Abstract: LeSelect is a mediator system which allows scientists to publish their resources (data and programs) so they can be transparently accessed. The scientists can typically issue queries which access distributed published data and involve the execution of expensive functions (corresponding to programs). Furthermore, the queries can involve large objects such as images (e.g., archived meteorological satellite data). In this context, the costs of transmitting large objects and invoking expensive functions are the dominant factors of execution time. In this paper, we first propose three query execution techniques which minimize these costs by taking full advantage of the distributed architecture of mediator systems like LeSelect. Then, we devise parallel processing strategies for queries including expensive functions. Based on experimentation, we show that it is hard to predict the optimal execution order when dealing with several functions. We propose a new hybrid parallel technique to solve this problem and give some experimental results.