A multi-frequency sampling method for the inverse source problems with sparse measurements

TL;DR

This paper introduces a novel multi-frequency sampling method for inverse source problems using sparse near field measurements, leveraging a new operator based on frequency integration and its factorization for source support reconstruction.

Contribution

The paper develops a multi-frequency near field operator and a sampling method based on its factorization, providing a new approach for inverse source problems with sparse data.

Findings

Effective reconstruction of source support demonstrated through numerical examples.

The method extends to far field cases with brief discussion.

Theoretical analysis supports the stability and accuracy of the approach.

Abstract

We consider the inverse source problems with multi-frequency sparse near field measurements. In contrast to the existing near field operator based on the integral over the space variable, a multi-frequency near field operator is introduced based on the integral over the frequency variable. A factorization of this multi-frequency near field operator is further given and analysed. Motivated by such a factorization, we introduce a multi-frequency sampling method to reconstruct the source support. Its theoretical foundation is then derived from the properties of the factorized operators and a properly chosen point spread function. Numerical examples are provided to illustrate the multi-frequency sampling method with sparse near field measurements. Finally we briefly discuss how to extend the near field case to the far field case.