Numerical simulation has become an inevitable tool in most industrial product development processes with simulations being used to understand the influence of design decisions (parameter configurations) on the structure and properties of the product. However, in order to allow the engineer to thoroughly explore the design space and fine-tune parameters, many - usually very time-consuming - simulation runs are necessary. Additionally, this results in a huge amount of data that cannot be analyzed in an efficient way without the support of appropriate tools. In this paper, we address the two-fold problem: First, instantly provide simulation results if the parameter configuration is changed, and, second, identify specific areas of the design space with concentrated change and thus importance. We propose the use of a hierarchical approach based on sparse grid interpolation or regression which acts as an efficient and cheap substitute for the simulation. Furthermore, we develop new visual representations based on the derivative information contained inherently in the hierarchical basis. They intuitively let a user identify interesting parameter regions even in higher-dimensional settings. This workflow is combined in an interactive visualization and exploration framework. We discuss examples from different fields of computational science and engineering and show how our sparse-grid-based techniques make parameter dependencies apparent and how they can be used to fine-tune parameter configurations.