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Convexification for a Coefficient Inverse Problem for a System of Two Coupled Nonlinear Parabolic Equations

Michael V. Klibanov, Jingzhi Li, Zhipeng Yang

Abstract

A system of two coupled nonlinear parabolic partial differential equations with two opposite directions of time is considered. In fact, this is the so-called "Mean Field Games System" (MFGS), which is derived in the mean field games (MFG) theory. This theory has numerous applications in social sciences. The topic of Coefficient Inverse Problems (CIPs) in the MFG theory is in its infant age, both in theory and computations. A numerical method for this CIP is developed. Convergence analysis ensures the global convergence of this method. Numerical experiments are presented.

Convexification for a Coefficient Inverse Problem for a System of Two Coupled Nonlinear Parabolic Equations

Abstract

A system of two coupled nonlinear parabolic partial differential equations with two opposite directions of time is considered. In fact, this is the so-called "Mean Field Games System" (MFGS), which is derived in the mean field games (MFG) theory. This theory has numerous applications in social sciences. The topic of Coefficient Inverse Problems (CIPs) in the MFG theory is in its infant age, both in theory and computations. A numerical method for this CIP is developed. Convergence analysis ensures the global convergence of this method. Numerical experiments are presented.
Paper Structure (19 sections, 100 equations, 6 figures)

This paper contains 19 sections, 100 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Statement of the Coefficient Inverse Problem
  3. Statement of the CIP
  4. Discussion
  5. The form of the kernel $K\left( x,y\right)$
  6. The Transformation Procedure
  7. Convexification
  8. The Carleman Weight Function (CWF) and some estimates
  9. The functional
  10. Convergence Analysis
  11. The central theorem
  12. Accuracy estimate and global convergence of the gradient descent method
  13. Accuracy estimate
  14. The gradient descent method
  15. Proof of Theorem 5.1
  16. ...and 4 more sections

Figures (6)

  • Figure 1: Test 1. The reconstructed coefficient $k(x)$, where the function $k(x)$ is given in \ref{['8.203']} with $c_{a}=2$ inside of the letter 'A'. We test different values of $\lambda$.
  • Figure 2: Test 1. The convergence of $|\nabla J_{\lambda ,\beta }\left( v, w, p, q \right) |$ in the iteration of fmincon with $\lambda=3$ and the function $k(x)$ is given in \ref{['8.203']} with $c_{a}=2$ inside of the letter 'A'.
  • Figure 3: Test 2. Exact (left) and reconstructed (right) coefficient $k(x)$ with $c_{a}=4$ (first row) and $c_{a}=8$ (second row) inside of the letter 'A' as in (\ref{['8.203']}). The inclusion/background contrasts in (\ref{['8.204']}) are respectively $4:1$ and $8:1$. The reconstructions of both shapes of inclusions and the inclusion/background contrasts ( \ref{['8.2040']}) are accurate.
  • Figure 4: Test 3. Exact (left) and reconstructed (right) coefficient $k(x)$, where the function $k(x)$ is given in \ref{['8.203']} with $c_{a}=2$ inside of the letter '$\Omega$'. The reconstructions of both the shape of the inclusions and the inclusion/background contrast (\ref{['8.2040']}) are accurate.
  • Figure 5: Test 4. Exact (left) and reconstructed (right) coefficient $k(x)$, where the function $k(x)$ is given in \ref{['8.203']} with $c_{a}=2$ inside of two letters 'SZ'. The reconstruction is worse than the one for the case of the single letter '$\Omega$' in Figure \ref{['plot_re_Omega']}. Nevertheless, the reconstructions of shapes of both letters are still accurate. In addition, the computed inclusion/background contrasts in \ref{['8.2040']} are accurately reconstructed in both letters.
  • ...and 1 more figures