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Increasing stability in the n-dimensional inverse source problem with multi-frequencies

Suliang Si

Abstract

In this paper, we show for the first time the increasing stability of the inverse source problem for the n-dimensional Helmholtz equation at multiple wave numbers, which is different from the two-or three-dimensional Helmholtz equation. In addition, we develop a new, unified approach to study increasing stability in any dimension. The method is based on the Fourier transform and explicit bounds for analytic continuation.

Increasing stability in the n-dimensional inverse source problem with multi-frequencies

Abstract

In this paper, we show for the first time the increasing stability of the inverse source problem for the n-dimensional Helmholtz equation at multiple wave numbers, which is different from the two-or three-dimensional Helmholtz equation. In addition, we develop a new, unified approach to study increasing stability in any dimension. The method is based on the Fourier transform and explicit bounds for analytic continuation.
Paper Structure (3 sections, 5 theorems, 44 equations)

This paper contains 3 sections, 5 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Increasing stability

Key Result

Theorem 2.1

Let $f\in\mathcal{C}_{M,d}$. Then there exists a constant $C>0$ depending on $n$, $d$ and $R$ such that where $K>1$, $4d>n$ and

Theorems & Definitions (10)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Lemma 3.4
  • proof