[Paper Review] Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems

Jérémi Dardé|arXiv (Cornell University)|Mar 30, 2015

Numerical methods in inverse problems42 references29 citations

TL;DR

This paper introduces an iterated quasi-reversibility method to stabilize ill-posed elliptic and parabolic data completion problems, such as Poisson's and heat equations with incomplete boundary data. By iteratively solving regularized variational problems with a fixed, well-conditioned parameter ε, the method achieves convergence to the true solution even with highly noisy data, using a Morozov discrepancy principle to stop iterations, and demonstrates high robustness in numerical tests on corrosion detection and 1D heat equations.

ABSTRACT

We study the iterated quasi-reversibility method to regularize ill-posed elliptic and parabolic problems: data completion problems for Poisson's and heat equations. We define an abstract setting to treat both equations at once. We demonstrate the convergence of the regularized solution to the exact one, and propose a strategy to deal with noise on the data. We present numerical experiments for both problems: a two-dimensional corrosion detection problem and the one-dimensional heat equation with lateral data. In both cases, the method prove to be efficient even with highly corrupted data.

Motivation & Objective

• To address the severe ill-posedness of data completion problems for Poisson’s and heat equations, where missing boundary data lead to unstable or non-existent solutions.
• To develop a regularization strategy that remains effective even with high levels of noise in measured data, particularly in inverse problems like corrosion detection.
• To establish a unified abstract framework applicable to both elliptic and parabolic inverse problems, enabling a common theoretical and numerical treatment.
• To demonstrate that using a fixed, large regularization parameter ε in the iterated method improves matrix conditioning without sacrificing reconstruction accuracy, unlike the standard quasi-reversibility method.

Proposed method

• Formulates the data completion problem abstractly as Ax = y with A a continuous, injective, non-surjective operator between Hilbert spaces, enabling unified treatment of elliptic and parabolic cases.
• Applies the standard quasi-reversibility method by introducing a higher-order, well-posed variational problem with a small parameter ε, regularizing the ill-posed problem.
• Introduces an iterative extension: at each step, the solution from the previous iteration is used as a source term in the next regularized problem, forming a sequence converging to the exact solution under exact data.
• Employs a Morozov discrepancy principle to stop iterations in the presence of noisy data, ensuring both stability and convergence by matching the residual to the noise level.
• Uses conforming finite elements (P2 for u, RT1 for p) and a tensor-product P1⊗P1 discretization for space-time problems, with precomputed matrix factorizations to maintain efficiency.
• Applies the method to two test cases: a 2D corrosion detection problem with Robin transmission conditions and a 1D heat equation with lateral data, using noisy Dirichlet and Neumann data.

Experimental results

Research questions

• RQ1 Can the iterated quasi-reversibility method achieve convergence to the true solution of an ill-posed data completion problem when exact data are provided, even with a large, fixed regularization parameter ε?
• RQ2 How does the iterated method perform numerically when the input data are corrupted by noise, and can it maintain accuracy despite high noise levels?
• RQ3 Can the Morozov discrepancy principle be effectively used to determine the optimal stopping iteration in the presence of noisy data, ensuring both stability and convergence?
• RQ4 Does the method remain robust and efficient when applied to both elliptic (Poisson) and parabolic (heat) equations under the same abstract framework?
• RQ5 Can the method reconstruct not only the solution but also derived quantities such as the Robin coefficient on inaccessible boundaries, as in corrosion detection problems?

Key findings

• The iterated quasi-reversibility method converges to the exact solution for both elliptic and parabolic data completion problems under exact data, even when using a fixed, large regularization parameter ε = 1, which improves the conditioning of the finite element system.
• For noisy data, the method achieves stable and accurate reconstructions, with acceptable error levels even at 5% relative noise in the Dirichlet data, as demonstrated in both the 2D corrosion detection and 1D heat equation problems.
• The residual quantity, measuring the violation of the PDE and boundary conditions, decreases monotonically during iterations and stabilizes at a level consistent with the noise amplitude, confirming the effectiveness of the stopping criterion.
• The reconstruction of the Robin coefficient on the inaccessible boundary Γc is accurate even under high noise, with the method successfully distinguishing between corroded (μ > 0) and healthy (μ = 0) regions.
• The relative error in the solution u over the domain remains below 10% for α = 1% and 2%, and below 15% for α = 5% in the 2D corrosion problem, indicating strong robustness.
• In the 1D heat equation case, the method achieves relative errors below 5% for α = 2% and below 10% for α = 5% in the solution reconstruction, confirming its effectiveness for parabolic problems with noisy lateral data.