We propose a dynamic working set method (DWS) for the problem \(\min _{\texttt{x} \in \mathbb {R}^n} \frac{1}{2}\Vert \texttt{Ax}-\texttt{b} \Vert ^2 + \eta \Vert \texttt{x} \Vert _1\) that arises from compressed sensing. DWS manages the working set while iteratively calling a regression solver to generate progressively better solutions. Our experiments show that DWS is more efficient than other state-of-the-art software in the context of compressed sensing. Scale space such that \(\Vert b \Vert =1\) . Let s be the number of non-zeros in the unknown signal. We prove that for any given \(\varepsilon > 0\) , DWS reaches a solution with an additive error \(\varepsilon /\eta ^2\) such that each call of the solver uses only \(O(\frac{1}{\varepsilon }s\log s \log \frac{1}{\varepsilon })\) variables, and each intermediate solution has \(O(\frac{1}{\varepsilon }s\log s\log \frac{1}{\varepsilon })\) non-zero coordinates.