchical bases, an adaptive weak Galerkin finite element method (WGFEM) is proposed for the elliptic problem with mixed boundary conditions. For the posteriori error estimator, we are only required to solve a linear algebraic system with diagonal entries corresponding to the degree of freedoms, which significantly reduces the computational cost. The upper and lower bounds of the error estimator are shown to addresses the reliability and efficiency of the adaptive approach. Numerical simulations are provided to demonstrate the effectiveness and robustness of the proposed method.

show abstract

Multi-level Monte Carlo weak Galerkin method for elliptic equations with stochastic jump coefficients

Cited by 20 publications

References 22 publications

A Study of Elliptic Partial Differential Equations with Jump Diffusion Coefficients

A Study of Elliptic Partial Differential Equations with Jump Diffusion Coefficients

Mixed Weak Galerkin Method for Heat Equation with Random Initial Condition

An adaptive weak Galerkin finite element method with hierarchical bases for the elliptic problem

Contact Info

Product

Resources

About