Abstract: We present a multithreaded method for supernodal sparse Cholesky factorizations on a hybrid multicore platform consisting of a multicore CPU and GPU. Our algorithm can utilize concurrentcy at differnt levels of the elimination tree by using multiple threads in both the CPU and the GPU. By factorizing multiple matrices in a batch our algorithm can generate better performance than previous implementations. Our experiments results on a platform consisting of an Intel multicore processor along with an Nvidia GPU indicate a significant improvement in performance over single-threaded supernodal algorithm.

Abstract: Value at Risk (VaR) is a statistical method of predicting market risk associated with financial portfolios. There are numerous statistical models which forecast VaR and out of those, Monte Carlo Simulation is a commonly used technique with a high accuracy though it is computationally intensive. Calculating VaR in real time is becoming a need of short term traders in current day markets and adapting Monte Carlo method of VaR computation for real time calculation poses a challenge due to the computational complexity involved with the simulation step of the Monte Carlo Simulation. The simulation process has an independent set of tasks. Hence a performance bottleneck occurs during the sequential execution of these independent tasks. By parallelizing these tasks, the time taken to calculate the VaR for a portfolio can be reduced significantly. In order to address this issue, we looked at utilizing the Advanced Vector Extensions (AVX) technology to parallelize the simulation process. We compared the performance of the AVX based solution against the sequential approach as well as against a multi threaded solution and a GPU based solution. The results showed that the AVX approach outperformed the GPU approach for up to an iteration count of 200000. Since such a number of iterations is generally not required to gain a sufficiently accurate VaR measure, it makes sense both computationally and economically to utilize AVX for Monte Carlo method of VaR computation.

Abstract: The Locally Linear Embedding (LLE) algorithm has proven useful for determining structure preserving, dimension reducing mappings of data on manifolds. We propose a modification to the LLE optimization problem that serves to minimize the number of neighbors required for the representation of each data point. The algorithm is shown to be robust over wide ranges of the sparsity parameter producing an average number of nearest neighbors that is consistent with the best performing parameter selection for LLE. Given the number of non-zero weights may be substantially reduced in comparison to LLE, Sparse LLE can be applied to larger data sets. We provide three numerical examples including a color image, the standard swiss roll, and a gene expression data set to illustrate the behavior of the method in comparison to LLE. The resulting algorithm produces comparatively sparse representations that preserve the neighborhood geometry of the data in the spirit of LLE.

Abstract: In this paper we present a comparison study for three different iterative Krylov methods that we have recently developed for the simultaneous numerical solution of wave propagation problems at multiple frequencies. The three approaches have in common that they require the application of a single shift-and-invert preconditioner at a suitable 'seed' frequency. The focus of the present work, however, lies on the performance of the respective iterative method. We conclude with numerical examples that provide guidance concerning the suitability of the three methods.

Abstract: Recently a method was presented to compute Lyapunov functions for nonlinear systems with multiple local attractors. This method was shown to succeed in delivering algorithmically a Lyapunov function giving qualitative information on the system's dynamics, including lower bounds on the attractors' basins of attraction. We suggest a simpler and faster algorithm to compute such a Lyapunov function if the attractors in question are exponentially stable equilibrium points. Just as in the earlier publication one can apply the algorithm and expect to obtain partial information on the system dynamics if the assumptions on the system at hand are only partially fulfilled. We give four examples of our method applied to different dynamical systems from the literature.