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Register a new userDecompositions and coalescing eigenvalues of symmetric definite pencils depending on parameters
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Added by
Alessandro Pugliese
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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In this work, we consider symmetric positive definite pencils depending on two parameters. That is, we are concerned with the generalized eigenvalue problem $A(x)-lambda B(x)$, where $A$ and $B$ are symmetric matrix valued functions in ${mathbb R}^{ntimes n}$, smoothly depending on parameters $xin Omegasubset {mathbb R}^2$; further, $B$ is also positive definite. In general, the eigenvalues of this multiparameter problem will not be smooth, the lack of smoothness resulting from eigenvalues being equal at some parameter values (conical intersections). We first give general theoretical results on the smoothness of eigenvalues and eigenvectors for the present generalized eigenvalue problem, and hence for the corresponding projections, and then perform a numerical study of the statistical properties of coalescing eigenvalues for pencils where $A$ and $B$ are either full or banded, for several bandwidths. Our numerical study will be performed with respect to a random matrix ensemble which respects the underlying engineering problems motivating our study.
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In an iterative approach for solving linear systems with ill-conditioned, symmetric positive definite (SPD) kernel matrices, both fast matrix-vector products and fast preconditioning operations are required. Fast (linear-scaling) matrix-vector products are available by expressing the kernel matrix in an $mathcal{H}^2$ representation or an equivalent fast multipole method representation. Preconditioning such matrices, however, requires a structured matrix approximation that is more regular than the $mathcal{H}^2$ representation, such as the hierarchically semiseparable (HSS) matrix representation, which provides fast solve operations. Previously, an algorithm was presented to construct an HSS approximation to an SPD kernel matrix that is guaranteed to be SPD. However, this algorithm has quadratic cost and was only designed for recursive binary partitionings of the points defining the kernel matrix. This paper presents a general algorithm for constructing an SPD HSS approximation. Importantly, the algorithm uses the $mathcal{H}^2$ representation of the SPD matrix to reduce its computational complexity from quadratic to quasilinear. Numerical experiments illustrate how this SPD HSS approximation performs as a preconditioner for solving linear systems arising from a range of kernel functions.
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