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    Krylov Subspace Methods with Fixed Memory Requirements: Nearly Hermitian Linear Systems and Subspace Recycling

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    Genre
    Thesis/Dissertation
    Date
    2012
    Author
    Soodhalter, Kirk McLane
    Advisor
    Szyld, Daniel
    Committee member
    Seibold, Benjamin
    Xue, Fei
    Parks, Michael L.
    Department
    Mathematics
    Subject
    Applied Mathematics
    Krylov Subspace
    Lattice Quantum Chromodynamics
    Linear Algebra
    Low-rank Modification
    Nearly Hermitian
    Subspace Recycling
    Permanent link to this record
    http://hdl.handle.net/20.500.12613/2438
    
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    DOI
    http://dx.doi.org/10.34944/dspace/2420
    Abstract
    Krylov subspace iterative methods provide an effective tool for reducing the solution of large linear systems to a size for which a direct solver may be applied. However, the problems of limited storage and speed are still a concern. Therefore, in this dissertation work, we present iterative Krylov subspace algorithms for non-Hermitian systems which do have fixed memory requirements and have favorable convergence characteristics. This dissertation describes three projects. The first project concerns short-term recurrence Krylov subspace methods for nearly-Hermitian linear systems. In 2008, Beckermann and Reichel introduced a short-term recurrence progressive GMRES algorithm for nearly-Hermitian linear systems. However, we have found this method to be unstable. We document the instabilities and introduce a different fixed-memory algorithm to treat nearly-Hermitian problems. We present numerical experiments demonstrating that the performance of this algorithm is competitive. The other two projects involve extending a strategy called Krylov subspace recycling, introduced by Parks and colleagues in 2005. This method requires more overhead than other subspace augmentation methods but offers the ability to recycle subspace information between cycles for a single linear system and recycle information between related linear systems. In the first project, we extend subspace recycling to the block Krylov subspace setting. A block Krylov subspace is a generalization of Krylov subspace where a single starting vector is replaced with a block of linearly independent starting vectors. We then apply our method to a sequence of matrices arising in a Newton iteration applied to fluid density functional theory and present some numerical experiments. In the second project, we extend the methods of subspace recycling to a family of linear systems differing only by multiples of the identity. These problems arise in the theory of quantum chromodynamics, a theory of the behavior of subatomic particles. We wish to build on the class of Krylov methods which allow the simultaneous solution of all shifted linear systems while generating only one subspace. However, the mechanics of subspace recycling complicates this situation and interferes with our ability to simultaneously solve all systems using these techniques. Therefore, we introduce an algorithm which avoids this complication and present some numerical experiments demonstrating its effectiveness.
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