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A novel model reduction method to solve inverse problems of parabolic type

Wenlong Zhang, Zhiwen Zhang

Abstract

In this paper, we propose novel proper orthogonal decomposition (POD)--based model reduction methods that effectively address the issue of inverse crime in solving parabolic inverse problems. Both the inverse initial value problems and inverse source problems are studied. By leveraging the inherent low-dimensional structures present in the data, our approach enables a reduction in the forward model complexity without compromising the accuracy of the inverse problem solution. Besides, we prove the convergence analysis of the proposed methods for solving parabolic inverse problems. Through extensive experimentation and comparative analysis, we demonstrate the effectiveness of our method in overcoming inverse crime and achieving improved inverse problem solutions. The proposed POD model reduction method offers a promising direction for improving the reliability and applicability of inverse problem-solving techniques in various domains.

A novel model reduction method to solve inverse problems of parabolic type

Abstract

In this paper, we propose novel proper orthogonal decomposition (POD)--based model reduction methods that effectively address the issue of inverse crime in solving parabolic inverse problems. Both the inverse initial value problems and inverse source problems are studied. By leveraging the inherent low-dimensional structures present in the data, our approach enables a reduction in the forward model complexity without compromising the accuracy of the inverse problem solution. Besides, we prove the convergence analysis of the proposed methods for solving parabolic inverse problems. Through extensive experimentation and comparative analysis, we demonstrate the effectiveness of our method in overcoming inverse crime and achieving improved inverse problem solutions. The proposed POD model reduction method offers a promising direction for improving the reliability and applicability of inverse problem-solving techniques in various domains.
Paper Structure (13 sections, 10 theorems, 77 equations, 9 figures)

This paper contains 13 sections, 10 theorems, 77 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Ajoint-POD method for parabolic inverse source problems
  3. Ajoint POD method
  4. Convergence of the Adjoint-POD method
  5. Convergence of inverse parabolic source problem
  6. Parabolic backward problem
  7. Convergence of the adjoint-POD method
  8. Convergence of backward problem
  9. Numerical examples
  10. Inverse source examples
  11. Examples for backward problem
  12. Conclusion
  13. Proper orthogonal decomposition (POD) method

Key Result

Proposition 2.1

Suppose $\Omega$ is a bounded domain in $\mathbb{R}^d$ and $a(x), c(x)\in C^0(\bar{\Omega})$, $c(x)\geq 0$, then the eigenvalue problem has a countable set of positive eigenvalues $\mu_1\le\mu_2\le\cdots$, with its corresponding eigenfunctions $\{\phi_k\}_{k=1}^\infty$ forming an orthogonal basis of $L^2(\Omega)$. Moreover, there exist constants $C_1,C_2>0$ such that $C_1 k^{2/d}\le \mu_k\le C_2k

Figures (9)

  • Figure 1: The importance of choice of POD basis
  • Figure 2: Comparison of basis between traditional POD and the adjoint POD for $f=\sin(2x)\sin(2y)$.
  • Figure 3: Comparison of basis between traditional POD and the adjoint POD for $f^*$ of Z shaped function.
  • Figure 4: Robustness of the adjoint POD against the noise for $f=sin(2x)sin(2y)e^{\frac{x+y}{\pi}}$.
  • Figure 5: The importance of choice of POD basis
  • ...and 4 more figures

Theorems & Definitions (14)

  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • Theorem 2.5
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 4 more