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Register a new userPole-swapping algorithms for alternating and palindromic eigenvalue problems
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Added by
David Watkins
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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Pole-swapping algorithms are generalizations of bulge-chasing algorithms for the generalized eigenvalue problem. Structure-preserving pole-swapping algorithms for the palindromic and alternating eigenvalue problems, which arise in control theory, are derived. A refinement step that guarantees backward stability of the algorithms is included. This refinement can also be applied to bulge-chasing algorithms that had been introduced previously, thereby guaranteeing their backward stability in all cases.
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Pole-swapping algorithms, which are generalizations of the QZ algorithm for the generalized eigenvalue problem, are studied. A new modular (and therefore more flexible) convergence theory that applies to all pole-swapping algorithms is developed. A key component of all such algorithms is a procedure that swaps two adjacent eigenvalues in a triangular pencil. An improved swapping routine is developed, and its superiority over existing methods is demonstrated by a backward error analysis and numerical tests. The modularity of the new convergence theory and the generality of the pole-swapping approach shed new light on bi-directional chasing algorithms, optimally packed shifts, and bulge pencils, and allow the design of novel algorithms.
Stochastic PDE eigenvalue problems often arise in the field of uncertainty quantification, whereby one seeks to quantify the uncertainty in an eigenvalue, or its eigenfunction. In this paper we present an efficient multilevel quasi-Monte Carlo (MLQMC) algorithm for computing the expectation of the smallest eigenvalue of an elliptic eigenvalue problem with stochastic coefficients. Each sample evaluation requires the solution of a PDE eigenvalue problem, and so tackling this problem in practice is notoriously computationally difficult. We speed up the approximation of this expectation in four ways: 1) we use a multilevel variance reduction scheme to spread the work over a hierarchy of FE meshes and truncation dimensions; 2) we use QMC methods to efficiently compute the expectations on each level; 3) we exploit the smoothness in parameter space and reuse the eigenvector from a nearby QMC point to reduce the number of iterations of the eigensolver; and 4) we utilise a two-grid discretisation scheme to obtain the eigenvalue on the fine mesh with a single linear solve. The full error analysis of a basic MLQMC algorithm is given in the companion paper [Gilbert and Scheichl, 2021], and so in this paper we focus on how to further improve the efficiency and provide theoretical justification of the enhancement strategies 3) and 4). Numerical results are presented that show the efficiency of our algorithm, and also show that the four strategies we employ are complementary.
Polynomial eigenvalue problems (PEPs) arise in a variety of science and engineering applications, and many breakthroughs in the development of classical algorithms to solve PEPs have been made in the past decades. Here we attempt to solve PEPs in a quantum computer. Firstly, for generalized eigenvalue problems (GEPs) $Ax = lambda Bx$ with $A,B$ symmetric, and $B$ positive definite, we give a quantum algorithm based on block-encoding and quantum phase estimation. In a more general case when $B$ is invertible, $B^{-1}A$ is diagonalizable and all the eigenvalues are real, we propose a quantum algorithm based on the Fourier spectral method to solve ordinary differential equations (ODEs). The inputs of our algorithms can be any desired states, and the outputs are superpositions of the eigenpairs. The complexities are polylog in the matrix size and linear in the precision. The dependence on precision is optimal. Secondly, we show that when $B$ is singular, any quantum algorithm uses at least $Omega(sqrt{n})$ queries to compute the eigenvalues, where $n$ is the matrix size. Thirdly, based on the linearization method and the connection between PEPs and higher-order ODEs, we provide two quantum algorithms to solve PEPs by extending the quantum algorithm for GEPs. We also give detailed complexity analysis of the algorithm for two special types of quadratic eigenvalue problems that are important in practice. Finally, under an extra assumption, we propose a quantum algorithm to solve PEPs when the eigenvalues are complex.
A novel orthogonalization-free method together with two specific algorithms are proposed to solve extreme eigenvalue problems. On top of gradient-based algorithms, the proposed algorithms modify the multi-column gradient such that earlier columns are decoupled from later ones. Global convergence to eigenvectors instead of eigenspace is guaranteed almost surely. Locally, algorithms converge linearly with convergence rate depending on eigengaps. Momentum acceleration, exact linesearch, and column locking are incorporated to further accelerate both algorithms and reduce their computational costs. We demonstrate the efficiency of both algorithms on several random matrices with different spectrum distribution and matrices from computational chemistry.
Several numerical tools designed to overcome the challenges of smoothing in a nonlinear and non-Gaussian setting are investigated for a class of particle smoothers. The considered family of smoothers is induced by the class of linear ensemble transform filters which contains classical filters such as the stochastic ensemble Kalman filter, the ensemble square root filter and the recently introduced nonlinear ensemble transform filter. Further the ensemble transform particle smoother is introduced and particularly highlighted as it is consistent in the particle limit and does not require assumptions with respect to the family of the posterior distribution. The linear update pattern of the considered class of linear ensemble transform smoothers allows one to implement important supplementary techniques such as adaptive spread corrections, hybrid formulations, and localization in order to facilitate their application to complex estimation problems. These additional features are derived and numerically investigated for a sequence of increasingly challenging test problems.
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