A s conventional oil and gas fields are maturing, our profession is challenged to come up with the nextgeneration of more and more sophisticated exploration tools. In exploration seismology this trend has led to the emergence of wave-equation-based inversion technologies such as reverse time migration and full-waveform inversion. While significant progress has been made in wave-equation-based inversion, major challenges remain in the development of robust and computationally feasible workflows that give reliable results in geophysically challenging areas that may include ultralow shear-velocity zones or high-velocity salt. Moreover, subsalt production carries risks that need mitigation, which raises the bar from creating subsalt images to inverting for subsalt overpressure.Among the many challenges that wave-equation-based inversion faces, we focus on reducing the excessive computational costs of full-waveform inversion (FWI). We accomplish these cost reductions by using modern techniques from machine learning and compressive sensing. Contrary to many implementations of wave-equation-based inversion, we propose a methodology where we do not insist on using all data (i.e., looping over all sources) to calculate the velocity model updates. Instead, we rely on a formulation that calls for more data and more accuracy in the wave simulations only as needed by the inversion. This dynamic selection of data and accuracy leads to major savings, in particular in the beginning when we are far from reaching the solution. Because this approach reduces the computational costs significantly, we open the way to test different scenarios for the purpose of quality control or to include more sophisticated regularization. Without this cost reduction, both would be computationally infeasible.The article is organized as follows. First, we introduce the basics of full-waveform inversion, followed by stochastic sampling techniques to reduce the computational costs. Next, we propose an adaptive scheme that dynamically selects the required sample size-i.e., the number of source experiments that partake in the inversion, and forward modeling accuracy. Recent results on the Chevron Gulf of Mexico data set are discussed next, followed by a discussion on challenges of FWI and possible roads ahead. Wave-equation-based inversionFull-waveform inversion (FWI) is a parameter estimation problem, seeking Earth models that explain observed data typically collected in shot records. FWI is generally cast as an optimization problem where Earth models are found by minimizing the misfit between observed and simulated data-i.e., we solve the following optimization problem: VAN Solutions of the above minimization problem are typically found by using iterative optimization techniques, which use local derivative information to compute descent directions that minimize the objective (m). At the bare minimum, these optimizations consist of the following steps for each experiment (Tarantola and Valette, 1982;Plessix, 2006): (1) data prediction by solving the w...