SCIENTIA MANU E T MENTE Pseudodifferential equations on spheres with spherical radial basis functions and spherical splines 31st March, 2011 A thesis presented to The School of Mathematics and Statistics The University of New South Wales in fulfilment of the thesis requirement for the degree of Doctor of Philosophy by Duong Thanh Pham 2 THE UNIVERSITY OF NEW SOUTH WALES Thesis/Dissertation Sheet Surname or Family name: PHAM Firstname: Duong Thanh Other names: Abbreviation for degree as given in the University calendar: PhD School: Mathematics and Statistics Faculty: Science Title: Pseudodifferential equations on spheres with spherical radial basis functions and spherical splines Abstract 350 words maximum n Pseudodifferential equations on the unit sphere in R , n ≥ 3, are considered. The class of pseudodiffrential operators have long been used as a modern and powerful tool to tackle linear boundary-value problems. These equations arise in geophysics, where the sphere of interest is the earth. Efficient solutions to these equations on the sphere become more demanding when given data are collected by satellites. In this dissertation, firstly we solve these equations by using spherical radial basis functions. The use of these functions results in meshless methods, which have recently become more and more popular. In this dissertation, the collocation and Galerkin methods are used to solve pseudodifferential equations. From the point of view of application, the collocation method is easier to implement, in particular when the given data are scattered. However, it is well-known that the collocation methods in general elicit a complicated error analysis. A salient feature of our work is that error estimates for collocation methods are obtained as a by-product of the analysis for the Galerkin method. This unified error analysis is thanks to an observation that the collocation equation can be viewed as a Galerkin equation, due to the reproducing kernel property of the space in use. Secondly, we solve these equations by using spherical splines with Galerkin methods. Our main result is an optimal convergence rate of the approximation. The key of the analysis is the approximation property of spherical splines as a subset of Sobolev spaces. Since the pseudodifferential operators to be studied can be of any order, it is necessary to obtain an approximation property in Sobolev norms of any real order, negative and positive. Solving pseudodifferential equations by using Galerkin methods with spherical splines results, in general, in ill- conditioned matrix equations. To tackle this ill-conditionedness arising when solving two special pseudodifferential equations, the Laplace–Beltrami and hypersingular integral equations, we solve them by using a preconditioner which is defined by using the additive Schwarz method. Bounds for condition numbers of the preconditioned systems are established. Declaration relating to disposition of project thesis/dissertation I hereby grant to the University of New South Wales or its agents the right to archive and to make available my thesis or dissertation in whole or in part in the University libraries in all forms of media, now or here after known, subject to the provisions of the Copyright Act 1968. I retain all property rights, such as patent rights. I also retain the right to use in future works (such as articles or books) all or part of this thesis or dissertation. I also authorise University Microfilms to use the 350 word abstract of my thesis in Dissertation Abstracts International (this is applicable to doctoral theses only) ..... Duong T. Pham ........ ............................. .......07/07/2011............ Signature Witness Date The University recognises that there may be exceptional circumstances requiring restrictions on copying or con- ditions on use. Requests for restriction for a period of up to 2 years must be made in writing. Requests for a longer period of restriction may be considered in exceptional circumstances and require the approval of the Dean of Graduate Research. FOR OFFICE USE ONLY Date of completion of requirements for Award: THIS SHEET IS TO BE GLUED TO THE INSIDE FRONT COVER OF THE THESIS i CERTIFICATE OF ORIGINALITY I hereby declare that this submission is my own work and to the best of my knowledge it contains no materials previously published or written by another person, or substantial proportions of material which have been accepted for the award of any other degree or diploma at UNSW or any other educational institution, except where due acknowledge- ment is made in the thesis. Any contribution made to the research by others, with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in the thesis. I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the project’s design and conception or in style, presentation and linguistic expression is acknowledged. Signed ..........Duong T. Pham....................... Date ............07/07/2011.......................... ii COPYRIGHT STATEMENT I hereby grant the University of New South Wales