With an increasing amount of data and user demands for fast query processing, the optimization of database operations continues to be a challenging task. A common optimization method is to leverage parallel hardware architectures. With the introduction of general-purpose GPU computing, massively parallel hardware has become available within commodity hardware. To efficiently exploit this technology, we introduce the method of speculative query processing. This speculative query processing works on index structures to efficiently support heavily used database operations. To show the benefits and opportunities of our approach, we present a fine and coarse grain implementation for multidimensional queries.