Reconstructing a signal from measurement data is a ubiquitous problem in science and engineering, called an inverse problem. Among many challenges in inverse problems, we focus on handling limited data in which the data cannot uniquely determine the unknown signal. We will review the compressive sensing technique in signal processing in the first part of this talk to address the issue. The first part is accessible to graduate and undergraduate students. In the second part, we will focus on implementing the idea of the compressive sensing technique in the framework of the ensemble Kalman inversion, a semi-probabilistic approach for optimization problems, and its application to PDE-constrained optimization problems.