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Lecture note on inverse problems and reconstruction methods

Manabu Machida

TL;DR

This lecture-note-style survey addresses inverse problems for PDEs, emphasizing ill-posedness and reconstruction strategies. It develops and compares regularization approaches, including SVD, truncated SVD, and Tikhonov, and demonstrates iterative schemes like Landweber and CG, using the heat equation as a central example. The text extends to optical tomography, inverse series (Born and Rytov), and classical imaging modalities (MRI, Radon transform, and X-ray CT), highlighting stability analyses via Carleman estimates and practical reconstruction algorithms. The work synthesizes theory and computation to show how regularization enables stable recovery of unknowns from indirect measurements across diverse imaging applications.

Abstract

The area of inverse problems in mathematics is highly interdisciplinary. In various fields of science, engineering, medicine, and industry, there arises a need to reconstruct information about unknown entities that cannot be directly observed. Examples include medical imaging techniques such as X-ray CT and optical tomography. Indeed, the mathematics of inverse problems has often originated from challenges posed by other fields. Inverse problems are often ill-posed and solutions are unstable. In this lecture, we will explore methods to solve such inverse problems.

Lecture note on inverse problems and reconstruction methods

TL;DR

This lecture-note-style survey addresses inverse problems for PDEs, emphasizing ill-posedness and reconstruction strategies. It develops and compares regularization approaches, including SVD, truncated SVD, and Tikhonov, and demonstrates iterative schemes like Landweber and CG, using the heat equation as a central example. The text extends to optical tomography, inverse series (Born and Rytov), and classical imaging modalities (MRI, Radon transform, and X-ray CT), highlighting stability analyses via Carleman estimates and practical reconstruction algorithms. The work synthesizes theory and computation to show how regularization enables stable recovery of unknowns from indirect measurements across diverse imaging applications.

Abstract

The area of inverse problems in mathematics is highly interdisciplinary. In various fields of science, engineering, medicine, and industry, there arises a need to reconstruct information about unknown entities that cannot be directly observed. Examples include medical imaging techniques such as X-ray CT and optical tomography. Indeed, the mathematics of inverse problems has often originated from challenges posed by other fields. Inverse problems are often ill-posed and solutions are unstable. In this lecture, we will explore methods to solve such inverse problems.
Paper Structure (43 sections, 14 theorems, 380 equations, 24 figures, 1 table)

This paper contains 43 sections, 14 theorems, 380 equations, 24 figures, 1 table.

Table of Contents

  1. What are inverse problems?
  2. Introduction
  3. Example 1: Heat equation
  4. Example 2: Contaminated water in a tank
  5. Stability
  6. Transport equation
  7. Linearization
  8. Carleman estimates
  9. Stability estimate
  10. Remarks
  11. The heat equation
  12. Derivation
  13. Solutions
  14. Smoothing effect and infinite propagation speed
  15. Singular value decomposition
  16. ...and 28 more sections

Key Result

Proposition 2.1

Assume that $w\in C_0^1([0,\ell]\times[-T,T])$ satisfies $w(0,t)=w(x,\pm T)=0$. Then there exist constants $C>0$, $s_0>0$ such that for all $s\ge s_0$.

Figures (24)

  • Figure 1: Contaminated water flows into a tank.
  • Figure 2: Cutoff function $\chi$.
  • Figure 3: The weight function $\varphi$ and cutoff function $\chi$.
  • Figure 4: The heat conduction in one dimension.
  • Figure 5: The heat conduction and the heat equation.
  • ...and 19 more figures

Theorems & Definitions (29)

  • Proposition 2.1
  • proof
  • Theorem 2.2
  • Theorem 5.1
  • proof
  • Theorem 6.1
  • proof
  • Proposition 6.2
  • proof
  • Remark 6.3
  • ...and 19 more