The Investigation of Efficiency of Physical Phenomena Modelling Using Differential Equations on Distributed Systems

This work is dedicated to development of mathematical modelling software. In this dissertation numerical methods and algorithms are investigated in software making context. While applying a numerical method it is important to take into account the limited computer resources, the architecture of these resources and how do methods affect software robustness. Three main aspects of this investigation are that software implementation must be efficient, robust and be able to utilize specific hardware resources. The hardware specificity in this work is related to distributed computations of different types: single CPU with multiple cores, multiple CPUs with multiple cores and highly parallel multithreaded GPU device. The investigation is done in three directions: GPU usage for 3D FDTD calculations, FVM method usage to implement efficient calculations of a very specific heat transferring problem, and development of special techniques for software for specific bacteria self organization problem when the results are sensitive to numerical methods, initial data and even computer round-off errors. All these directions are dedicated to create correct technological components that make a software implementation robust and efficient. The time prediction model for 3D FDTD calculations is proposed, which lets to evaluate the efficiency of different GPUs. A reasonable speedup with GPU comparing to CPU is obtained. For FVM implementation the OpenFOAM open source software is selected as a basis for implementation of calculations and a few algorithms and their modifications to solve efficiency issues are proposed. The FVM parallel solver is implemented and analyzed, it is adapted to heterogeneous cluster Vilkas. To create robust software for simulation of bacteria self organization mathematically robust methods are applied and results are analyzed, the algorithm is modified for parallel computations

A Three-Level Parallelisation Scheme and Application to the Nelder-Mead Algorithm

We consider a three-level parallelisation scheme. The second and third levels define a classical two-level parallelisation scheme and some load balancing algorithm is used to distribute tasks among processes. It is well-known that for many applications the efficiency of parallel algorithms of the second and third level starts to drop down after some critical parallelisation degree is reached. This weakness of the two-level template is addressed by introduction of one additional parallelisation level. As an alternative to the basic solver some new or modified algorithms are considered on this level. The idea of the proposed methodology is to increase the parallelisation degree by using less efficient algorithms in comparison with the basic solver. As an example we investigate two modified Nelder-Mead methods. For the selected application, a few partial differential equations are solved numerically on the second level, and on the third level the parallel Wang's algorithm is used to solve systems of linear equations with tridiagonal matrices. A greedy workload balancing heuristic is proposed, which is oriented to the case of a large number of available processors. The complexity estimates of the computational tasks are model-based, i.e. they use empirical computational data

The analysis of parallel OpenFOAM solver for the heat transfer in electrical power cables

Proceedings of the First PhD Symposium on Sustainable Ultrascale Computing Systems (NESUS PhD 2016) Timisoara, Romania. February 8-11, 2016.Here we present the part of results obtained in PhD thesis “The investigation of efficiency of physical phenomena modelling using differential equations on distributed systems” by Andrej Bugajev. This work is dedicated to development of mathematical modelling software. While applying a numerical method it is important to take into account the limited computer resources, the architecture of these resources and how do methods affect software robustness. Three main aspects of this investigation are that software implementation must be efficient, robust and be able to utilize specific hardware resources. The hardware specificity in this work is related to distributed computations. The investigation is done for FVM method usage to implement efficient calculations of a very specific heat transferring problem. That lets to create technological components that make a software implementation robust and efficient. OpenFOAM open source software is selected as a basis for implementation of calculations and a few algorithms to solve efficiency issues are proposed. The FVM parallel solver is implemented and analyzed, it is adapted to heterogeneous cluster Vilkas.European Cooperation in Science and Technology. COSTThe paper was supported by NESUS project Winter School & PhD Symposium 2016

An algorithm for modelling the impact of the judicial conflict-resolution process on construction investment

In this article, the modelling of the judicial conflict-resolution process is considered from a construction investor’s point of view. Such modelling is important for improving the risk management for construction investors and supporting sustainable city development by supporting the development of rules regulating the construction process. Thus, this raises the problem of evaluation of different decisions and selection of the optimal one followed by distribution extraction. First, the example of such a process is analysed and schematically represented. Then, it is formalised as a graph, which is described in the form of a decision graph with cycles. We use some natural problem properties and provide the algorithm to convert this graph into a tree. Then, we propose the algorithm to evaluate profits for different scenarios with estimation of time, which is done by integration of an average daily costs function. Afterwards, the optimisation problem is solved and the optimal investor strategy is obtained—this allows one to extract the construction project profit distribution, which can be used for further analysis by standard risk (and other important information)-evaluation techniques. The overall algorithm complexity is analysed, the computational experiment is performed and conclusions are formulated

Parallel algorithms for three-dimensional parabolic and pseudoparabolic problems with different boundary conditions

