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On the Non-Linear Integral Equation Approach for an Inverse Boundary Value Problem for the Heat Equation

Roman Chapko, Leonidas Mindrinos

Abstract

We consider the inverse problem of reconstructing the interior boundary curve of a doubly connected domain from the knowledge of the temperature and the thermal flux on the exterior boundary curve. The use of the Laguerre transform in time leads to a sequence of stationary inverse problems. Then, the application of the modified single-layer ansatz, reduces the problem to a sequence of systems of non-linear boundary integral equations. An iterative algorithm is developed for the numerical solution of the obtained integral equations. We find the Fréchet derivative of the corresponding integral operator and we show the unique solvability of the linearized equation. Full discretization is realized by a trigonometric quadrature method. Due to the inherited ill-possedness of the derived system of linear equations we apply the Tikhonov regularization. The numerical results show that the proposed method produces accurate and stable reconstructions.

On the Non-Linear Integral Equation Approach for an Inverse Boundary Value Problem for the Heat Equation

Abstract

We consider the inverse problem of reconstructing the interior boundary curve of a doubly connected domain from the knowledge of the temperature and the thermal flux on the exterior boundary curve. The use of the Laguerre transform in time leads to a sequence of stationary inverse problems. Then, the application of the modified single-layer ansatz, reduces the problem to a sequence of systems of non-linear boundary integral equations. An iterative algorithm is developed for the numerical solution of the obtained integral equations. We find the Fréchet derivative of the corresponding integral operator and we show the unique solvability of the linearized equation. Full discretization is realized by a trigonometric quadrature method. Due to the inherited ill-possedness of the derived system of linear equations we apply the Tikhonov regularization. The numerical results show that the proposed method produces accurate and stable reconstructions.

Paper Structure

This paper contains 7 sections, 1 theorem, 67 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Laguerre transform and Boundary Integral Equations
  3. Numerical Implementation
  4. Numerical solution of the "field" system
  5. Numerical solution of the "data" equation
  6. Numerical Results
  7. Conclusion

Key Result

Theorem 3.1

The Fréchet derivative operator $\mathcal{D}_n [\varphi , r;q]$ is injective at the exact solution.

Figures (3)

  • Figure 1: Reconstructions of the peanut-shaped boundary $\Gamma_1$ for exact data (left) and data with $3\%$ noise (right).
  • Figure 2: Reconstructions of the apple-shaped boundary $\Gamma_1$ for exact data (left) and noisy data (right).
  • Figure 3: Reconstructions of the kite-shaped boundary $\Gamma_1$ for exact data (left) and data with $3\%$ noise (right).

Theorems & Definitions (1)

  • Theorem 3.1