Solving non-linear systems of equations on graphics processing units

• Authors: Lubomir T. Dechevsky;Børre Bang;Joakim Gundersen;Arne Lakså;Arnt R. Kristoffersen
• Venue: LSSC'09 Proceedings of the 7th international conference on Large-Scale Scientific Computing
• Year: 2009

In [6], [7] a method for isometric immersion of smooth m-variate n-dimensional vector fields, m=1,2,3,4,..., n=1,2,3,4,.. onto fractal curves and surfaces was developed, thereby creating an opportunity to process high-dimensional geometric data on graphics processing units (GPUs) which in this case are used as relatively simple parallel computing architectures (with relatively very low price) For this construction, the structure of multivariate tensor-product orthonormal wavelet bases was of key importance In the two afore-mentioned papers, one of the topics discussed was the spatial localization of points in high dimensional space and their images in the plane (corresponding to pixels in the images when processed by the GPU) In the present work we show how to compute approximately on the GPU multivariate intersection manifolds, using a new orthonormal-wavelet scaling-function basis-matching algorithm which offers considerable simplifications compared to the original proposed in [6], [7] This algorithm is not as general as the Cantor diagonal type of algorithm considered in [6], [7], but is much simpler to implement and use for global mapping of the wavelet basis indices in one and several dimensions This new, simpler, approach finds also essential use in the results obtained in [5] which can be considered as continuation of the present paper, extending the range of applications of the present simplified approach to GPU-based computation of multivariate orthogonal wavelet transforms The new method can be also used to accelerate the initial phase of the so-called Marching Simplex algorithm (or any other 'marching' algorithm for numerical solution of nonlinear systems of equations).