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Numerical Reconstruction of Coefficients in Elliptic Equations Using Continuous Data Assimilation

Peiran Zhang

TL;DR

The paper tackles reconstructing the spatially varying conductivity $q(x)$ and source $f(x)$ in a 2D elliptic PDE from interior observations by embedding the problem into a continuous data assimilation framework with a feedback term. It derives approximated gradients for updating $(q,f)$ that avoid adjoint computations and proves local Lipschitz properties of the coefficient-to-solution maps, enabling stable gradient-based updates. The authors establish new $L^2$ error estimates (sublinear for $q$ and linear for $f$ in discretization) and provide explicit gradient formulas, supported by numerical experiments on a unit square that confirm theory and show robustness to coefficient perturbations and data noise. The results offer a practical, tunable approach for coefficient identification in elliptic problems and set the stage for parabolic extensions and simultaneous reconstruction tasks.

Abstract

We consider the numerical reconstruction of the spatially dependent conductivity coefficient and the source term in elliptic partial differential equations in a two-dimensional convex polygonal domain, with the homogeneous Dirichlet boundary condition and given interior observations of the solution. Using data assimilation, we derive approximated gradients of the error functional to update the reconstructed coefficients. New $L^2$ error estimates are provided for the spatially discretized reconstructions. Numerical examples are given to illustrate the effectiveness of the method and demonstrate the error estimates. The numerical results also show that the reconstruction is very robust to the errors in specific inputted coefficients.

Numerical Reconstruction of Coefficients in Elliptic Equations Using Continuous Data Assimilation

TL;DR

The paper tackles reconstructing the spatially varying conductivity and source in a 2D elliptic PDE from interior observations by embedding the problem into a continuous data assimilation framework with a feedback term. It derives approximated gradients for updating that avoid adjoint computations and proves local Lipschitz properties of the coefficient-to-solution maps, enabling stable gradient-based updates. The authors establish new error estimates (sublinear for and linear for in discretization) and provide explicit gradient formulas, supported by numerical experiments on a unit square that confirm theory and show robustness to coefficient perturbations and data noise. The results offer a practical, tunable approach for coefficient identification in elliptic problems and set the stage for parabolic extensions and simultaneous reconstruction tasks.

Abstract

We consider the numerical reconstruction of the spatially dependent conductivity coefficient and the source term in elliptic partial differential equations in a two-dimensional convex polygonal domain, with the homogeneous Dirichlet boundary condition and given interior observations of the solution. Using data assimilation, we derive approximated gradients of the error functional to update the reconstructed coefficients. New error estimates are provided for the spatially discretized reconstructions. Numerical examples are given to illustrate the effectiveness of the method and demonstrate the error estimates. The numerical results also show that the reconstruction is very robust to the errors in specific inputted coefficients.

Paper Structure

This paper contains 15 sections, 15 theorems, 105 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Estimates for assimilation error and the some properties of the coefficient-to-solution maps
  3. Estimates for assimilation error
  4. Lipschitz continuity of the coefficient-to-solution maps
  5. Approximated gradients for coefficient reconstruction and the error estimates
  6. The approximated gradients for the error functional
  7. An approximated gradient formula for $q$
  8. An approximated gradient formula for $f$
  9. Application to the elliptic case
  10. Error estimates for reconstruction of $q$
  11. Error estimates for reconstruction of $f$
  12. Numerical results
  13. Example 1
  14. Example 2
  15. Conclusions

Key Result

Lemma 2.1

Denote the error by $w = u - v$. For $t>0$, if $\mu$ is taken large enough and $h$ is taken small enough such that $(\frac{3}{4}-2C_P^{-2}C_A^{2}h^2)\mu \geq 2C_P^{-2}(b_1 C_P -q_0)-2c_0$, we have

Figures (9)

  • Figure 1: True data for Example 1.
  • Figure 2: Firs row: the true $q$ and reconstructed $q$ with exact $b$ and $c$ or with $\tilde{b}= (3.5,3.5)^\intercal$ and $\tilde{c} = 3$; second row: the true $f$ and reconstructed $f$ with exact $b$ and $c$ or with $\tilde{b}= (3.5,3.5)^\intercal$ and $\tilde{c} = 3$. $\mu=1{,}000$ in all reconstructions here. The wireframes in the plots of reconstructions reflect the mesh sizes $h_q = h_f = 1/8$, while those in the plots of true coefficients are for the clarity of their profiles.
  • Figure 3: Relative $L^2$ errors for $q$ and $f$ with noisy data produced by (\ref{['eq:obs-error']}), Example 1.
  • Figure 4: Relative reconstruction errors for $q$ and $f$ with noisy data produced by (\ref{['eq:obs-error-2']}), Example 1.
  • Figure 5: True coefficients and the reconstructed coefficients using $h_q = h_f = 1/16$, Example 1.
  • ...and 4 more figures

Theorems & Definitions (25)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.1
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 15 more