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Increasing stability for inverse acoustic source problems

Suliang Si

Abstract

In this paper, we show the increasing stability of the inverse source problems for the acoustic wave equation in the full space R3.The goal is to understand increasing stability for wave equation in the time domain. If the time and spatial variables of the source term can be separated with compact support, the increasing stability estimates of the $L^2$-norm of the acoustic source function can be established. The stability estimates consist of two parts: the Lipschitz type data discrepancy and the high time tail of the source functions. As the time increases, the latter decreases and thus becomes negligible.

Increasing stability for inverse acoustic source problems

Abstract

In this paper, we show the increasing stability of the inverse source problems for the acoustic wave equation in the full space R3.The goal is to understand increasing stability for wave equation in the time domain. If the time and spatial variables of the source term can be separated with compact support, the increasing stability estimates of the -norm of the acoustic source function can be established. The stability estimates consist of two parts: the Lipschitz type data discrepancy and the high time tail of the source functions. As the time increases, the latter decreases and thus becomes negligible.

Paper Structure

This paper contains 3 sections, 6 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Known results and Motivations
  3. Proof of Theorem \ref{['1']}

Key Result

Theorem 1.1

Let $\|f\|_{L^2{({\mathbb R}^3)}}\leq M$ and $g(t)=e^{-\gamma t}$ for constant $M, \gamma>0$. Assume that $U$ is the solution of the scattering problem (eq1)–(eq2). Then there exist constant $\alpha>0$ and $C>0$ depending on $R$, $\gamma$, $\alpha$ such that where $T>1$ and

Theorems & Definitions (11)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • ...and 1 more