An extended GS method for dense linear systems

AbstractDavey and Rosindale [K. Davey, I. Rosindale, An iterative solution scheme for systems of boundary element equations, Internat. J. Numer. Methods Engrg. 37 (1994) 1399–1411] derived the GSOR method, which uses an upper triangular matrix Ω in order to solve dense linear systems. By applying functional analysis, the authors presented an expression for the optimum Ω. Moreover, Davey and Bounds [K. Davey, S. Bounds, A generalized SOR method for dense linear systems of boundary element equations, SIAM J. Comput. 19 (1998) 953–967] also introduced further interesting results. In this note, we employ a matrix analysis approach to investigate these schemes, and derive theorems that compare these schemes with existing preconditioners for dense linear systems. We show that the convergence rate of the Gauss–Seidel method with preconditioner PG is superior to that of the GSOR method. Moreover, we define some splittings associated with the iterative schemes. Some numerical examples are reported to confirm the theoretical analysis. We show that the EGS method with preconditioner PG(γopt) produces an extremely small spectral radius in comparison with the other schemes considered

Explicit alternating direction methods for problems in fluid dynamics

Recently an iterative method was formulated employing a new splitting strategy for the solution of tridiagonal systems of difference equations. The method was successful in solving the systems of equations arising from one dimensional initial boundary value problems, and a theoretical analysis for proving the convergence of the method for systems whose constituent matrices are positive definite was presented by Evans and Sahimi [22]. The method was known as the Alternating Group Explicit (AGE) method and is referred to as AGE-1D. The explicit nature of the method meant that its implementation on parallel machines can be very promising. The method was also extended to solve systems arising from two and three dimensional initial-boundary value problems, but the AGE-2D and AGE-3D algorithms proved to be too demanding in computational cost which largely reduces the advantages of its parallel nature. In this thesis, further theoretical analyses and experimental studies are pursued to establish the convergence and suitability of the AGE-1D method to a wider class of systems arising from univariate and multivariate differential equations with symmetric and non symmetric difference operators. Also the possibility of a Chebyshev acceleration of the AGE-1D algorithm is considered. For two and three dimensional problems it is proposed to couple the use of the AGE-1D algorithm with an ADI scheme or an ADI iterative method in what is called the Explicit Alternating Direction (EAD) method. It is then shown through experimental results that the EAD method retains the parallel features of the AGE method and moreover leads to savings of up to 83 % in the computational cost for solving some of the model problems. The thesis also includes applications of the AGE-1D algorithm and the EAD method to solve some problems of fluid dynamics such as the linearized Shallow Water equations, and the Navier Stokes' equations for the flow in an idealized one dimensional Planetary Boundary Layer. The thesis terminates with conclusions and suggestions for further work together with a comprehensive bibliography and an appendix containing some selected programs

Pole Placement and Reduced-Order Modelling for Time-Delayed Systems Using Galerkin Approximations

