Adaptive Algebraic Multigrid for Finite Element Elliptic Equations with Random Coefficients

This thesis presents a two-grid algorithm based on Smoothed Aggregation Spectral Element Agglomeration Algebraic Multigrid (SA-{rho}AMGe) combined with adaptation. The aim is to build an efficient solver for the linear systems arising from discretization of second-order elliptic partial differential equations (PDEs) with stochastic coefficients. Examples include PDEs that model subsurface flow with random permeability field. During a Markov Chain Monte Carlo (MCMC) simulation process, that draws PDE coefficient samples from a certain distribution, the PDE coefficients change, hence the resulting linear systems to be solved change. At every such step the system (discretized PDE) needs to be solved and the computed solution used to evaluate some functional(s) of interest that then determine if the coefficient sample is acceptable or not. The MCMC process is hence computationally intensive and requires the solvers used to be efficient and fast. This fact that at every step of MCMC the resulting linear system changes, makes an already existing solver built for the old problem perhaps not as efficient for the problem corresponding to the new sampled coefficient. This motivates the main goal of our study, namely, to adapt an already existing solver to handle the problem (with changed coefficient) with the objective to achieve this goal to be faster and more efficient than building a completely new solver from scratch. Our approach utilizes the local element matrices (for the problem with changed coefficients) to build local problems associated with constructed by the method agglomerated elements (a set of subdomains that cover the given computational domain). We solve a generalized eigenproblem for each set in a subspace spanned by the previous local coarse space (used for the old solver) and a vector, component of the error, that the old solver cannot handle. A portion of the spectrum of these local eigen-problems (corresponding to eigenvalues close to zero) form the coarse basis used to define the new two-level method of our interest. We illustrate the performance of this adaptive two-level procedure with a large set of numerical experiments that demonstrate its efficiency over building the solvers from scratch

Cytotoxic and genotoxic effects of Br-containing oxaphosphole on Allium cepa L. root tip cells and mouse bone marrow cells

The continuous production and release of chemicals into the environment has led to the need to assess their genotoxicity. Numerous organophosphorus compounds with different structures have been synthesized in recent years, and several oxaphosphole derivatives are known to possess biological activity. Such chemical compounds may influence proliferating cells and cause disturbances of the genetic material. In this study, we examined the cytotoxicity and genotoxicity of 4-bromo-N,N-diethyl-5,5-dimethyl-2,5-dihydro-1,2-oxaphosphol-2-amine 2-oxide (Br-oxph). In A. cepa cells, Br-oxph (10-9 M, 10 -6 M and 10 -3 M) reduced the mitotic index 48 h after treatment with the two highest concentrations, with no significant effect at earlier intervals. Mitotic cells showed abnormalities 24 h and 48 h after treatment with the two lowest concentrations but there were no consistent changes in interphase cells. Bone marrow cells from mice treated with Br-oxph (2.82 x 10 -3 μg/kg) also showed a reduced mitotic index after 48 h and a greater percentage of cells with aberrations (principally chromatid and isochromatid breaks). These findings indicate the cytotoxicity and genotoxicity of Br-oxph in the two systems studied

Comparative Performance Analysis of Coarse Solvers for Algebraic Multigrid on Multicore and Manycore Architectures

We study the performance of a two-level algebraic-multigrid algorithm, with a focus on the impact of the coarse-grid solver on performance. We consider two algorithms for solving the coarse-space systems: the preconditioned conjugate gradient method and a new robust HSS-embedded low-rank sparse-factorization algorithm. Our test data comes from the SPE Comparative Solution Project for oil-reservoir simulations. We contrast the performance of our code on one 12-core socket of a Cray XC30 machine with performance on a 60-core Intel Xeon Phi coprocessor. To obtain top performance, we optimized the code to take full advantage of fine-grained parallelism and made it thread-friendly for high thread count. We also developed a bounds-and-bottlenecks performance model of the solver which we used to guide us through the optimization effort, and also carried out performance tuning in the solver’s large parameter space. As a result, significant speedups were obtained on both machines

Adaptive Algebraic Multigrid for Finite Element Elliptic Equations with Random Coefficients

