Milena Veneva

Milena Veneva, Alexander Ayriyan

This paper presents an experimental performance study of implementations of three different types of algorithms for solving band matrix systems of linear algebraic equations (SLAEs) after parabolic nonlinear partial differential equations -- direct, symbolic, and iterative, the former two of which were introduced in Veneva and Ayriyan (arXiv:1710.00428v2). An iterative algorithm is presented -- the strongly implicit procedure (SIP), also known as the Stone method. This method uses the incomplete LU (ILU(0)) decomposition. An application of the Hotelling-Bodewig iterative algorithm is suggested as a replacement of the standard forward-backward substitutions. The upsides and the downsides of the SIP method are discussed. The complexity of all the investigated methods is presented. Performance analysis of the implementations is done using the high-performance computing (HPC) clusters "HybriLIT" and "Avitohol". To that purpose, the experimental setup and the results from the conducted computations on the individual computer systems are presented and discussed.

Milena Veneva, Alexander Ayriyan

A class of models of heat transfer processes in a multilayer domain is considered. The governing equation is a nonlinear heat-transfer equation with different temperature-dependent densities and thermal coefficients in each layer. Homogeneous Neumann boundary conditions and ideal contact ones are applied. A finite difference scheme on a special uneven mesh with a second-order approximation in the case of a piecewise constant spatial step is built. This discretization leads to a pentadiagonal system of linear equations (SLEs) with a matrix which is neither diagonally dominant, nor positive definite. Two different methods for solving such a SLE are developed -- diagonal dominantization and symbolic algorithms.

Milena Veneva, Alexander Ayriyan

This paper presents an experimental performance study of implementations of three symbolic algorithms for solving band matrix systems of linear algebraic equations with heptadiagonal, pentadiagonal, and tridiagonal coefficient matrices. The only assumption on the coefficient matrix in order for the algorithms to be stable is nonsingularity. These algorithms are implemented using the GiNaC library of C++ and the SymPy library of Python, considering five different data storing classes. Performance analysis of the implementations is done using the high-performance computing (HPC) platforms "HybriLIT" and "Avitohol". The experimental setup and the results from the conducted computations on the individual computer systems are presented and discussed. An analysis of the three algorithms is performed.

Milena Veneva, Alexander Ayriyan

This paper presents a symbolic algorithm for solving band matrix systems of linear algebraic equations with heptadiagonal coefficient matrices. The algorithm is given in pseudocode. A theorem which gives the condition for the algorithm to be stable is formulated and proven.

Milena Veneva, Toshiyuki Imamura

This paper presents a heuristic for finding the optimum number of CUDA streams by using tools common to the modern AI-oriented approaches and applied to the parallel partition algorithm. A time complexity model for the GPU realization of the partition method is built. Further, a refined time complexity model for the partition algorithm being executed on multiple CUDA streams is formulated. Computational experiments for different SLAE sizes are conducted, and the optimum number of CUDA streams for each of them is found empirically. Based on the collected data a model for the sum of the times for the non-dominant GPU operations (that take part in the stream overlap) is formulated using regression analysis. A fitting non-linear model for the overhead time connected with the creation of CUDA streams is created. Statistical analysis is done for all the built models. An algorithm for finding the optimum number of CUDA streams is formulated. Using this algorithm, together with the two models mentioned above, predictions for the optimum number of CUDA streams are made. Comparing the predicted values with the actual data, the algorithm is deemed to be acceptably good.

Milena Veneva

This paper presents a generalised symbolic algorithm for solving systems of linear algebraic equations with multi-diagonal coefficient matrices. The algorithm is given in a pseudocode. A theorem which gives the condition for correctness of the algorithm is formulated and proven. Formula for the complexity of the multi-diagonal numerical algorithm is obtained.

Milena Veneva

This paper presents a machine learning (ML)-based heuristic for finding the optimum sub-system size for the CUDA implementation of the parallel partition algorithm. Computational experiments for different system of linear algebraic equation (SLAE) sizes are conducted, and the optimum sub-system size for each of them is found empirically. To estimate a model for the sub-system size, we perform the k-nearest neighbors (kNN) classification method. Statistical analysis of the results is done. By comparing the predicted values with the actual data, the algorithm is deemed to be acceptably good. Next, the heuristic is expanded to work for the recursive parallel partition algorithm as well. An algorithm for determining the optimum sub-system size for each recursive step is formulated. A kNN model for predicting the optimum number of recursive steps for a particular SLAE size is built.