13,161 research outputs found

    Multilevel quasiseparable matrices in PDE-constrained optimization

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    Optimization problems with constraints in the form of a partial differential equation arise frequently in the process of engineering design. The discretization of PDE-constrained optimization problems results in large-scale linear systems of saddle-point type. In this paper we propose and develop a novel approach to solving such systems by exploiting so-called quasiseparable matrices. One may think of a usual quasiseparable matrix as of a discrete analog of the Green's function of a one-dimensional differential operator. Nice feature of such matrices is that almost every algorithm which employs them has linear complexity. We extend the application of quasiseparable matrices to problems in higher dimensions. Namely, we construct a class of preconditioners which can be computed and applied at a linear computational cost. Their use with appropriate Krylov methods leads to algorithms of nearly linear complexity

    Computational linear algebra over finite fields

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    We present here algorithms for efficient computation of linear algebra problems over finite fields

    Efficient Transition Probability Computation for Continuous-Time Branching Processes via Compressed Sensing

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    Branching processes are a class of continuous-time Markov chains (CTMCs) with ubiquitous applications. A general difficulty in statistical inference under partially observed CTMC models arises in computing transition probabilities when the discrete state space is large or uncountable. Classical methods such as matrix exponentiation are infeasible for large or countably infinite state spaces, and sampling-based alternatives are computationally intensive, requiring a large integration step to impute over all possible hidden events. Recent work has successfully applied generating function techniques to computing transition probabilities for linear multitype branching processes. While these techniques often require significantly fewer computations than matrix exponentiation, they also become prohibitive in applications with large populations. We propose a compressed sensing framework that significantly accelerates the generating function method, decreasing computational cost up to a logarithmic factor by only assuming the probability mass of transitions is sparse. We demonstrate accurate and efficient transition probability computations in branching process models for hematopoiesis and transposable element evolution.Comment: 18 pages, 4 figures, 2 table

    Efficient approximation of functions of some large matrices by partial fraction expansions

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    Some important applicative problems require the evaluation of functions Ψ\Psi of large and sparse and/or \emph{localized} matrices AA. Popular and interesting techniques for computing Ψ(A)\Psi(A) and Ψ(A)v\Psi(A)\mathbf{v}, where v\mathbf{v} is a vector, are based on partial fraction expansions. However, some of these techniques require solving several linear systems whose matrices differ from AA by a complex multiple of the identity matrix II for computing Ψ(A)v\Psi(A)\mathbf{v} or require inverting sequences of matrices with the same characteristics for computing Ψ(A)\Psi(A). Here we study the use and the convergence of a recent technique for generating sequences of incomplete factorizations of matrices in order to face with both these issues. The solution of the sequences of linear systems and approximate matrix inversions above can be computed efficiently provided that A1A^{-1} shows certain decay properties. These strategies have good parallel potentialities. Our claims are confirmed by numerical tests

    Sparse approximate inverse preconditioners on high performance GPU platforms

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    Simulation with models based on partial differential equations often requires the solution of (sequences of) large and sparse algebraic linear systems. In multidimensional domains, preconditioned Krylov iterative solvers are often appropriate for these duties. Therefore, the search for efficient preconditioners for Krylov subspace methods is a crucial theme. Recent developments, especially in computing hardware, have renewed the interest in approximate inverse preconditioners in factorized form, because their application during the solution process can be more efficient. We present here some experiences focused on the approximate inverse preconditioners proposed by Benzi and Tůma from 1996 and the sparsification and inversion proposed by van Duin in 1999. Computational costs, reorderings and implementation issues are considered both on conventional and innovative computing architectures like Graphics Programming Units (GPUs)
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