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    A new block Krylov subspace framework with applications to functions of matrices acting on multiple vectors

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    Genre
    Thesis/Dissertation
    Date
    2018
    Author
    Lund, Kathryn
    Advisor
    Szyld, Daniel
    Frommer, Andreas
    Committee member
    Seibold, Benjamin
    Queisser, Gillian
    Bolten, Matthias
    Department
    Mathematics
    Subject
    Applied Mathematics
    Block Krylov Subspace Methods
    Functions of Matrices
    Functions of Tensors
    Matrix Polynomials
    Shifted Linear Systems
    Tensor T-product
    Permanent link to this record
    http://hdl.handle.net/20.500.12613/3212
    
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    DOI
    http://dx.doi.org/10.34944/dspace/3194
    Abstract
    We propose a new framework for understanding block Krylov subspace methods, which hinges on a matrix-valued inner product. We can recast the ``classical" block Krylov methods, such as O'Leary's block conjugate gradients, global methods, and loop-interchange methods, within this framework. Leveraging the generality of the framework, we develop an efficient restart procedure and error bounds for the shifted block full orthogonalization method (Sh-BFOM(m)). Regarding BFOM as the prototypical block Krylov subspace method, we propose another formalism, which we call modified BFOM, and show that block GMRES and the new block Radau-Lanczos method can be regarded as modified BFOM. In analogy to Sh-BFOM(m), we develop an efficient restart procedure for shifted BGMRES with restarts (Sh-BGMRES(m)), as well as error bounds. Using this framework and shifted block Krylov methods with restarts as a foundation, we formulate block Krylov subspace methods with restarts for matrix functions acting on multiple vectors f(A)B. We obtain convergence bounds for \bfomfom (BFOM for Functions Of Matrices) and block harmonic methods (i.e., BGMRES-like methods) for matrix functions. With various numerical examples, we illustrate our theoretical results on Sh-BFOM and Sh-BGMRES. We also analyze the matrix polynomials associated to the residuals of these methods. Through a variety of real-life applications, we demonstrate the robustness and versatility of B(FOM)^2 and block harmonic methods for matrix functions. A particularly interesting example is the tensor t-function, our proposed definition for the function of a tensor in the tensor t-product formalism. Despite the lack of convergence theory, we also show that the block Radau-Lanczos modification can reduce the number of cycles required to converge for both linear systems and matrix functions.
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