In this paper, three-dimensional parabolic and pseudo-parabolic equations with classical, periodic and nonlocal boundary conditions are approximated by the full approximation backward Euler method, locally one dimensional and Douglas ADI splitting schemes. The stability with respect to initial conditions is investigated. We note that the stability of the proposed numerical algorithms can be proved only if the matrix of discrete operator can be diagonalized and eigenvectors make a complete basis system.Parallel versions of all algorithms are constructed and scalability analysis is done. It is shown that discrete one-dimensional problems with periodic and nonlocal boundary conditions can be efficiently solved with similar modifications of the parallel Wang algorithm

On Efficiency of the OpenFOAM-based Parallel Solver for the Heat Transfer in Electrical Power Cables

Proceedings of: First International Workshop on Sustainable Ultrascale Computing Systems (NESUS 2014). Porto (Portugal), August 27-28, 2014.In this work, we study the efficiency of the OpenFOAM-based parallel solver for the heat conduction in electrical power cables. The 2D benchmark problem with three cables is used for our numerical tests. We study and compare the efficiency of conjugate gradient solver with diagonal incomplete Cholesky (DIC) preconditioner and generalized geometric algebraic multigrid solver (GAMG), which is available in Open- FOAM. The convergence and parallel scalability of the solvers are presented and analyzed. Parallel numerical tests are performed on the cluster of multicore computers.The work of authors was supported by Eureka project E!6799 POWEROPT "Mathematical modelling and optimization of electrical power cables for an improvement of their design rules". The work presented in this paper has been partially supported by EU under the COST programme Action IC1305, ’Network for Sustainable Ultrascale Computing (NESUS)’

Comparison of adaptive meshes for a singularly perturbed reaction–diffusion problem

We consider a singular second-order boundary value problem. The differential problem is approximated by the Galerkin finite element scheme. The main goal is to compare the well known apriori Bakhvalov and Shishkin meshes with the adaptive mesh based on the aposteriori dual error estimators. Results of numerical experiments are presented

A three-level parallelisation scheme and application to the Nelder-Mead algorithm

We consider a three-level parallelisation scheme. The second and third levels define a classical two-level parallelisation scheme and some load balancing algorithm is used to distribute tasks among processes. It is well-known that for many applications the efficiency of parallel algorithms of these two levels starts to drop down after some critical parallelisation degree is reached. This weakness of the twolevel template is addressed by introduction of one additional parallelisation level. As an alternative to the basic solver some new or modified algorithms are considered on this level. The idea of the proposed methodology is to increase the parallelisation degree by using possibly less efficient algorithms in comparison with the basic solver. As an example we investigate two modified Nelder-Mead methods. For the selected application, a Schro¨dinger equation is solved numerically on the second level, and on the third level the parallel Wang’s algorithm is used to solve systems of linear equations with tridiagonal matrices. A greedy workload balancing heuristic is proposed, which is oriented to the case of a large number of available processors. The complexity estimates of the computational tasks are model-based, i.e. they use empirical computational data

The Modelling of Roof Installation Projects Using Decision Trees and the AHP Method

In this work, the process of roofing projects’ execution is considered. The proper analysis of this process is important to optimise the behaviour of a project’s participants and to perform risk evaluation. The main result of this work is methodology, which can be used to optimise a project owner’s decisions and potentially can be applied for risk control or integrated into expert systems. This methodology includes the application of a decision tree and AHP (analytic hierarchy process) method to perform the modelling for roof installation project selection. In the proposed approach, a decision tree describes the process with nodes representing the states of a project. The tree includes the decision on whether to sell the project results or not, which requires the estimation of the subjective opinion of the project owner. These subjective values are used in the decision tree leaves. We propose to perform this estimation with the AHP method and describe how to do it in this paper. A particular example was considered. The proposed methodology was applied to that case, and all details of the process and results are provided. Using the proposed methodology, the adapted version of a specific, current situation model of project participants’ behaviours can be formed, allowing one to make the most efficient decisions in the light of the existing constraints. The application of results can increase the investor protection and contribute to the general sustainability of investments.This article belongs to the Special Issue Sustainable Construction Engineering and Managemen

On Stability Analysis of Finite Difference Schemes for Generalized Kuramoto-Tsuzuki Equation with Nonlocal Boundary Conditions

A general methodology for the stability analysis of discrete approximations of nonstationary PDEs is applied to solve the Kuramoto-Tsuzuki equation, including also the Schr¨odinger problem. Stability regions are constructed for the explicit, backward and symmetrical Euler schemes. The obtained results are applied to solve the Kuramoto-Tsuzuki problem with a non-local integral boundary condition. Results of computational experiments are provided