The scale of nowadays High Performance Computing (HPC) systems is the key element that determines the achievement of impressive performance, as well as the reason for their relatively limited reliability. Over the last decade, specific areas of the HPC research field have addressed the issue at different levels, by enriching the infrastructure, the platforms, or the algorithms with fault tolerance features.In this work, we focus on the rather-pervasive task of computing the solution of a dense, unstructured linear system and we propose an algorithm-based technique to obtain fault tolerance to multiple anywhere-located faults during the parallel computation. We particularly study the ways to boost the performance of the rollback-free recovery, and we provide an extensive evaluation of our technique w.r.t. to other state-of-the-art algorithm-based methods.