Abstract: A sub-sea mud pump system includes a plurality of pump units, and each pumping unit includes a plurality of pumping elements. Each pumping element includes a pressure vessel with a first and a second chamber, a separating member between the first and second chambers, a measurement device adapted to measure the volume of at least one of the first and second chambers, a hydraulic inlet control valve and a hydraulic outlet control valve coupled to the first chamber, a mud suction valve and a mud discharge valve coupled to the second chamber. The first chamber is hydraulically coupled to receive and discharge a hydraulic fluid, and the second chamber is hydraulically coupled to receive and discharge a drilling fluid. The separating member is adapted to move within its the pressure vessel in response to a pressure differential between the first and second chambers. The pump system also includes a hydraulic control unit adapted to control the plurality of pump units.