This thesis presents a two-grid algorithm based on Smoothed Aggregation Spectral Element Agglomeration Algebraic Multigrid (SA-{rho}AMGe) combined with adaptation. The aim is to build an efficient solver for the linear systems arising from discretization of second-order elliptic partial differential equations (PDEs) with stochastic coefficients. Examples include PDEs that model subsurface flow with random permeability field. During a Markov Chain Monte Carlo (MCMC) simulation process, that draws PDE coefficient samples from a certain distribution, the PDE coefficients change, hence the resulting linear systems to be solved change. At every such step the system (discretized PDE) needs to be solved and the computed solution used to evaluate some functional(s) of interest that then determine if the coefficient sample is acceptable or not. The MCMC process is hence computationally intensive and requires the solvers used to be efficient and fast. This fact that at every step of MCMC the resulting linear system changes, makes an already existing solver built for the old problem perhaps not as efficient for the problem corresponding to the new sampled coefficient. This motivates the main goal of our study, namely, to adapt an already existing solver to handle the problem (with changed coefficient) with the objective to achieve this goal to be faster and more efficient than building a completely new solver from scratch. Our approach utilizes the local element matrices (for the problem with changed coefficients) to build local problems associated with constructed by the method agglomerated elements (a set of subdomains that cover the given computational domain). We solve a generalized eigenproblem for each set in a subspace spanned by the previous local coarse space (used for the old solver) and a vector, component of the error, that the old solver cannot handle. A portion of the spectrum of these local eigen-problems (corresponding to eigenvalues close to zero) form the coarse basis used to define the new two-level method of our interest. We illustrate the performance of this adaptive two-level procedure with a large set of numerical experiments that demonstrate its efficiency over building the solvers from scratch

Investigation on some biotic factors in carp fish ponds

Abstract. Three years studies (2004 – 2006) on the main biotic parameters (chlorophyll-a, phytoplankton biomass, zooplankton biomass and bacterioplankton biomass) in carp fish ponds were carried out. The aim of the study was to investigate the biotic factors and the effect of manuring on the fish ponds. The relative -1 changes in these factors in case of fertilization with manure 3000 kg.ha or without fertilization were determined. The impact of fertilization as bottom-up melioration on some biotic factors was proven by means of paired non-parametric Wilcoxon test with following significant differences: higher levels of chlorophyll-a and higher phytoplankton biomass in fertilized ponds. Zooplankton biomass was higher in fertilized ponds, but the differences were statistically insignificant. Bacterioplankton biomass was higher in the fertilized ponds, which is an indication that the applied melioration does not lead to overload of organic matter in the ponds

Tuning the coarse space construction in a spectral AMG solver

In this paper, we discuss strategies for computing subsets of eigenvectors of matrices corresponding to subdomains of finite element meshes achieving compromise between two contradicting goals. The subset of eigenvectors is required in the construction of coarse spaces used in algebraic multigrid methods (AMG) as well as in certain domain decomposition (DD) methods. The quality of the coarse spaces depends on the number of eigenvectors, which improves the approximation properties of the coarse space and impacts the overall performance and convergence of the associated AMG or DD algorithms. However, a large number of eigenvectors affects negatively the sparsity of the corresponding coarse matrices, which can become fairly dense. The sparsity of the coarse matrices can be controlled to a certain extent by the size of the subdomains (union of finite elements) referred to as agglomerates. If the size of the agglomerates is too large, then the cost of the eigensolvers increases and eventually can become unacceptable for the purpose of constructing the AMG or DD solvers. This paper investigates strategies to optimize the solution of the partial eigenproblems of interest. In particular, we examine direct and iterative eigensolvers for computing those subsets. Our experiments with a well-known model of an oil-reservoir simulation benchmark indicate that iterative eigensolvers can lead to significant improvements in the overall performance of an AMG solver that exploits such spectral construction of coarse spaces

Comparative performance analysis of coarse solvers for algebraic multigrid on multicore and manycore architectures

We study the performance of a two-level algebraic-multigrid algorithm, with a focus on the impact of the coarse-grid solver on performance. We consider two algorithms for solving the coarse-space systems: the preconditioned conjugate gradient method and a new robust HSS-embedded low-rank sparse-factorization algorithm. Our test data comes from the SPE Comparative Solution Project for oil-reservoir simulations. We contrast the performance of our code on one 12-core socket of a Cray XC30 machine with performance on a 60-core Intel Xeon Phi coprocessor. To obtain top performance, we optimized the code to take full advantage of fine-grained parallelism and made it thread-friendly for high thread count. We also developed a bounds-and-bottlenecks performance model of the solver which we used to guide us through the optimization effort, and also carried out performance tuning in the solver’s large parameter space. As a result, significant speedups were obtained on both machines