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Global Convergence and Uniqueness for an Inverse Problem Posed by Gelfand

Michael V. Klibanov, Jingzhi Li, Tian Niu, Vladimir G. Romanov

Abstract

The first globally convergent numerical method is developed for a coefficient inverse problem (CIP) for the $n-$d, $n\geq 2$ wave equation with the unknown potential in the most challenging case when the $δ-$ function is present in the initial condition with a single location of the point source. In fact, an approximate mathematical model for that CIP is derived. That globally convergent numerical method is developed for this model. This is a new version of the so-called convexification numerical method. Uniqueness theorem is proven as well within the framework of that approximate mathematical model. The question about uniqueness of this CIP was first posed by a famous mathematician I. M. Gelfand in 1954 as an $n-$d ($n=2,3$) extension of the fundamental theorem of V.A. Marchenko in the 1-d case (1950). Based on a Carleman estimate, convergence analysis is carried out. This analysis ensures the global convergence of the proposed numerical method, i.e. it is not necessary to have a good first guess for the solution. Exhaustive computational experiments with noisy data demonstrate a high reconstruction accuracy of complicated structures. In particular, this accuracy points towards a high adequacy of that approximate mathematical model.

Global Convergence and Uniqueness for an Inverse Problem Posed by Gelfand

Abstract

The first globally convergent numerical method is developed for a coefficient inverse problem (CIP) for the d, wave equation with the unknown potential in the most challenging case when the function is present in the initial condition with a single location of the point source. In fact, an approximate mathematical model for that CIP is derived. That globally convergent numerical method is developed for this model. This is a new version of the so-called convexification numerical method. Uniqueness theorem is proven as well within the framework of that approximate mathematical model. The question about uniqueness of this CIP was first posed by a famous mathematician I. M. Gelfand in 1954 as an d () extension of the fundamental theorem of V.A. Marchenko in the 1-d case (1950). Based on a Carleman estimate, convergence analysis is carried out. This analysis ensures the global convergence of the proposed numerical method, i.e. it is not necessary to have a good first guess for the solution. Exhaustive computational experiments with noisy data demonstrate a high reconstruction accuracy of complicated structures. In particular, this accuracy points towards a high adequacy of that approximate mathematical model.

Paper Structure

This paper contains 18 sections, 225 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Forward and Inverse Problems
  3. Statements of forward and inverse problems
  4. Asymptotic behavior of the function $u\left( \mathbf{x},t\right)$ at $t\rightarrow 0^{+}$
  5. Transformation Procedure
  6. The first step of the transformation procedure
  7. The second step of the transformation procedure
  8. Convexification Functional for the Numerical Solution of Problem ( \ref{['3.25']})-(\ref{['3.35']})
  9. Convergence Analysis
  10. The strong convexity of the functional $J_{\lambda , \alpha }$ on the set $\overline{B\left( R\right) }$
  11. Accuracy estimates in the case of noisy data
  12. Accuracy estimates in the case of noiseless data
  13. Global convergence of the gradient descent method
  14. Partially Addressing the Conjecture of Gelfand
  15. Numerical Studies
  16. ...and 3 more sections

Figures (8)

  • Figure 1: A typical convergence behavior of $\left\Vert \left\vert \nabla J_{\lambda ,\alpha }^{disc}\left( V_{m}\right) \right\vert \right\Vert _{L_{2}^{disc}\left( \Omega \right) }$ with respect to the iteration number n of iterations of the L-BFGS algorithm. Note that the value of the norm $\left\Vert \left\vert \nabla J_{\lambda ,\alpha }^{disc}\left( V_{m}\right) \right\vert \right\Vert _{L_{2}^{disc}\left( \Omega \right) }$ decreases by the factor of 100 due to the global convergence. This figure explains our stopping criterion ( \ref{['9.100']}).
  • Figure 2: Reconstruction results for different numbers of time steps $N_{t}=T/h=4/h,$ where $T=4$ as in (\ref{['9.90']}), and $h$ is as in (\ref{['3.16']}). Clearly, $N_{t}=20$ is the optimal number.
  • Figure 3: Reconstruction results illustrating the choice of an optimal value of the parameter $\varepsilon$ in (\ref{['3.2']}). Obviously, $\varepsilon =0.01$ is the best one out of five values $\{0.001,0.01,0.03,0.05,0.1\}$. Hence, we assign the optimal value of this parameter $\varepsilon =0.01$ in all numerical tests below.
  • Figure 4: Reconstruction results illustrating the choice of an optimal value of the parameter $\lambda$ in Carleman Weight Function $\varphi _{\lambda }\left( x_{1}\right)$ in (\ref{['4.4']}). Obviously, $\lambda =3$ is the best one out of five values $\{1,2,3,4,5\}$. Hence, we assign the optimal value of the parameter $\lambda =3$ in all numerical tests below.
  • Figure 5: Reconstruction results illustrating the sensitivity to the noise level. Even for the highest noise levels of 5% shapes of both inclusions are recognizable and the maximal value $\max a\left( \mathbf{x}\right) =2$ inside these letters is reconstructed accurately.
  • ...and 3 more figures