We present a method for using programmable graphics hardware to solve a variety of boundary value problems. The time-evolution of such problems is frequently governed by partial differential equations, which are used to describe a wide range of dynamic phenomena including heat transfer and fluid mechanics. The need to solve these equations efficiently arises in many areas of computational science. Finite difference methods are commonly used for solving partial differential equations; we show that this approach can be mapped onto a modern graphics processor. We demonstrate an implementation of the multigrid method, a fast and popular approach to solving boundary value problems, on two modern graphics architectures. Our initial tests with available hardware indicate a 15x speedup compared to traditional software implementation. This work presents a novel use of computer hardware and raises the intriguing possibility that we can make the inexpensive power of modern commodity graphics hardware accessible to and useful for the simulation community. Index Terms-Boundary value problems, partial differential equations, multigrid method, graphics hardware.
Modern graphics architectures have replaced stages of the graphics pipeline with fully programmable modules. Therefore, it is now possible to perform fairly general computation on each vertex or fragment in a scene. In addition, the nature of the graphics pipeline makes substantial computational power available if the programs have a suitable structure. In this paper, we show that it is possible to cleanly map a state-of-the-art tone mapping algorithm to the pixel processor. This allows an interactive application to achieve higher levels of realism by rendering with physically based, unclamped lighting values and high dynamic range texture maps. We also show that the tone mapping operator can easily be extended to include a time-dependent model, which is crucial for interactive behavior. Finally, we describe the ways in which the graphics hardware limits our ability to compress dynamic range efficiently, and discuss modifications to the algorithm that could alleviate these problems.
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