[Paper Review] Elliptic PDEs on log-Gaussian Shapes: Sparsity and Finite Element Discretization

Dinh Phuong Dung, Helmut Harbrecht+2 more|arXiv (Cornell University)|Mar 25, 2026

Probabilistic and Robust Engineering Design0 citations

TL;DR

The paper studies elliptic diffusion on random domains generated by log-Gaussian radial deformations, proves existence, regularity, and develops FEM and sparse grid/QMC discretizations in the parametric setting.

ABSTRACT

In this article, we consider the solution to elliptic diffusion problems on a class of random domains obtained by log-Gaussian random homothety of the unit disk respectively an annulus. We model the problem under consideration and verify the existence and uniqueness of the random solution by path-wise pullback to the nominal unit disk respectively annulus. We prove the analytic regularity of the solution with respect to the random input parameter. We consider the numerical approximation of the random diffusion problem by means of continuous, piecewise linear Lagrangian Galerkin Finite Elements with numerical quadrature in the nominal domain, and by sparse grid interpolation and quadrature of Gauss-Hermite Smolyak and Quasi-Monte Carlo type in the parameter domain. The theoretical findings are complemented by numerical results.

Motivation & Objective

• Model random domains via log-Gaussian, star-shaped deformations of the unit disk/annulus.
• Prove existence and uniqueness of the solution by pulling back to a fixed reference domain.
• Establish analytic regularity and parametric holomorphy of the solution with respect to random inputs.
• Develop and analyze numerical discretizations: continuous FEM in space and sparse grids/Quasi-Monte Carlo in the parameter domain.
• Provide numerical experiments to validate the theoretical findings.

Proposed method

• Model the domain as D_kappa(a) with a(theta) = exp(sum_k y_k psi_k(theta)).
• Pull back the PDE to the reference domain D_ref,kappa via F(a) and derive the variable-coefficient PDE with diffusion matrix M(a).
• Show coercivity and boundedness of B(v,v;a) and derive bounds on the inverse of the minimum eigenvalue lambda_min(a).
• Prove holomorphy of the parametric solution u_hat(a) with respect to the complexified parameters and derive derivative bounds.
• Establish derivative estimates for partial derivatives of the solution with respect to the parameter vector y, under summability conditions on rho.
• Implement numerical schemes: continuous Lagrangian FEM in the physical domain mapped to the reference domain, sparse grid interpolation, and Halton-based quasi-Monte Carlo quadrature.

Experimental results

Research questions

• RQ1 How does a log-Gaussian, Fourier-series-based domain deformation influence the well-posedness of the elliptic diffusion problem?
• RQ2 What regularity (holomorphy and derivative bounds) can be established for the pullback solution with respect to the random parameter vector?
• RQ3 How effective are sparse grid and Quasi-Monte Carlo methods for approximating statistics (e.g., expectations) of the random solution?
• RQ4 How can finite element discretization be applied after domain mapping, and what are the resulting error/complexity properties?
• RQ5 What are practical conditions on the domain deformation that ensure coercivity and stable numerical approximation?

Key findings

• Existence and uniqueness of the pullback solution on the reference domain are established under a set of domain regularity assumptions.
• The pullback solution is holomorphic in the parameter region, with explicit bounds on derivatives with respect to the random variables.
• Coercivity and stability bounds are derived for the parametric problem, leading to norm estimates for the solution in terms of the data.
• Explicit derivative estimates for the parametric solution are obtained, showing controlled growth governed by summability weights and analytic constants.
• The framework supports continuous, piecewise linear FEM in space and sparse-grid/Quasi-Monte Carlo techniques for the parameter integration, with numerical experiments corroborating the theory.