The dynamics of time-delayed systems (TDS) are governed by delay differential equa- tions (DDEs), which are infinite dimensional and pose computational challenges. The Galerkin approximation method is one of several techniques to obtain the spectrum of DDEs for stability and stabilization studies. In the literature, Galerkin approximations for DDEs have primarily dealt with second-order TDS (second-order Galerkin method), and the for- mulations have resulted in spurious roots, i.e., roots that are not among the characteristic roots of the DDE. Although these spurious roots do not affect stability studies, they never- theless add to the complexity and computation time for control and reduced-order modelling studies of DDEs. A refined mathematical model, called the first-order Galerkin method, is proposed to avoid spurious roots, and the subtle differences between the two formulations (second-order and first-order Galerkin methods) are highlighted with examples. For embedding the boundary conditions in the first-order Galerkin method, a new pseudoinverse-based technique is developed. This method not only gives the exact location of the rightmost root but also, on average, has a higher number of converged roots when compared to the existing pseudospectral differencing method. The proposed method is combined with an optimization framework to develop a pole-placement technique for DDEs to design closed-loop feedback gains that stabilize TDS. A rotary inverted pendulum system apparatus with inherent sensing delays as well as deliberately introduced time delays is used to experimentally validate the Galerkin approximation-based optimization framework for the pole placement of DDEs. Optimization-based techniques cannot always place the rightmost root at the desired location; also, one has no control over the placement of the next set of rightmost roots. However, one has the precise location of the rightmost root. To overcome this, a pole- placement technique for second-order TDS is proposed, which combines the strengths of the method of receptances and an optimization-based strategy. When the method of receptances provides an unsatisfactory solution, particle swarm optimization is used to improve the location of the rightmost pole. The proposed approach is demonstrated with numerical studies and is validated experimentally using a 3D hovercraft apparatus. The Galerkin approximation method contains both converged and unconverged roots of the DDE. By using only the information about the converged roots and applying the eigenvalue decomposition, one obtains an r-dimensional reduced-order model (ROM) of the DDE. To analyze the dynamics of DDEs, we first choose an appropriate value for r; we then select the minimum value of the order of the Galerkin approximation method system at which at least r roots converge. By judiciously selecting r, solutions of the ROM and the original DDE are found to match closely. Finally, an r-dimensional ROM of a 3D hovercraft apparatus in the presence of delay is validated experimentally

Solution Methods for Re-entrant Corners in Elliptic Partial Differential Equations

Computing and Information Science

Molecular Genetic Analysis of Drought Resistance and Productivity Traits of Rice Genotypes

Rice (Oryza sativa L.) is the staple food for a majority of the world’s population, and uses 30% of the global fresh water during its life cycle. Drought at the reproductive stage is the most important abiotic stress factor limiting grain yield. The United States is the third largest exporter of rice, and Arkansas is the top rice-producing state. The Arkansas rice-growing region in the Lower Mississippi belt is among the 10 areas with the highest risk of water scarcity. Adapted U.S. rice cultivars were screened for drought resistant (DR) traits to find sources for breeding U.S. rice cultivars for a water saving agricultural system. A recombinant inbred line (RIL) population, derived from varieties Kaybonnet (DR) and ZHE733 (drought sensitive), termed K/Z RILs was chosen for genetic analysis of DR traits. The objectives of this research were to 1) analyze the phenotypic and grain yield components of the K/Z RIL rice population for drought-resistance-related traits, 2) evaluate the Abscisic Acid (ABA) response of the K/Z RIL rice population on root architectural traits in relation to drought stress resistance, 3) screen polymorphic molecular markers to identify genes linked to productivity traits of grain yield under drought stress, measured by number of filled grain per panicle using bulk segregant analysis (BSA), and 4) identify QTLs and candidate genes in the K/Z RIL population for drought resistance associated with vegetative morphological traits, grain yield components under drought stress and well-watered conditions, and root architectural traits related to ABA response. The RIL population was screened in the field at Fayetteville (AR) by controlled drought stress (DS) treatment at the reproductive stage, and the effect of DS quantified by measuring drought-related traits. ABA sensitivity was quantified by measuring root architectural traits at the V3 stage. Based on the filled grain per panicle number, 13.13% of K/Z RIL population and parent Kaybonnet were highly drought resistant, while 75.75% of RILs and parent ZHE733 were drought sensitive. Under ABA conditions, Kaybonnet and 48 drought resistant lines exhibit ABA sensitivity, implying regulation of osmotic stress tolerance via ABA-mediated cell signaling. Based on BSA screening, 13 polymorphic markers potentially linked to DR traits were identified. QTL analysis was performed with 4133 SNPs markers by using QTL IciMapping. A total of 213 QTLs and 628 candidate genes within the QTL regions were identified for drought-related traits. The RT-qPCR analysis of the candidate genes revealed that a high number of drought resistance genes were up-regulated in Kaybonnet as the drought-resistant parent. Information from this research will serve an important step towards improvement of adapted Arkansas rice cultivars for higher grain production under DS conditions