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Register a new userFlexible subspace iteration with moments for an effective contour integration-based eigensolver
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Added by
Sarah Huber
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Contour integration schemes are a valuable tool for the solution of difficult interior eigenvalue problems. However, the solution of many large linear systems with multiple right hand sides may prove a prohibitive computational expense. The number of right hand sides, and thus, computational cost may be reduced if the projected subspace is created using multiple moments. In this work, we explore heuristics for the choice and application of moments with respect to various other important parameters in a contour integration scheme. We provide evidence for the expected performance, accuracy, and robustness of various schemes, showing that good heuristic choices can provide a scheme featuring good properties in all three of these measures.
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Subspace recycling techniques have been used quite successfully for the acceleration of iterative methods for solving large-scale linear systems. These methods often work by augmenting a solution subspace generated iteratively by a known algorithm with a fixed subspace of vectors which are ``useful for solving the problem. Often, this has the effect of inducing a projected version of the original linear system to which the known iterative method is then applied, and this projection can act as a deflation preconditioner, accelerating convergence. Most often, these methods have been applied for the solution of well-posed problems. However, they have also begun to be considered for the solution of ill-posed problems. In this paper, we consider subspace augmentation-type iterative schemes applied to linear ill-posed problems in a continuous Hilbert space setting, based on a recently developed framework describing these methods. We show that under suitable assumptions, a recycling method satisfies the formal definition of a regularization, as long as the underlying scheme is itself a regularization. We then develop an augmented subspace version of the gradient descent method and demonstrate its effectiveness, both on an academic Gaussian blur model and on problems arising from the adaptive optics community for the resolution of large sky images by ground-based extremely large telescopes.
We propose a new approach to the numerical solution of radiative transfer equations with certified a posteriori error bounds. A key role is played by stable Petrov--Galerkin type variational formulations of parametric transport equations and corresponding radiative transfer equations. This allows us to formulate an iteration in a suitable, infinite dimensional function space that is guaranteed to converge with a fixed error reduction per step. The numerical scheme is then based on approximately realizing this iteration within dynamically updated accuracy tolerances that still ensure convergence to the exact solution. To advance this iteration two operations need to be performed within suitably tightened accuracy tolerances. First, the global scattering operator needs to be approximately applied to the current iterate within a tolerance comparable to the current accuracy level. Second, parameter dependent linear transport equations need to be solved, again at the required accuracy of the iteration. To ensure that the stage dependent error tolerances are met, one has to employ rigorous a posteriori error bounds which, in our case, rest on a Discontinuous Petrov--Galerkin (DPG) scheme. These a posteriori bounds are not only crucial for guaranteeing the convergence of the perturbed iteration but are also used to generate adapted parameter dependent spatial meshes. This turns out to significantly reduce overall computational complexity. Since the global operator is only applied, we avoid the need to solve linear systems with densely populated matrices. Moreover, the approximate application of the global scatterer accelerated through low-rank approximation and matrix compression techniques. The theoretical findings are illustrated and complemented by numerical experiments with non-trivial scattering kernels.
Traditional numerical methods for calculating matrix eigenvalues are prohibitively expensive for high-dimensional problems. Randomized iterative methods allow for the estimation of a single dominant eigenvalue at reduced cost by leveraging repeated random sampling and averaging. We present a general approach to extending such methods for the estimation of multiple eigenvalues and demonstrate its performance for problems in quantum chemistry with matrices as large as 28 million by 28 million.
Many Krylov subspace methods for shifted linear systems take advantage of the invariance of the Krylov subspace under a shift of the matrix. However, exploiting this fact in the non-Hermitian case introduces restrictions; e.g., initial residuals must be collinear and this collinearity must be maintained at restart. Thus we cannot simultaneously solve shifted systems with unrelated right-hand sides using this strategy, and all shifted residuals cannot be simultaneously minimized over a Krylov subspace such that collinearity is maintained. It has been shown that this renders them generally incompatible with techniques of subspace recycling [Soodhalter et al. APNUM 14]. This problem, however, can be overcome. By interpreting a family of shifted systems as one Sylvester equation, we can take advantage of the known shift invariance of the Krylov subspace generated by the Sylvester operator. Thus we can simultaneously solve all systems over one block Krylov subspace using FOM or GMRES type methods, even when they have unrelated right-hand sides. Because residual collinearity is no longer a requirement at restart, these methods are fully compatible with subspace recycling techniques. Furthermore, we realize the benefits of block sparse matrix operations which arise in the context of high-performance computing applications. In this paper, we discuss exploiting this Sylvester equation point of view which has yielded methods for shifted systems which are compatible with unrelated right-hand sides. From this, we propose a recycled GMRES method for simultaneous solution of shifted systems.Numerical experiments demonstrate the effectiveness of the methods.
In this paper we present a novel matrix method for polynomial rootfinding. By exploiting the properties of the QR eigenvalue algorithm applied to a suitable CMV-like form of a companion matrix we design a fast and computationally simple structured QR iteration.
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