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.

Understanding the influence of multiple parameters in a complex simulation setting is a difficult task. In the ideal case, the scientist can freely steer such a simulation and is immediately presented with the results for a certain configuration of the input parameters. Such an exploration process is however not possible if the simulation is computationally too expensive. For these cases we present in this paper a scalable computational steering approach utilizing a fast surrogate model as substitute for the time-consuming simulation. The surrogate model we propose is based on the sparse grid technique, and we identify the main computational tasks associated with its evaluation and its extension. We further show how distributed data management combined with the specific use of accelerators allows us to approximate and deliver simulation results to a high-resolution visualization system in real-time. This significantly enhances the steering workflow and facilitates the interactive exploration of large datasets.

The sparse grid discretization technique enables a compressed representation of higher-dimensional functions. In its original form, it relies heavily on recursion and complex data structures, thus being far from well-suited for GPUs. In this paper, we describe optimizations that enable us to implement compression and decompression, the crucial sparse grid algorithms for our application, on Nvidia GPUs. The main idea consists of a bijective mapping between the set of points in a multi-dimensional sparse grid and a set of consecutive natural numbers. The resulting data structure consumes a minimum amount of memory. For a 10-dimensional sparse grid with approximately 127 million points, it consumes up to 30 times less memory than trees or hash tables which are typically used. Compared to a sequential CPU implementation, the speedups achieved on GPU are up to 17 for compression and up to 70 for decompression, respectively. We show that the optimizations are also applicable to multicore CPUs.