arXiv subjects

Tomáš Gergelits

Publications and source records attributed to Tomáš Gergelits.

4 recordsLinked to original sources

arXiv2021

Mixed Precision $s$-step Lanczos and Conjugate Gradient Algorithms

Compared to the classical Lanczos algorithm, the $s$-step Lanczos variant has the potential to improve performance by asymptotically decreasing the synchronization cost per iteration. However, this comes at a cost. Despite being mathematically equivalent, the $s$-step variant is known to behave quite differently in finite precision, with potential for greater loss of accuracy and a decrease in the convergence rate relative to the classical algorithm. It has previously been shown that the errors that occur in the $s$-step version follow the same structure as the errors in the classical algorithm, but with the addition of an amplification factor that depends on the square of the condition number of the $O(s)-$dimensional Krylov bases computed in each outer loop. As the condition number of these $s$-step bases grows (in some cases very quickly) with $s$, this limits the parameter $s$ that can be chosen and thus limits the performance that can be achieved. In this work we show that if a select few computations in $s$-step Lanczos are performed in double the working precision, the error terms then depend only linearly on the conditioning of the $s$-step bases. This has the potential for drastically improving the numerical behavior of the algorithm with little impact on per-iteration performance. Our numerical experiments demonstrate the improved numerical behavior possible with the mixed precision approach, and also show that this improved behavior extends to the $s$-step CG algorithm in mixed precision.

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arXiv2021

Numerical approximation of the spectrum of self-adjoint continuously invertible operators

This paper deals with the generalized spectrum of continuously invertible linear operators defined on infinite dimensional Hilbert spaces. More precisely, we consider two bounded, coercive, and self-adjoint operators $\bc{A, B}: V\mapsto V^{\#}$, where $V^{\#}$ denotes the dual of $V$, and investigate the conditions under which the whole spectrum of $\bc{B}^{-1}\bc{A}:V\mapsto V$ can be approximated to an arbitrary accuracy by the eigenvalues of the finite dimensional discretization $\bc{B}_n^{-1}\bc{A}_n$. Since $\bc{B}^{-1}\bc{A}$ is continuously invertible, such an investigation cannot use the concept of uniform (normwise) convergence, and it relies instead on the pointwise (strong) convergence of $\bc{B}_n^{-1}\bc{A}_n$ to $\bc{B}^{-1}\bc{A}$. The paper is motivated by operator preconditioning which is employed in the numerical solution of boundary value problems. In this context, $\bc{A}, \bc{B}: H_0^1(Ω) \mapsto H^{-1}(Ω)$ are the standard integral/functional representations of the differential operators $ -\nabla \cdot (k(x)\nabla u)$ and $-\nabla \cdot (g(x)\nabla u)$, respectively, and $k(x)$ and $g(x)$ are scalar coefficient functions. The investigated question differs from the eigenvalue problem studied in the numerical PDE literature which is based on the approximation of the eigenvalues within the framework of compact operators. This work follows the path started by the two recent papers published in [SIAM J. Numer. Anal., 57 (2019), pp.~1369-1394 and 58 (2020), pp.~2193-2211] and addresses one of the open questions formulated at the end of the second paper.

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arXiv2020

Generalized spectrum of second order differential operators

We analyze the spectrum of the operator $Δ^{-1} [\nabla \cdot (K\nabla u)]$, where $Δ$ denotes the Laplacian and $K=K(x,y)$ is a symmetric tensor. Our main result shows that this spectrum can be derived from the spectral decomposition $K=Q ΛQ^T$, where $Q=Q(x,y)$ is an orthogonal matrix and $Λ=Λ(x,y)$ is a diagonal matrix. More precisely, provided that $K$ is continuous, the spectrum equals the convex hull of the ranges of the diagonal function entries of $Λ$. The involved domain is assumed to be bounded and Lipschitz, and both homogeneous Dirichlet and homogeneous Neumann boundary conditions are considered. We study operators defined on infinite dimensional Sobolev spaces. Our theoretical investigations are illuminated by numerical experiments, using discretized problems. The results presented in this paper extend previous analyses which have addressed elliptic differential operators with scalar coefficient functions. Our investigation is motivated by both preconditioning issues (efficient numerical computations) and the need to further develop the spectral theory of second order PDEs (core analysis).

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arXiv2018

Laplacian preconditioning of elliptic PDEs: Localization of the eigenvalues of the discretized operator

In the paper \textit{Preconditioning by inverting the {L}aplacian; an analysis of the eigenvalues. IMA Journal of Numerical Analysis 29, 1 (2009), 24--42}, Nielsen, Hackbusch and Tveito study the operator generated by using the inverse of the Laplacian as preconditioner for second order elliptic PDEs $\nabla \cdot (k(x) \nabla u) = f$. They prove that the range of $k(x)$ is contained in the spectrum of the preconditioned operator, provided that $k$ is continuous. Their rigorous analysis only addresses mappings defined on infinite dimensional spaces, but the numerical experiments in the paper suggest that a similar property holds in the discrete case. % Motivated by this investigation, we analyze the eigenvalues of the matrix $\bf{L}^{-1}\bf{A}$, where $\bf{L}$ and ${\bf{A}}$ are the stiffness matrices associated with the Laplace operator and general second order elliptic operators, respectively. Without any assumption about the continuity of $k(x)$, we prove the existence of a one-to-one pairing between the eigenvalues of $\bf{L}^{-1}\bf{A}$ and the intervals determined by the images under $k(x)$ of the supports of the FE nodal basis functions. As a consequence, we can show that the nodal values of $k(x)$ yield accurate approximations of the eigenvalues of $\bf{L}^{-1}\bf{A}$. Our theoretical results are illuminated by several numerical experiments.

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