Cycle Gmres Convergence Behavior Of The Algorithm On Wavelet - Boundary Element Sparse System Research

The restart GMRES is a variant of the generalized minimal residual method,also known as GMRES(m) algorithm; a lot of computing practice and theoreticalanalysis show that GMRES(m) algorithm is an effective iterative method tosolve non-symmetric linear systems. On the other hand, integral equation is acommon mathematical model in physical and engineering fields, which can bediscretized into linear equations by virtue of boundary element method. Theresulting linear system generally has a dense and asymmetric coefficient matrix,so that the resolution procedure could be restricted by the memory and operationspeed of the computer in large scale practical problems. This paper introduceswavelet transform into boundary integral equation which is discretized by theboundary element method; the convergence behaviors of the GMRES(m) solverare studied on a two-dimensional BEM system with wavelet shrinkage. Themain contributions of this work are listed as follows:1. We use the 'SD-RE curves' as a metric tool, in order to conduct ananalysis on the compressibility of the BEM system subjected to a feworthogonal wavelet filters due to Haar and Daubechies, and study the relativeerrors of the approximate boundary solutions and the inner potentials calculatedwith the approximate boundary solutions for the given sparse degree,respectively; we also relate them to the relative residual for the approximateboundary solutions, which could be hopefully used to estimate the inner relativeerror without actually computing the inner potential itself, making the studymore efficient.2. We investigate the convergence behavior of the GMRES(m) solver onthe BEM systems under consideration, without involvement of the wavelettransform, and seek the optimal configurations of the Krylov dimension m andthe convergence tolerance for the GMRES(m) solver, in order to set up abenchmark for the wavelet solver.3. The fast wavelet transform based on the binary partition techniques isintroduced into the boundary element system, with an implementation of the compact compression algorithm in Matlab. Combining with the sparse matrixtechnologies, we investigate the convergence behavior of the GMRES(m) solveron the BEM system with wavelet shrinkage, which could be compared with thebenchmark. Experiments show that, when the number of iterations exceedingcertain critical number, the convergence curve of the sparse boundary solutionpresents the phenomena of stagnation or reverse, which indicates that a biggeriteration number (or a smaller value) does not necessarily lead to a betteraccuracy; instead, a relatively bigger value is more desirable. In particular,even the relative residue norm of the boundary solution is as big as~0.3, wecould still calculate the inner potential with an acceptable accuracy.4. Inspired by the systems thinking, we regard the main procedures of therestart GMRES algorithm as two sub-systems, with special concerns on theeffects to the whole system due to different coupling patterns of the sub-systems.In particular, we propose a week-coupling implementation for the restartGMRES. Experimental results show that, under vector programmingenvironment, the weak-coupling implementation is more efficient than theoriginal implementation, with the merit of easier programming.