Fast Iterative Methods for Wiener-Hopf Equations

This dissertation, "Fast Iterative Methods for Wiener-Hopf Equations" by 林福榮, Fu-rong, Lin, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: Abstract of thesis entitled 'FAST ITERATIVE METHODS FOR WIENER-HOPF EQUATIONS' submitted by Fu-Rong Lin for the degree of Doctor of Philosophy at The University of Hong Kong in May, 1995 Inthisthesis, westudythenumericalsolutionsofWiener-Hopfequationsdefined on the half-line 3/4x(t)+ a(ts)x(s)ds=g(t); 0-twhere 3/4 > 0, a(t) 2 L (1;1) and g(t) 2 L [0;1). We use the preconditioned 1 2 conjugate gradient (PCG) method to solve finite-section Wiener-Hopf equations: 3/4x(t)] a(ts)x(s)ds=g(t); 0-t- This thesis consists of two parts. In the first part, we construct several types of preconditioners for finite-section Wiener-Hopf equations. We then prove the clus- tering property of the preconditioned operators. In order to analyze the circulant integral preconditioners we constructed, we extend the definition of the optimal preconditionerP(B ) to integral operators B that are not of convolution-type and prove some of the properties of P. We then construct the super-optimal circulant integral preconditioner and genuine-optimal circulant integral preconditioner. We also construct preconditioners that are not circulant integral operators. These in- cludethepreconditionerobtainedbysplittingthefinite-sectionconvolutionoperator iinto a sum of f!g-circulant integral operators and the preconditioner by using the Fourier transform of a(t). When the rectangular rule is used, the discretization matrix systems of the pre- conditioned integral equations can be solved by the PCG method efficiently. How- ever, there is a drawback of using the rectangular rule, i.e. the order of accuracy of the discretized solution depends only linearly on the step-size of quadrature points. In order to obtain high order accuracy, one can use higher order quadratures, for instance, the trapezoidal rule and Simpson's rule. In this case, special techniques are required in order to apply the PCG method to the resulting matrix systems. This is the main focus of the second part of the thesis. Basedonthepreconditionerswehaveconstructed, wederiveourfastalgorithms. We test the algorithms with the rectangular rule, trapezoidal rule and Simpson's rule. Wealsotestthealgorithmsformatrixsystemsresultingfromthediscretization offinite-sectionWiener-Hopfequationswiththecombinationofthetrapezoidalrule and Simpson's rule. Numerical results show that if the kernel function a(t) and the righthandsidefunctiong(t)aresmoothenough, thentheuseofthetrapezoidalrule and Simpson's rule will produce accuracy of second and fourth order respectively. ii DOI: 10.5353/th_b3017148 Subjects: Wiener-Hopf equations