Circulant embedding with QMC: analysis for elliptic PDE with lognormal coefficients

Area of Science:

• Computational mathematics
• Numerical analysis
• Scientific computing

Background:

• A previous study introduced a practical method for computing expected values of functionals for elliptic partial differential equations with random coefficients.
• This method combined quasi-Monte Carlo (QMC) and circulant embedding for sampling random fields, proving effective for high-dimensional stochastic problems like fluid flow in heterogeneous media.
• However, the prior work lacked a rigorous convergence theory.

Purpose of the Study:

• To provide a convergence analysis for the previously proposed computational method.
• To investigate the method's performance when using a specific type of QMC: randomly shifted lattice rules.
• To establish convergence rates that may be independent of the number of stochastic variables.

Main Methods:

• The study employs a convergence analysis framework for the combined QMC and circulant embedding method.
• The analysis focuses on randomly shifted lattice rules as the quasi-Monte Carlo component.
• It incorporates error analysis for the finite element method, accommodating locally refined meshes.

Main Results:

• A convergence result is established for the method when using randomly shifted lattice rules.
• The convergence rate is shown to depend on the eigenvalues of the underlying nested block circulant matrix.
• Under specific assumptions, the convergence can be independent of the number of stochastic variables, and the QMC analysis extends to general covariance matrix factorizations.

Conclusions:

• The paper successfully provides a theoretical convergence analysis for the computational method.
• The findings support the method's efficiency and applicability to complex problems, including those with high stochastic dimensions.
• Numerical validation on 2D and 3D domains, including those with singularities, demonstrates the method's practical utility.