or its agents the right to archive and to make available my thesis or dissertation in whole or part in the University libraries in all forms of media, now or here after known, subject to the provisions of the Copyright Act 1968. I retain all proprietary rights, such as patent rights. I also retain the right to use in future works (such as articles or books) all or part of this thesis or dissertation. I also authorise University Microfilms to use the 350 word abstract of my thesis in Dis- sertation Abstract International (this is applicable to doctoral theses only). I have either used no substantial portions of copyright material in my thesis or I have obtained permis- sion to use copyright material; where permission has not been granted I have applied/will apply for a partial restriction of the digital copy of my thesis or dissertation. Signed ..........Duong T. Pham....................... Date ............07/07/2011.......................... AUTHENTICITY STATEMENT I certify that the Library deposit digital copy is a direct equivalent of the final officially approved version of my thesis. No emendation of content has occurred and if there are any minor variations in formatting, they are the result of the conversion to digital format. Signed ..........Duong T. Pham....................... Date ............07/07/2011.......................... iii ABSTRACT n Pseudodifferential equations on the unit sphere in R , n ≥ 3, are considered. The class of pseudodiffrential operators have long been used as a modern and powerful tool to tackle linear boundary-value problems. These equations arise in geophysics, where the sphere of interest is the earth. Efficient solutions to these equations on the sphere become more demanding when given data are collected by satellites. In this dissertation, firstly we solve these equations by using spherical radial basis functions. The use of these functions results in meshless methods, which have recently become more and more popular. In this dissertation, the collocation and Galerkin methods are used to solve pseudodifferential equations. From the point of view of application, the collocation method is easier to implement, in particular when the given data are scattered. However, it is well-known that the collocation methods in general elicit a complicated error analysis. A salient feature of our work is that error estimates for collocation methods are obtained as a by-product of the analysis for the Galerkin method. This unified error analysis is thanks to an observation that the collocation equation can be viewed as a Galerkin equation, due to the reproducing kernel property of the space in use. Secondly, we solve these equations by using spherical splines with Galerkin methods. Our main result is an optimal convergence rate of the approximation. The key of the analysis is the approximation property of spherical splines as a subset of Sobolev spaces. Since the pseudodifferential operators to be studied can be of any order, it is necessary to obtain an approximation property in Sobolev norms of any real order, negative and positive. Solving pseudodifferential equations by using Galerkin methods with spherical splines results, in general, in ill-conditioned matrix equations. To tackle this ill-conditionedness arising when solving two special pseudodifferential equations, the Laplace–Beltrami and hypersingular integral equations, we solve them by using a preconditioner which is defined by using the additive Schwarz method. Bounds for condition numbers of the precondi- tioned systems are established. iv ACKNOWLEDGEMENT I would like to thank my supervisor Doctor Thanh Tran for introducing the field of computational mathematics to me, for his generous supports and guidance, and for sharing with me his vision and passion of computational mathematics in particular and mathematics in general. I also wish to express my gratitude to him for his encouragement and patience during my three PhD years. This dissertation has benefited greatly from his invaluable comments and suggestions. I am thankful to my co-supervisor Doctor William McLean for being generous to let me sit in his Fortran class during my first PhD semester and for always being ready to give me helpful advices in Fortran. I am very grateful to Thong Le Gia for always being willing to answer my questions, and giving me helpful advices in Fortran. I also wish to express my gratitude to him and his family for treating me as a close friend during my three years studying in Sydney. Writing this thesis, I have had the privilege of working with Professor Alexey Chernov. Also, I would like to thank him for inviting me to visit him in Germany and giving me the opportunity to present my research there. These were great learning opportunities for me for which I am very grateful. Many thanks to my fellows students Simon Crothers, Jan Baldeaux, Ngan Le and Ilya Tregubov for the fun and good times over the past few years. I also would like to thank Simon for his many useful advices in Fortran and Jan for his help with Latex.