mathematical description distance estimation algorithm is a delicate matter to establish, even in the case of julia sets over the complex numbers. depends upon the riemann mapping theorem and a number of special results in complex analysis. we have found that the exact analog of this distance estimate works very well (empirically) for quaternionic julia and mandelbrot sets in three and four dimensions. this experimental work demands both new theoretical work and a finer grained approach to the experimentation. on the theoretical side, collaborator yumei dang is pursuing a program to analyze the distance estimate using theorems in analysis over the quaternions. to complement this, she is pursuing the best ways to push the distance estimate algorithm and corresponding rendering processes on good computer systems. since these images at the present time are quite time consuming to create, this limits the flexibility of the experiments. this is the main mathematical reason for wanting the best supercomputer help for this problem. it is to be expected that a good and flexible source of images for higher-dimensional julia and mandelbrot sets will raise many mathematical questions of whose nature we have not yet dreamed.
ray tracing is one of the realistic methods for rendering the julias. it is not a practical method to sample each point at given resolution along each ray. with the use of the distance estimate formula, the amount of sample points per ray is greatly reduced, but it is still time consuming. hence high-performance supercomputing becomes essential to the visualization of julia sets. the ray-tracing algorithm has been tried on a cray y-mp and it is being run on the dec alpha at the pittsburgh supercomputing center, and better results are expected. the alpha supercluster is a layered system comprised of front-end queue controllers and dedicated machines for computations (the alpha system consists of two front-ends, four parallel development nodes, and eight dedicated compute nodes). it can execute a variety of single-threaded codes, such as the ray-tracing algorithm used in this project, with the parallel virtual machine (pvm) software library. the implementation utilizes advanced supercomputing techniques, such as parallelization and vectorization.
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