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Fiber product homotopy method for multiparameter eigenvalue problems

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Abstract

We develop a new homotopy method for solving multiparameter eigenvalue problems (MEPs) called the fiber product homotopy method. For a k-parameter eigenvalue problem with matrices of sizes \(n_1,\ldots ,n_k = O(n)\), fiber product homotopy method requires deformation of O(1) linear equations, while existing homotopy methods for MEPs require O(n) nonlinear equations. We show that the fiber product homotopy method theoretically finds all eigenpairs of an MEP with probability one. It is especially well-suited for a class of problems we call dimension-deficient singular problems that are generic with respect to intrinsic dimension, as the fiber product homotopy method is provably convergent with probability one for such problems as well, a fact borne out by numerical experiments. More generally, our numerical experiments indicate that the fiber product homotopy method significantly outperforms the standard Delta method in terms of accuracy, with consistent backward errors on the order of \(10^{-16}\) without any use of extended precision. In terms of speed, it significantly outperforms previous homotopy-based methods on all problems and outperforms the Delta method on larger problems, and is also highly parallelizable. We show that the fiber product MEP that we solve in the fiber product homotopy method, although mathematically equivalent to a standard MEP, is typically a much better conditioned problem.

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Notes

  1. More precisely, we solve \(H_{i}(\varvec{\lambda }_{i})\varvec{x}_{i}=0\), \(G_1 = \cdots = G_k = 0\), \(i=1,\ldots ,k\), where each \(G_i :{\mathbb {C}}^{k^2} \rightarrow {\mathbb {C}}^{k-1}\) is a linear function of \(\varvec{\lambda }_{1},\ldots ,\varvec{\lambda }_{k}\), chosen so that the resulting system is equivalent to (4). See Sect. 5.2.

  2. Here “generic” is used in its usual sense in algebraic geometry. Those unfamiliar with this notion may assume that it is synonymous with “random.”

  3. This is an algebraic geometry term that implies the polynomial \(\det H_i(\varvec{\lambda }_i)\) is square-free.

  4. https://github.com/JoseMath/MEP_Homotopy.

  5. We have also used a Macaulay2 [9] package [4] to produce the relevant input files.

  6. The example is implemented in our GitHub codes at https://github.com/JoseMath/MEP_Homotopy/blob/master/MATLAB/examples/demo_flutter1.m for aerospace engineering [19], which remains a challenge for our method. In general, when the solutions to (3) are not the same as the regular eigenpairs of the MEP, the fiber product homotopy method may lose some regular eigenpairs. To address this issue, one has to know the connection between the \(H_i (\lambda )\) and the corresponding system of generalized eigenvalue problems on \(\varDelta \)-matrices but this is unfortunately not well-understood.

  7. These are respectively right and left eigenvectors of the polynomial matrix.

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Acknowledgements

We are extremely grateful for the reviewer’s exceptionally helpful comments and suggestions that led to a much improved article.

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Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin, Madison, WI, USA

    Jose Israel Rodriguez

  2. Department of Statistics, University of Chicago, Chicago, IL, USA

    Jin-Hong Du

  3. Department of Mathematics, University of California, Berkeley, CA, USA

    Yiling You

  4. Computational and Applied Mathematics Initiative, University of Chicago, Chicago, IL, USA

    Lek-Heng Lim

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  1. Jose Israel Rodriguez
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  2. Jin-Hong Du
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  4. Lek-Heng Lim
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Correspondence to Jose Israel Rodriguez.

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The work in this article is generously supported by DARPA D15AP00109, HR00112190040, and NSF IIS 1546413, DMS 1854831. LHL has also received support from the Eckhardt Faculty Fund. The work of JIR is supported by the College of Letters and Science, UW-Madison.

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Rodriguez, J.I., Du, JH., You, Y. et al. Fiber product homotopy method for multiparameter eigenvalue problems. Numer. Math. 148, 853–888 (2021). https://doi.org/10.1007/s00211-021-01215-6

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  • DOI: https://doi.org/10.1007/s00211-021-01215-6

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